The document discusses linear approximations of functions. It provides examples of determining:
1) The value of a function f(x) at a point x0 and the derivative f'(x0)
2) The linear approximation L(x) = f(x0) + f'(x0)(x - x0)
3) Using L(x) to estimate the value of f(x) near x0
Profº. Marcelo Santos Chaves - Cálculo I (Limites e Continuidades) - Exercíci...MarcelloSantosChaves
1. The document discusses limits and continuities. It provides solutions to calculating the limits of 6 different functions as x approaches certain values.
2. The solutions involve algebraic manipulations such as factoring, simplifying, and applying limit properties. Various limit results are obtained such as 1, -6, 0.
3. The techniques demonstrated include making substitutions to simplify indeterminate forms, factoring, and taking limits of rational functions as the variables approach certain values.
This document provides a calculus cheat sheet covering key topics in limits, derivatives, and integrals. It defines limits, including one-sided limits and limits at infinity. Properties of limits are listed. Derivatives are defined and basic rules like the power, constant multiple, sum, difference, and chain rules are covered. Common derivatives are provided. Higher order derivatives and the second derivative are defined. Evaluation techniques like L'Hospital's rule, polynomials at infinity, and piecewise functions are summarized.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Profº. Marcelo Santos Chaves - Cálculo I (Limites e Continuidades) - Exercíci...MarcelloSantosChaves
1. The document discusses limits and continuities. It provides solutions to calculating the limits of 6 different functions as x approaches certain values.
2. The solutions involve algebraic manipulations such as factoring, simplifying, and applying limit properties. Various limit results are obtained such as 1, -6, 0.
3. The techniques demonstrated include making substitutions to simplify indeterminate forms, factoring, and taking limits of rational functions as the variables approach certain values.
This document provides a calculus cheat sheet covering key topics in limits, derivatives, and integrals. It defines limits, including one-sided limits and limits at infinity. Properties of limits are listed. Derivatives are defined and basic rules like the power, constant multiple, sum, difference, and chain rules are covered. Common derivatives are provided. Higher order derivatives and the second derivative are defined. Evaluation techniques like L'Hospital's rule, polynomials at infinity, and piecewise functions are summarized.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
This document contains a 7 page exam for the course CS-601: Differential and Integral Calculus with Applications. The exam contains 8 questions testing a variety of calculus concepts:
1) Part a contains 6 multiple choice questions testing derivatives, integrals, limits, and monotonicity. Part b contains 6 fill in the blank questions testing derivatives, integrals, and equations of tangents.
2) Questions 2-5 contain additional multiple choice or short answer problems testing continuity, derivatives, integrals, Rolle's theorem, and partial derivatives.
3) Questions 6-8 contain free response problems on geometry, differential equations, and approximating an area using Simpson's rule. The exam tests a comprehensive understanding of
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Profº Marcelo Santos Chaves Cálculo I (limites trigonométricos)MarcelloSantosChaves
The document provides solutions to 12 limit problems involving trigonometric functions. Each problem is solved in 3 steps or less. The solutions show that:
1) Many of the limits evaluate to simple numeric values like 1, 0, or constants like a.
2) Trigonometric limits are often solved by factorizing the expressions and applying standard trigonometric limits like lim(sinx/x) = 1 as x approaches 0.
3) More complex problems are broken down into composite limits and simplified through algebraic manipulation and properties of limits.
The document is a math worksheet containing calculus problems involving functions. It includes 21 problems involving operations on functions such as composition, inversion and transformations of function graphs. The problems involve determining expressions for composed functions, inverses, graphs of related functions obtained through transformations of an original function graph. The document also provides answers to the problems.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document provides a summary of key concepts that must be known for AP Calculus, including:
- Curve sketching and analysis of critical points, local extrema, and points of inflection
- Common differentiation and integration rules like product rule, quotient rule, trapezoidal rule
- Derivatives of trigonometric, exponential, logarithmic, and inverse functions
- Concepts of limits, continuity, intermediate value theorem, mean value theorem, fundamental theorem of calculus
- Techniques for solving problems involving solids of revolution, arc length, parametric equations, polar curves
- Series tests like ratio test and alternating series error bound
- Taylor series approximations and common Maclaurin series
Here are the problems from the slides with their solutions:
1. Find the slope of the line tangent to the graph of the function f(x) = x^2 - 5x + 8 at the point P(1,4).
Slope = -3
2. Find the equation of the tangent line to the curve f(x) = 2x^2 - 3 at the point P(1,-1) using point-slope form.
y - (-1) = 4(x - 1)
3. Find the equation of the tangent line to the curve f(x) = x + 6 at the point P(3,3) using point-slope form.
y
The document provides a summary of mathematics formulae for Form 4 students. It includes:
1) Common functions and their derivatives such as absolute value, inverse, quadratic, and fractional functions.
