The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. The rules for exponents such as b0, b-k, √b, fractional exponents, and real number exponents are explained. Examples are provided to illustrate calculating exponential functions and applications to compound interest. Exponential functions appear in many fields including finance, science, and engineering.
2.2 exponential function and compound interestmath123c
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. Some key points made in the document include:
- The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained.
- Exponential functions are defined for all real numbers x.
- Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents.
- Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y
The document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. The base is A and the exponent is N. Some key exponent rules covered include: the multiply-add rule where AnAk = An+k; the divide-subtract rule where An/Ak = An-k; the power-multiply rule where (An)k = Ank. Special exponents rules include: A0 = 1 when A ≠ 0; A-k = 1/Ak; A1/n = nth root of A. Examples are provided to demonstrate applying these exponent rules.
24 exponential functions and periodic compound interests pina xmath260
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. It provides examples of calculating exponential expressions using rules for positive integer, fractional, and real number exponents. Exponential functions are important in fields like finance, science, and computing. Common exponential functions include y = 10x, y = ex, and y = 2x. An example shows how to calculate compound interest monthly over several periods using the exponential function formulation.
The document discusses periodic compound interest and continuous compound interest formulas. It provides an example to calculate the accumulation in an account over 20 years with an annual 8% interest rate compounded 100, 1000, and 10000 times per year. Compounding more frequently results in a larger return, approaching the continuous compound interest formula value. Compounding 10000 times per year yields the highest return of $4953.
4.2 exponential functions and compound interestsmath260
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. The rules for exponents such as b0, b-k, √b, fractional exponents, and real number exponents are explained. Examples are provided to illustrate calculating exponential functions. Exponential functions appear in various fields such as finance, science, and are important as they model growth rates. The most common exponential functions y = 10x, y = ex, and y = 2x are noted. An example of compound interest calculation is given to demonstrate the application of exponential growth.
This document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. It then provides several exponent rules including:
- Multiply-Add Rule: AnAk = An+k
- Divide-Subtract Rule: An/Ak = An-k
- Power-Multiply Rule: (An)k = Ank
Special exponents are also discussed such as A0=1 if A≠0, A-k=1/Ak, and calculating fractional exponents by extracting the root first then raising it to the numerator power. Examples are provided to demonstrate applying these exponent rules and calculating fractional exponents.
2.1 reviews of exponents and the power functionsmath123c
The document discusses solving power equations and proper calculator input format for expressions involving powers and fractions. It provides examples of solving various power equations by taking the reciprocal of the power. It also emphasizes the need for precise text input, such as using parentheses and the caret symbol "^", to evaluate expressions correctly on a calculator. Common mistakes like incorrect ordering of operations when inputting a fraction or power are highlighted.
The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are:
- The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K
- The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K
Examples are provided to demonstrate calculating exponents using these rules.
2.2 exponential function and compound interestmath123c
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. Some key points made in the document include:
- The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained.
- Exponential functions are defined for all real numbers x.
- Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents.
- Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y
The document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. The base is A and the exponent is N. Some key exponent rules covered include: the multiply-add rule where AnAk = An+k; the divide-subtract rule where An/Ak = An-k; the power-multiply rule where (An)k = Ank. Special exponents rules include: A0 = 1 when A ≠ 0; A-k = 1/Ak; A1/n = nth root of A. Examples are provided to demonstrate applying these exponent rules.
24 exponential functions and periodic compound interests pina xmath260
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. It provides examples of calculating exponential expressions using rules for positive integer, fractional, and real number exponents. Exponential functions are important in fields like finance, science, and computing. Common exponential functions include y = 10x, y = ex, and y = 2x. An example shows how to calculate compound interest monthly over several periods using the exponential function formulation.
The document discusses periodic compound interest and continuous compound interest formulas. It provides an example to calculate the accumulation in an account over 20 years with an annual 8% interest rate compounded 100, 1000, and 10000 times per year. Compounding more frequently results in a larger return, approaching the continuous compound interest formula value. Compounding 10000 times per year yields the highest return of $4953.
4.2 exponential functions and compound interestsmath260
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. The rules for exponents such as b0, b-k, √b, fractional exponents, and real number exponents are explained. Examples are provided to illustrate calculating exponential functions. Exponential functions appear in various fields such as finance, science, and are important as they model growth rates. The most common exponential functions y = 10x, y = ex, and y = 2x are noted. An example of compound interest calculation is given to demonstrate the application of exponential growth.