2) Key concepts in algebra including the quadratic formula, nature of roots, and forming quadratic equations from roots.
3) Essential statistics measures like mean, median, variance, and standard deviation.
4) Formulas for coordinate geometry topics like distance, gradient, parallel and perpendicular lines, and locus equations.
5) Rules for differentiation including algebraic, fractional, and chain rule.
The document provides an introduction to evaluating limits, including:
1. The limit of a constant function is the constant.
2. Common limit laws can be used to evaluate limits of sums, differences, products, and quotients if the individual limits exist.
3. Special techniques may be needed to evaluate limits that involve indeterminate forms, such as 0/0, infinity/infinity, or limits approaching infinity. These include factoring, graphing, and rationalizing.
This document contains examples and explanations of limits involving various functions. Some key points covered include:
- Substitution can be used to evaluate limits, such as substituting 2 into -2x^3.
- Left and right hand limits must agree for the overall limit to exist.
- The limit of a piecewise function exists if the left and right limits are the same.
- Graphs can help verify limit calculations and show discontinuities.
- Special limits involving trigonometric and greatest integer functions are evaluated.
This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.
This document discusses linear equations and curve fitting. It provides 18 examples of using a linear system to solve for the coefficients of linear, quadratic, and cubic polynomials that fit given data points. It also provides examples of using a linear system to solve for the coefficients of circle and central conic equations that fit given points. The linear systems are set up and solved, providing the resulting equations that fit the data in each example.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
The document discusses equations of planes in 3D space. It introduces the point-normal form of a plane equation: A(x-r) + B(y-s) + C(z-t) = 0, where <A,B,C> is a normal vector to the plane and <r,s,t> is a point on the plane. This equation is analogous to the point-slope form for a line, and can be derived by setting the dot product of the normal vector and a position vector from the point <r,s,t> to a generic point <x,y,z> on the plane equal to zero.
Linear algebra-solutions-manual-kuttler-1-30-11-otckjalili
This document contains 17 exercises involving complex numbers and operations on complex numbers:
1) Find the inverse, product, sum, square, and quotient of various complex numbers.
2) Find the complete solution to polynomial equations like x4 + 16 = 0 by finding the roots.
3) De Moivre's theorem can be used to derive trigonometric identities and extends to negative integer exponents.
4) The complex field is not an ordered field since i2 = -1 violates trichotomy.
This document discusses limits of functions. It begins by defining the limit of a function f(x) as x approaches a number c as the value that f(x) approaches as x gets closer to c. It provides examples of limits, including one-sided limits and limits at infinity. Key theorems are presented for computing limits, including properties of limits and the sandwich theorem. The document focuses on conceptual understanding and applying techniques to evaluate a variety of limit examples.
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
1. The set of all functions f: R → R with f(0) = 0 is a vector space, as the linear combination of such functions will also satisfy f(0) = 0.
2. The set of all odd functions is a vector space, as any linear combination of odd functions will also be odd.
3. The solution space to the differential equation y''(x) - 5y'(x) = 0 is 2-dimensional with basis {1, e^5x}, as the general solution is Ce^5x + D.
This document contains a 7 page exam for the course CS-601: Differential and Integral Calculus with Applications. The exam contains 8 questions testing a variety of calculus concepts:
1) Part a contains 6 multiple choice questions testing derivatives, integrals, limits, and monotonicity. Part b contains 6 fill in the blank questions testing derivatives, integrals, and equations of tangents.
2) Questions 2-5 contain additional multiple choice or short answer problems testing continuity, derivatives, integrals, Rolle's theorem, and partial derivatives.
3) Questions 6-8 contain free response problems on geometry, differential equations, and approximating an area using Simpson's rule. The exam tests a comprehensive understanding of
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Profº Marcelo Santos Chaves Cálculo I (limites trigonométricos)MarcelloSantosChaves
The document provides solutions to 12 limit problems involving trigonometric functions. Each problem is solved in 3 steps or less. The solutions show that:
1) Many of the limits evaluate to simple numeric values like 1, 0, or constants like a.
2) Trigonometric limits are often solved by factorizing the expressions and applying standard trigonometric limits like lim(sinx/x) = 1 as x approaches 0.
3) More complex problems are broken down into composite limits and simplified through algebraic manipulation and properties of limits.
The document is a math worksheet containing calculus problems involving functions. It includes 21 problems involving operations on functions such as composition, inversion and transformations of function graphs. The problems involve determining expressions for composed functions, inverses, graphs of related functions obtained through transformations of an original function graph. The document also provides answers to the problems.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document provides a summary of key concepts that must be known for AP Calculus, including:
- Curve sketching and analysis of critical points, local extrema, and points of inflection
- Common differentiation and integration rules like product rule, quotient rule, trapezoidal rule
- Derivatives of trigonometric, exponential, logarithmic, and inverse functions
- Concepts of limits, continuity, intermediate value theorem, mean value theorem, fundamental theorem of calculus
- Techniques for solving problems involving solids of revolution, arc length, parametric equations, polar curves
- Series tests like ratio test and alternating series error bound
- Taylor series approximations and common Maclaurin series
Here are the problems from the slides with their solutions:
1. Find the slope of the line tangent to the graph of the function f(x) = x^2 - 5x + 8 at the point P(1,4).
Slope = -3
2. Find the equation of the tangent line to the curve f(x) = 2x^2 - 3 at the point P(1,-1) using point-slope form.
y - (-1) = 4(x - 1)
3. Find the equation of the tangent line to the curve f(x) = x + 6 at the point P(3,3) using point-slope form.
y
The document provides a summary of mathematics formulae for Form 4 students. It includes:
1) Common functions and their derivatives such as absolute value, inverse, quadratic, and fractional functions.
2) Key concepts in algebra including the quadratic formula, nature of roots, and forming quadratic equations from roots.
3) Essential statistics measures like mean, median, variance, and standard deviation.
4) Formulas for coordinate geometry topics like distance, gradient, parallel and perpendicular lines, and locus equations.
5) Rules for differentiation including algebraic, fractional, and chain rule.
The document provides an introduction to evaluating limits, including:
1. The limit of a constant function is the constant.
2. Common limit laws can be used to evaluate limits of sums, differences, products, and quotients if the individual limits exist.
3. Special techniques may be needed to evaluate limits that involve indeterminate forms, such as 0/0, infinity/infinity, or limits approaching infinity. These include factoring, graphing, and rationalizing.
This document contains examples and explanations of limits involving various functions. Some key points covered include:
- Substitution can be used to evaluate limits, such as substituting 2 into -2x^3.
- Left and right hand limits must agree for the overall limit to exist.
- The limit of a piecewise function exists if the left and right limits are the same.
- Graphs can help verify limit calculations and show discontinuities.
- Special limits involving trigonometric and greatest integer functions are evaluated.
This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.
This document discusses linear equations and curve fitting. It provides 18 examples of using a linear system to solve for the coefficients of linear, quadratic, and cubic polynomials that fit given data points. It also provides examples of using a linear system to solve for the coefficients of circle and central conic equations that fit given points. The linear systems are set up and solved, providing the resulting equations that fit the data in each example.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
The document discusses equations of planes in 3D space. It introduces the point-normal form of a plane equation: A(x-r) + B(y-s) + C(z-t) = 0, where <A,B,C> is a normal vector to the plane and <r,s,t> is a point on the plane. This equation is analogous to the point-slope form for a line, and can be derived by setting the dot product of the normal vector and a position vector from the point <r,s,t> to a generic point <x,y,z> on the plane equal to zero.
Linear algebra-solutions-manual-kuttler-1-30-11-otckjalili
This document contains 17 exercises involving complex numbers and operations on complex numbers:
1) Find the inverse, product, sum, square, and quotient of various complex numbers.
2) Find the complete solution to polynomial equations like x4 + 16 = 0 by finding the roots.
3) De Moivre's theorem can be used to derive trigonometric identities and extends to negative integer exponents.
4) The complex field is not an ordered field since i2 = -1 violates trichotomy.
This document discusses limits of functions. It begins by defining the limit of a function f(x) as x approaches a number c as the value that f(x) approaches as x gets closer to c. It provides examples of limits, including one-sided limits and limits at infinity. Key theorems are presented for computing limits, including properties of limits and the sandwich theorem. The document focuses on conceptual understanding and applying techniques to evaluate a variety of limit examples.
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
1. The set of all functions f: R → R with f(0) = 0 is a vector space, as the linear combination of such functions will also satisfy f(0) = 0.
2. The set of all odd functions is a vector space, as any linear combination of odd functions will also be odd.
3. The solution space to the differential equation y''(x) - 5y'(x) = 0 is 2-dimensional with basis {1, e^5x}, as the general solution is Ce^5x + D.
This document contains a tutorial on calculating limits, derivatives, and slopes from graphs and equations. It works through multiple examples of finding:
1) The limit of a function as x approaches a number from the left and right, and determining if the limit exists.
2) The slope of a secant line using the formula for average velocity.
3) The slope of a tangent line using the formula for instantaneous velocity.
4) Using slopes to find equations of lines tangent to a curve at a point.