This document discusses exponents and exponent rules. It defines exponents as multiplying a base A by itself repeatedly N times, written as AN. It then provides several exponent rules including:
- Multiply-Add Rule: AnAk = An+k
- Divide-Subtract Rule: An/Ak = An-k
- Power-Multiply Rule: (An)k = Ank
Special exponents are also discussed such as A0=1 if A≠0, A-k=1/Ak, and calculating fractional exponents by extracting the root first then raising it to the numerator power. Examples are provided to demonstrate applying these exponent rules and calculating fractional exponents.
2.1 reviews of exponents and the power functionsmath123c
The document discusses solving power equations and proper calculator input format for expressions involving powers and fractions. It provides examples of solving various power equations by taking the reciprocal of the power. It also emphasizes the need for precise text input, such as using parentheses and the caret symbol "^", to evaluate expressions correctly on a calculator. Common mistakes like incorrect ordering of operations when inputting a fraction or power are highlighted.
The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are:
- The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K
- The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K
Examples are provided to demonstrate calculating exponents using these rules.
The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are:
- The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K
- The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K
Examples are provided to demonstrate calculating exponents using these rules.
The document discusses solving numerical equations involving logarithmic and exponential functions in base 10 or base e. It provides examples of solving both log equations, by rewriting them in exponential form, and exponential equations, by rewriting them in logarithmic form. The key steps are to isolate the part containing the unknown, then rewrite the equation by "bringing down" exponents or taking the logarithm/exponential to solve for the unknown.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then provides examples to illustrate the multiplication rule (ANAK = AN+K), division rule (AN/AK = AN-K), power rule ((AN)K = ANK), 0-power rule (A0 = 1), and negative power rule (A-K = 1/AK).
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
I am Piers L. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their assignments for the past 6 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
The document discusses solving numerical equations involving logarithmic and exponential functions in base 10 or base e. It provides examples of solving log and exponential equations by isolating the part containing the unknown, then rewriting the equation in the opposite form (log to exponential or exponential to log). The key steps outlined are: 1) isolate the exponential/log part containing the unknown, 2) rewrite the equation by "bringing down" exponents as logarithms or vice versa. Several examples are worked through demonstrating these steps.
The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
This document discusses two types of log and exponential equations: those that do not require calculators and numerical equations that do. Equations that do not require calculators have related bases on both sides and can be simplified using the law of uniqueness of log and exponential functions. These equations are solved by consolidating bases or logs and then dropping the common base. Numerical equations require using calculators to evaluate logarithms and exponents. Examples of solving each type of equation without calculators are provided.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
I am Duncan V. I am a Calculus Homework Expert at mathshomeworksolver.com. I hold a Master's in Mathematics from Manchester, United Kingdom. I have been helping students with their homework for the past 8 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
Growth of Functions
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 6, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
I am Piers L. I am a Calculus Homework Expert at mathhomeworksolver.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving both log equations and exponential equations. To solve log equations, the problems are rewritten in exponential form by removing the log. To solve exponential equations, the exponents are brought down by rewriting the problem in logarithmic form. Steps include isolating the exponential or logarithmic term containing the unknown, rewriting the equation accordingly, and then solving for the unknown variable. Practice problems with solutions are provided to illustrate the process.
The document discusses rules and procedures for adding and subtracting rational expressions. It states that fractions can only be directly added or subtracted if they have the same denominator. It provides an example of adding and subtracting fractions with the same denominator and simplifying the results. It also discusses how to convert fractions with different denominators to have a common denominator before adding or subtracting them, using the least common multiple (LCM) of the denominators. It provides an example problem that demonstrates converting fractions to equivalent forms with a specified common denominator.
The document discusses substituting expressions into formulas and evaluating them. It provides examples of substituting expressions for variables in A^2 - B^2 and evaluating, as well as identifying the expressions for A and B given the output of an evaluation. It also discusses evaluating expressions of the form A^3 - B^3 and the reverse process of identifying A and B given the output. Factoring formulas for difference of squares and cubes are presented.
Decreasing and increasing functions by arun umraossuserd6b1fd
Function analysis - characteristics of increasing and decreasing functions. How "sign" either positive or negative tells about the nature of the function, i.e. where it is increasing and where it is decreasing.
The document discusses different types of equations including:
1) Simple equations with one unknown variable in the form of ax + b = 0.
2) Simultaneous linear equations with two unknown variables in the form of ax + by + c = 0. These can be solved using elimination or cross-multiplication methods.
3) Quadratic equations in the form of ax2 + bx + c = 0. The nature of the roots depends on the discriminant b2 - 4ac.
I am Piers L. I am a Calculus Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From, the University of Adelaide. I have been helping students with their homework for the past 7 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
* Evaluate exponential functions.
* Find the equation of an exponential function.
* Use compound interest formulas.
* Evaluate exponential functions with base e.
The document discusses exponential and logarithmic functions. It defines logarithms as exponents and explains that logarithms were once used to simplify calculations before calculators. It then covers several topics related to exponential functions including:
- Basic laws of exponents using integral exponents
- Examples of applying the order of operations to exponents
- Extending the rules of exponents to include rational exponents
- Exponential growth and decay models and examples
- Graphing and properties of exponential functions
- The number e and the natural exponential function ex
- Compound interest formulas including continuous compounding
The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are:
- The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K
- The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K
Examples are provided to demonstrate calculating exponents using these rules.
The document discusses solving numerical equations involving logarithmic and exponential functions in base 10 or base e. It provides examples of solving both log equations, by rewriting them in exponential form, and exponential equations, by rewriting them in logarithmic form. The key steps are to isolate the part containing the unknown, then rewrite the equation by "bringing down" exponents or taking the logarithm/exponential to solve for the unknown.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then provides examples to illustrate the multiplication rule (ANAK = AN+K), division rule (AN/AK = AN-K), power rule ((AN)K = ANK), 0-power rule (A0 = 1), and negative power rule (A-K = 1/AK).
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
I am Piers L. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their assignments for the past 6 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
The document discusses solving numerical equations involving logarithmic and exponential functions in base 10 or base e. It provides examples of solving log and exponential equations by isolating the part containing the unknown, then rewriting the equation in the opposite form (log to exponential or exponential to log). The key steps outlined are: 1) isolate the exponential/log part containing the unknown, 2) rewrite the equation by "bringing down" exponents as logarithms or vice versa. Several examples are worked through demonstrating these steps.
The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
This document discusses two types of log and exponential equations: those that do not require calculators and numerical equations that do. Equations that do not require calculators have related bases on both sides and can be simplified using the law of uniqueness of log and exponential functions. These equations are solved by consolidating bases or logs and then dropping the common base. Numerical equations require using calculators to evaluate logarithms and exponents. Examples of solving each type of equation without calculators are provided.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
I am Duncan V. I am a Calculus Homework Expert at mathshomeworksolver.com. I hold a Master's in Mathematics from Manchester, United Kingdom. I have been helping students with their homework for the past 8 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
Growth of Functions
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 6, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
I am Piers L. I am a Calculus Homework Expert at mathhomeworksolver.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving both log equations and exponential equations. To solve log equations, the problems are rewritten in exponential form by removing the log. To solve exponential equations, the exponents are brought down by rewriting the problem in logarithmic form. Steps include isolating the exponential or logarithmic term containing the unknown, rewriting the equation accordingly, and then solving for the unknown variable. Practice problems with solutions are provided to illustrate the process.
The document discusses rules and procedures for adding and subtracting rational expressions. It states that fractions can only be directly added or subtracted if they have the same denominator. It provides an example of adding and subtracting fractions with the same denominator and simplifying the results. It also discusses how to convert fractions with different denominators to have a common denominator before adding or subtracting them, using the least common multiple (LCM) of the denominators. It provides an example problem that demonstrates converting fractions to equivalent forms with a specified common denominator.
The document discusses substituting expressions into formulas and evaluating them. It provides examples of substituting expressions for variables in A^2 - B^2 and evaluating, as well as identifying the expressions for A and B given the output of an evaluation. It also discusses evaluating expressions of the form A^3 - B^3 and the reverse process of identifying A and B given the output. Factoring formulas for difference of squares and cubes are presented.
Decreasing and increasing functions by arun umraossuserd6b1fd
Function analysis - characteristics of increasing and decreasing functions. How "sign" either positive or negative tells about the nature of the function, i.e. where it is increasing and where it is decreasing.
The document discusses different types of equations including:
1) Simple equations with one unknown variable in the form of ax + b = 0.
2) Simultaneous linear equations with two unknown variables in the form of ax + by + c = 0. These can be solved using elimination or cross-multiplication methods.
3) Quadratic equations in the form of ax2 + bx + c = 0. The nature of the roots depends on the discriminant b2 - 4ac.
I am Piers L. I am a Calculus Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From, the University of Adelaide. I have been helping students with their homework for the past 7 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
* Evaluate exponential functions.
* Find the equation of an exponential function.
* Use compound interest formulas.
* Evaluate exponential functions with base e.
The document discusses exponential and logarithmic functions. It defines logarithms as exponents and explains that logarithms were once used to simplify calculations before calculators. It then covers several topics related to exponential functions including:
- Basic laws of exponents using integral exponents
- Examples of applying the order of operations to exponents
- Extending the rules of exponents to include rational exponents
- Exponential growth and decay models and examples
- Graphing and properties of exponential functions
- The number e and the natural exponential function ex
- Compound interest formulas including continuous compounding
This document provides information on mathematical concepts and formulas relevant to economics, including:
- Exponential functions such as y=ex and their graphs showing exponential growth and decay
- Quadratic functions of the form y=ax2+bx+c and total cost functions
- Differentiation rules for common functions like exponentials, logarithms, and the product, quotient and chain rules
- Integration basics and formulas for integrating common functions
- Concepts like inverse functions, the mean, variance and standard deviation in statistics
- Information is also provided on fractions, ratios, percentages, and algebraic rules involving exponents, logarithms and sigma notation.
4.2 exponential functions and periodic compound interests pina tmath260
This document discusses compound interest concepts and formulas. It contains:
1) Examples of calculating compound interest with different periodic rates and time periods.
2) Formulas for calculating principal (P), accumulation (A), periodic interest rate (i), and the relationship between annual (r) and periodic rates.
3) Exercises involving using the compound interest formulas to calculate principal, accumulation, and converting between annual and periodic rates for different time periods and rates.
This document covers key concepts about exponential functions including:
1) Properties of exponents that can be used to simplify expressions.
2) The definition of an exponential function f(x)=abx and characteristics of its graph.
3) Solving exponential equations by setting exponents equal to each other after rewriting both sides with a common base.
4) The compound interest formula A=P(1+r/n)nt and examples of calculating interest compounded over time.
This document provides a summary of key concepts from a college algebra textbook, including:
Rational expressions involve fractions of polynomials. The domain of an algebraic expression is the set of real numbers for which the expression is defined. Compound fractions contain fractions in the numerator, denominator, or both. Rational expressions can be simplified by factoring and canceling common factors. Adding and subtracting rational expressions requires finding a common denominator. The denominator of a fraction can be rationalized by multiplying the numerator and denominator by the conjugate radical. Common errors involve applying properties of multiplication to addition incorrectly.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
Determinants provide a scalar quantity associated with square matrices. There are several properties of determinants, including that the determinant of a matrix does not change if rows or columns are interchanged. The determinant can be expressed as the sum of the products of each element and its corresponding cofactor. Examples show how to evaluate determinants by expanding along rows or columns and applying properties such as identical rows resulting in a determinant of zero.
The document discusses exponential equations and their applications. Exponential equations take the form y=abx, where a and b are constants. When b>1, b is the growth factor, and when 0<b<1, b is the decay factor. Examples are provided to demonstrate how to write an exponential equation that passes through two given points. The concepts are then applied to modeling growth and decay scenarios using exponential equations. Compound interest is also discussed, with the formula A(t)=P(1+r/n)nt provided, where n is the number of times interest is compounded per year.
This document discusses solving numerical equations involving logarithmic and exponential functions. It provides examples of solving:
1) Logarithmic equations by dropping the log and writing the equation in exponential form.
2) Exponential equations by isolating the exponential term containing the unknown, then taking the log of both sides to write it in logarithmic form.
3) The document demonstrates solving sample equations of each type step-by-step and explains the differences between logarithmic and exponential equations.
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
Introduction:
[Start with a brief introduction about yourself, including your profession or main area of expertise.]
Background:
[Discuss your background, education, and any relevant experiences that have shaped your journey.]
Accomplishments:
[Highlight notable achievements, awards, or significant projects you've been involved in.]
Expertise:
[Detail your areas of expertise, skills, or specific knowledge that sets you apart in your field.]
Passions and Interests:
[Share your passions, hobbies, or interests outside of your professional life, adding depth to your personality.]
Vision or Mission:
[If applicable, articulate your vision, mission, or goals in your chosen field or in life in general.]
Closing Statement:
[End with a closing statement that summarizes your essence or leaves a lasting impression.]
Feel free to customize each section with your own personal details and experiences. If you need further assistance or have specific points you'd like to include, feel free to let me know!
1) The document contains calculations related to interest rates, compounding periods, and time value of money. Various interest rates, discount rates, and yields are computed.
2) Quadratic and logarithmic equations are set up and solved to relate future and present values under different interest rates and time periods.
3) Formulas are provided and derived for simple and compound interest, continuous compounding, and calculating time periods between dates.
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
This document provides an introduction to matrices and their arithmetic operations. It defines what a matrix is, with m rows and n columns. It introduces basic matrix operations like addition, which is done element-wise, and multiplication by scalars. Matrix multiplication is defined as the sum of the products of corresponding entries of the first matrix's rows and second matrix's columns. Several examples are provided to illustrate these matrix operations.
The document provides an introduction to the binomial theorem. It defines binomial coefficients through the Pascal triangle and gives an explicit formula for computing them using factorials. The binomial theorem is then derived and stated, providing a formula for expanding expressions of the form (a + b)^n in terms of binomial coefficients. Several examples are worked out to demonstrate expanding expressions and finding coefficients using the binomial theorem. Applications to estimating interest calculations are also briefly discussed.
The document discusses number systems and conversions between different bases. It explains that computers use the binary system with bits representing 0s and 1s. 8 bits form a byte. Decimal, binary, octal and hexadecimal numbering systems are covered. Methods for converting between these bases are provided using division and remainders or grouping bits. Common powers and units used in computing like kilo, mega and giga are also defined. Exercises on converting values between the different number systems are included.
Economics
Curve Fitting
macroeconomics
Curve fitting helps in capturing the trend in the data by assigning a single function
across the entire range.
If the functional relationship between the two quantities being graphed is known to be
within additive or multiplicative constants, it is common practice to transform the data in
such a way that the resulting line is a straight line.(by plotting) A process of quantitatively
estimating the trend of the outcomes, also known as regression or curve fitting, therefore
becomes necessary.
For a series of data, curve fitting is used to find the best fit curve. The produced equation is
used to find points anywhere along the curve. It also uses interpolation (exact fit to the data)
and smoothing.
Some people also refer it as regression analysis instead of curve fitting. The curve fitting
process fits equations of approximating curves to the raw field data. Nevertheless, for a
given set of data, the fitting curves of a given type are generally NOT unique.
Smoothing of the curve eliminates components like seasonal, cyclical and random
variations. Thus, a curve with a minimal deviation from all data points is desired. This
best-fitting curve can be obtained by the method of least squares.
What is curve fitting Curve fitting?
Curve fitting is the process of constructing a curve, or mathematical functions, which possess closest
proximity to the series of data. By the curve fitting we can mathematically construct the functional
relationship between the observed fact and parameter values, etc. It is highly effective in mathematical
modelling some natural processes.
What is a fitting model?
A fit model (sometimes fitting model) is a person who is used by a fashion designer or
clothing manufacturer to check the fit, drape and visual appearance of a design on a
'real' human being, effectively acting as a live mannequin.
What is a model fit statistics?
The goodness of fit of a statistical model describes how well it fits a set of
observations. Measures of goodness of fit typically summarize the discrepancy
between observed values and the values expected under the model in question.
What is a commercial model?
Commercial modeling is a more generalized type of modeling. There are high
fashion models, and then there are commercial models. ... They can model for
television, commercials, websites, magazines, newspapers, billboards and any other
type of advertisement. Most people who tell you they are models are “commercial”
models.
What is the exponential growth curve?
Growth of a system in which the amount being added to the system is proportional to the
amount already present: the bigger the system is, the greater the increase. ( See geometric
progression.) Note : In everyday speech, exponential growth means runaway expansion, such
as in population growth.
Why is population exponential?
Exponential population growth: When resources are unlimited, populations
exhibit exponential growth, resulting in a J-shaped curve.
Similar to 4.2 exponential functions and periodic compound interests pina x (20)
The document introduces matrices and matrix operations. Matrices are rectangular tables of numbers that are used for applications beyond solving systems of equations. Matrix notation defines a matrix with R rows and C columns as an R x C matrix. The entry in the ith row and jth column is denoted as aij. Matrices can be added or subtracted if they are the same size by adding or subtracting the corresponding entries. There are two types of matrix multiplication: scalar multiplication multiplies a matrix by a constant, and matrix multiplication involves multiplying corresponding rows and columns where the number of columns of the left matrix equals the rows of the right matrix.
35 Special Cases System of Linear Equations-x.pptxmath260
The document discusses special cases of systems of linear equations, including inconsistent/contradictory systems where the equations are impossible to satisfy simultaneously, and dependent systems where there are infinitely many solutions. An inconsistent system is shown with equations x + y = 2 and x + y = 3, which has no solution since they cannot both be true. A dependent system is shown with equations x + y = 2 and 2x + 2y = 4, which has infinitely many solutions like (2,0) and (1,1). The row-reduced echelon form (rref) of a matrix is also discussed, which puts a system of equations in a standard form to help determine if it is consistent, dependent, or has
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
The document discusses sign charts for factorable formulas. It provides examples of determining the sign (positive or negative) of expressions when evaluated at given values of x by factoring the expressions into their factored forms. The key steps to create a sign chart are: 1) solve for values where the expression is equal to 0, 2) mark these values on a number line, 3) select points in each segment to test the sign, 4) indicate the sign (positive or negative) in each segment based on the testing. Sign charts show the regions where an expression is positive, negative or equal to 0.
19 more parabolas a& hyperbolas (optional) xmath260
After dividing the general quadratic equation Ax2 + By2 + Cx + Dy = E by A, three types of conic sections can be obtained:
1) Parabolas occur when B = 0, resulting in equations of the form 1x2 + #x + #y = #.
2) Circles occur when A = B, resulting in the equation 1x2 + 1y2 = 1.
3) Hyperbolas occur when A and B have opposite signs, resulting in equations of the form 1x2 + ry2 + #x + #y = # with r < 0. Hyperbolas have two foci and asymptotes, and points on the hyperbola have
The document discusses conic sections, specifically circles and ellipses. It defines an ellipse as the set of points where the sum of the distances to two fixed foci is a constant. An ellipse has a center, two axes (semi-major and semi-minor), and can be represented by the standard form (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. Examples are provided to demonstrate finding attributes of ellipses from their equations.
This document discusses conic sections and first degree equations. It begins by introducing conic sections as the shapes formed by slicing a cone at different angles. It then covers first degree equations, noting that their graphs are straight lines that can be written in the form of y=mx+b. Specific examples of first degree equations and their graphs are shown. The document ends by introducing the four types of conic sections - circles, ellipses, parabolas, and hyperbolas - and how graphs of second degree equations can represent these shapes.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
14 graphs of factorable rational functions xmath260
The document discusses graphs of rational functions. It defines rational functions as functions of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials. It describes how vertical asymptotes occur where the denominator Q(x) is zero. The graph runs along either side of vertical asymptotes, going up or down depending on the sign chart. There are four cases for how the graph behaves at a vertical asymptote. The document uses examples to illustrate graphing rational functions and determining vertical asymptotes. It also mentions horizontal asymptotes will be discussed.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses the basic language of functions. It defines a function as a procedure that assigns each input exactly one output. Functions can be represented by formulas using typical variables like f(x) = x^2 - 2x + 3, where x is the input and f(x) is the output. Functions have a domain, which is the set of all possible inputs, and a range, which is the set of all possible outputs. Functions can be depicted graphically or via tables listing inputs and outputs.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Information and Communication Technology in EducationMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 2)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐈𝐂𝐓 𝐢𝐧 𝐞𝐝𝐮𝐜𝐚𝐭𝐢𝐨𝐧:
Students will be able to explain the role and impact of Information and Communication Technology (ICT) in education. They will understand how ICT tools, such as computers, the internet, and educational software, enhance learning and teaching processes. By exploring various ICT applications, students will recognize how these technologies facilitate access to information, improve communication, support collaboration, and enable personalized learning experiences.
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐫𝐞𝐥𝐢𝐚𝐛𝐥𝐞 𝐬𝐨𝐮𝐫𝐜𝐞𝐬 𝐨𝐧 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐫𝐧𝐞𝐭:
-Students will be able to discuss what constitutes reliable sources on the internet. They will learn to identify key characteristics of trustworthy information, such as credibility, accuracy, and authority. By examining different types of online sources, students will develop skills to evaluate the reliability of websites and content, ensuring they can distinguish between reputable information and misinformation.
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
3. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
The Exponential Functions
K
N
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
4. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 =
8 =
The Exponential Functions
K
N
3
2
3
2
8 –2 =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
5. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 =
The Exponential Functions
K
N
3
2
3
2
8 –2 =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
6. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 =
82
1
64
1
The Exponential Functions
K
N
3
2
3
2
8 –2 = =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
7. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
64
1
The Exponential Functions
K
N
3
2
3
2
8 –2 = =
8 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
8. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
8 –2 = =
8 = ( ) = 1/4
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
9. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
8 –2 = =
8 = ( ) = 1/4
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
10. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 =
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 =
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
11. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 =
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 =
3
2
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
12. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 =
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 = (9 ) = 27
3
2
3
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
13. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 = 10
61
50
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 = (9 ) = 27
3
2
3
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
14. b0 = 1 b–K =
b = ( b ) b = ( )
K
N
K N
bK
1
K
N
b
1
Example A.
80 = 1
8 = ( 8 ) = 4
3 2
82
1
3
2
8
1
64
1
The Exponential Functions
K
N
3
2
3
2
Decimal exponents are well defined since decimals may be
represented as reduced fractions.
b. 101.22 = 10 = ( 10 ) 16.59586….
61
50
50 61
8 –2 = =
8 = ( ) = 1/4
Example B.
a. 91.50 = 9 = (9 ) = 27
3
2
3
The meaning positive integral exponents such as x2 is clear.
Below are the rules for other special exponents:
15. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159..
10
Example C.
The Exponential Functions
16. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10
Example C.
The Exponential Functions
17. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10
Example C.
31
10
The Exponential Functions
≈1258.9..
18. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10
Example C.
31
10
314
100
The Exponential Functions
≈1258.9.. ≈1380.3..
19. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10
Example C.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
20. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
21. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b 1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
22. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b 1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
Exponential functions show up in finance, bio science,
computer science and physical sciences.
23. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b 1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
Exponential functions show up in finance, bio science,
computer science and physical sciences.
The most used exponential functions are
y = 10x, y = ex and y = 2x.
24. For a real-number-exponent such as , we approximate the
real number with fractions then use the fractional powers to
approximate the result.
3.14159.. 3.1 3.14 3.141 3.1415
10 10 10 10 10 10≈1385.45..
Example C.
31
10
314
100
3141
1000
31415
10000
The Exponential Functions
Hence exponential functions or functions of the form
f(x) = bx (b > 0 and b 1) are defined for all real numbers x.
≈1258.9.. ≈1380.3.. ≈1383.5.. ≈1385.1..
Exponential functions show up in finance, bio science,
computer science and physical sciences.
The most used exponential functions are
y = 10x, y = ex and y = 2x.
Let’s use $ growth as applications below.
25. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
Compound Interest
26. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
27. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
28. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01)
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
29. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
30. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
31. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
after 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
32. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
33. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
After 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
34. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01)
= 1000(1 + 0.01)4 = $1040.60
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
After 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
35. Example D. We deposit $1,000 in an account that gives
1% interest compounded monthly. How much money is there
after 1 month? 2 months? 3 months? and after 4 months?
After 1 month: 1000(1 + 0.01) = $1010.
After 2 months: 1010(1 + 0.01) = 1000(1 + 0.01)(1 + 0.01)
= 1000(1 + 0.01)2 = $1020.10
After 3 months: 1020(1 + 0.01) = 1000(1 + 0.01)2(1 + 0.01)
= 1000(1 + 0.01)3 = $1030.30
After 4 months: 1030(1 + 0.01) = 1000(1 + 0.01)3(1 + 0.01)
= 1000(1 + 0.01)4 = $1040.60
Compound Interest
Let P = principal, i = (periodic) interest rate, A = accumulation.
After 1 period A = P(1 + i)
After 2 periods A = P(1 + i)(1 + i) = P(1 + i)2
After 3 periods A = P(1 + i)2(1 + i) = P(1 + i)3
Continue the pattern, after N periods, we obtain the
exponential periodic-compound formula (PINA): P(1 + i)N = A.
36. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
The PINA Formula (Periodic Interest)
37. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
38. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
0 1 2 3 Nth periodN–1
39. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
40. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i)
41. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2
42. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3
43. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1
44. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
45. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
46. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N =
47. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N = 60 *12 = 720 months
48. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N = 60 *12 = 720 months
so by PINA, there will be 1000(1 + 0.01) 720
49. Compound Interest
Let P = principal
i = (periodic) interest rate,
N = number of periods
A = accumulation
then P(1 + i) N = A
The PINA Formula (Periodic Interest)
We use the following time line to see what is happening.
P
0 1 2 3 Nth periodN–1
Rule: Multiply (1 + i) each period forward
P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) N - 1 P(1 + i) N
Example E. $1,000 is in an account that has a monthly interest
rate of 1%. How much will be there after 60 years?
We have P = $1,000, i = 1% = 0.01, N = 60 *12 = 720 months
so by PINA, there will be 1000(1 + 0.01) 720 = $1,292,376.71
after 60 years.
50. Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
51. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year.
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
52. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year.
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
53. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year.
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12.
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
54. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12.
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
55. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000
Compound Interest
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
56. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000, so by PINA:
Compound Interest
P (1 + ) 480 = 250,0000.09
12
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
57. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000, so by PINA:
Compound Interest
P (1 + ) 480 = 250,0000.09
12 or
(1 + ) 480
P = 250,000
0.09
12
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
58. Example F. We open an account with annual rate r = 9%,
compounded monthly, i.e. 12 times a year. After 40 years the
total return is $250,000, what was the initial principal?
We have r = 9% = 0.09 for one year,
and f = 12 is the number of times of compounding in one year,
so the periodic or monthly rate i = 0.09/12. After 40 years, i.e.
N = 40(12) = 480 months the return A = 250,000, so by PINA:
Compound Interest
P (1 + ) 480 = 250,0000.09
12 or
(1 + ) 480
P = 250,000
0.09
12
P = $6,923.31
by calculator
Hence the initial deposit is $6,923.31.
In practice, compound interests are usually quoted in
annual interest rate r and the frequency f, the number of times
of compounding in one year, so the periodic rate i = .
r
f
59. x -4 -3 -2 -1 0 1 2 3 4
y=2x 1/16 1/8 1/4 1/2 1 2 4 8 16
Graphs of the Exponential Functions
Here is a table of y = 2x for plotting its graph.
61. (0,1)
(1,2)
(2,4)
(3,8)
(-1,1/2)(-2,1/4)
y=2x
Graph of y = 2x
x -4 -3 -2 -1 0 1 2 3 4
y=2x 1/16 1/8 1/4 1/2 1 2 4 8 16
Graphs of the Exponential Functions
Graph of y = bx where b>1
Here is a table of y = 2x for plotting its graph.
This is the shape of the graphs of y = bx for b > 1.
62. x -4 -3 -2 -1 0 1 2 3 4
y=(½)x 16 8 4 2 1 1/2 1/4 1/8 1/16
Here is a table of y = (½)x for plotting its graph.
Graphs of the Exponential Functions
64. (0,1)
(-1,2)
(-2,4)
(-3,8)
(1,1/2) (2,1/4)
y= (½)x
Graph of y = bx where 0<b<1Graph of y = (½)x
x -4 -3 -2 -1 0 1 2 3 4
y=(½)x 16 8 4 2 1 1/2 1/4 1/8 1/16
Here is a table of y = (½)x for plotting its graph.
Graphs of the Exponential Functions
This is the shape of the graphs of y = bx for b < 1.
65. The graphs shown here are the different returns with r = 20%
with different compounding frequencies.
Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)
Compound Interest
66. The graphs shown here are the different returns with r = 20%
with different compounding frequencies. We observe that
I. the more frequently we compound, the bigger the return
Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)
Compound Interest
67. The graphs shown here are the different returns with r = 20%
with different compounding frequencies. We observe that
I. the more frequently we compound, the bigger the return
II. but the returns do not go above the blue-line
the continuous compound return, which is the next topic.
Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)
Compound Interest
68. Compound Interest
B. Given the monthly compounded periodic rate i, find the
principal needed to obtain a return of $1,000 after the given
amount the time.
1. i = 1%, time = 60 months.
Exercise A. Given the monthly compounded periodic rate i and
the amount the time, find the return with a principal of $1,000.
2. i = 1%, time = 60 years.
3. i = ½ %, time = 60 years 4. i = ½ %, time = 60 months.
5. i = 1¼ %, time = 6 months. 6. i = 1¼ %, time = 5½ years.
.7. i = 3/8%, time = 52 months. 8. i = 2/3%, time = 27 months.
1. i = 1%, time = 60 months. 2. i = 1%, time = 60 years.
3. i = ½ %, time = 60 years 4. i = ½ %, time = 60 months.
5. i = 1¼ %, time = 60 months. 6. i = 1¼ %, time = 60 years.
7. i = 3/8%, time = 60 years 8. i = 2/3%, time = 60 months.
69. Compound Interest
D. Given the annual rate r, convert it into the monthly
compounded periodic rate i and find the principal needed to
obtain $1,000 after the given amount the time.
1. r = 1%, time = 60 months.
C. Given the annual rate r, convert it into the monthly
compounded periodic rate i and find the return with a principal
of $1,000 after the given amount the time.
2. r = 1%, time = 60 years.
3. r = 3 %, time = 60 years 4. r = 3½ %, time = 60 months.
1. r = 1%, time = 60 months. 2. r = 1%, time = 60 years.
3. r = 3 %, time = 60 years 4. r = 3½ %, time = 60 months.
5. r = 1¼ %, time = 6 months. 6. r = 1¼ %, time = 5½ years.
.7. r = 3/8%, time = 52 months. 8. r = 2/3%, time = 27 months.
5. r = 1¼ %, time = 6 months. 6. r = 1¼ %, time = 5½ years.
.7. r = 3/8%, time = 52 months. 8. r = 2/3%, time = 27 months.
70. Exercise B.
1. 𝐴 ≈ 1816.7
(Answers to the odd problems) Exercise A.
3. 𝐴 ≈ 36271.41 5. 𝐴 ≈ 1077.39
7. 𝐴 ≈ 1214.87
1. P ≈ 550.45 3. P ≈ 27.57 5. P ≈ 474.57
7. P ≈ 67.55
1. 𝐴 ≈ 1051.25
Exercise C.
3. 𝐴 ≈ 6036.07 5. 𝐴 ≈ 1006.27
7. 𝐴 ≈ 1016.39
Exercise D.
1. 𝑃 ≈ 951.25 3. 𝑃 ≈ 165.67 5. 𝑃 ≈ 993.78
7. 𝑃 ≈ 983.88
Compound Interest