The document explains the relevant formulas and step-by-step workings through examples to demonstrate how to apply the concepts.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 26: The Fundamental Theorem of Calculus (slides)Mel Anthony Pepito
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the anti-derivative F of f. Examples are provided to illustrate how to use the Fundamental Theorem to find derivatives and integrals.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
The document discusses polynomial functions, including how to graph common polynomials, find zeros of polynomials, and write polynomials given their roots. It provides examples of matching polynomial equations to their graphs, finding the real zeros of polynomials by factoring, and writing polynomials when given the roots. The document also covers how to use a graphing calculator to find the zeros of polynomials.
The document discusses techniques for sketching graphs of functions, including:
- Using the increasing/decreasing test to determine if a function is increasing or decreasing based on the sign of the derivative
- Using the concavity test to determine if a graph is concave up or down based on the second derivative
- A checklist for completely graphing a function, including finding critical points, inflection points, asymptotes, and putting together the information about monotonicity and concavity.
The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
Exponential and logarithm functions are important in both theory and practice. They examine the graphs of exponential functions f(x)=ax where a>0 and logarithm functions f(x)=loga(x) where a>0. It is important to practice these functions so their properties become intuitive. Key properties include exponential functions where a>1 increase rapidly for positive x and 0<a<1 increase for decreasing negative x, and both pass through (0,1). The natural logarithm function f(x)=ln(x) is particularly important.
This document defines and lists several common parent functions including: constant, linear, quadratic, cubic, absolute value, greatest integer, square root, cube root, exponential, logarithmic, reciprocal, rational, and trigonometric functions. The parent functions are basic building blocks used to model real world phenomena through transformations and combinations.
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
The document discusses limits and continuity of functions. It provides examples of computing one-sided limits, limits at points of discontinuity, and limits involving algebraic, trigonometric, exponential and logarithmic functions. The key rules for limits include the properties of limits, the sandwich theorem, and limits of compositions of functions. Continuity of functions is defined as a function having a limit equal to its value at a point. Polynomials, trigonometric functions and exponentials are shown to be continuous everywhere they are defined.
1. A differential equation is an equation that relates an unknown function with some of its derivatives. The document provides a step-by-step example of solving a differential equation to find the xy-equation of a curve with a given gradient condition.
2. The key steps are: (1) write the derivative term as a fraction, (2) integrate both sides, (3) apply the initial condition to determine the constant term, (4) write the final function relationship.
3. Common types of differential equations discussed are separable first order equations, where the derivative terms can be isolated by dividing both sides.
This section introduces general and particular solutions to differential equations of the form y' = f(x) through direct integration and evaluation of constants. Examples provided include:
1) Integrating y' = 2x + 1 and applying the initial condition x = 0, y = 3 yields the general solution y(x) = x^2 + x + 3.
2) Integrating y' = (x - 2)^2 and applying x = 2, y = 1 yields y(x) = (1/3)(x - 2)^3.
3) Six more examples of first-order differential equations are worked through to find their general solutions.
The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.
This document provides mathematical formulas and definitions for topics in algebra, geometry, trigonometry, and other areas of mathematics. It includes 3 or fewer sentences summarizing key information about triangles, circles, factoring polynomials, exponents, trigonometric functions, and other concepts. Diagrams illustrate formulas for areas of geometric shapes, trigonometric functions, and other visual representations. Tables list trigonometric function values at common angles in both radians and degrees.
This document provides examples of different sections from a larger work, specifically examples 7.3.5 through 7.3.9. The examples discuss various topics but no other context or details are provided about the content or purpose of the examples.
This document provides a tutorial on convergence tests for series, including:
1) Comparison tests 1 and 2, where a series is compared to a convergent or divergent series.
2) The ratio test, where the limit of successive terms is evaluated.
3) The root test, where the limit of the nth root of terms is evaluated.
4) The integral test, where a series is compared to an integral.
Several examples are worked through applying these tests to determine if various series converge or diverge. Advice is given to practice more problems and remember God.
This document contains a tutorial on limits of sequences. It provides examples of calculating the limit of several sequences as n approaches infinity. It also gives examples of finding the limit supremum and limit infirmum of sequences. The document concludes by wishing the reader good luck on their exercises and reminds them to remember Allah.
The document provides examples of using Lagrange multipliers to find the extremum of a function subject to a constraint. In example 8, the critical point and extremum are found for f(x,y,z) = x + y + z with the constraint x + y + z = 1. In example 9, the critical point (0, 0, 0) is identified as minimizing the distance from any point (x,y,z) to the origin. Example 10 finds the critical point (0, √2, √2) minimizes the distance function with the constraint x^2 + y^2 + z^2 = 2.
This tutorial discusses three ways to use Green's theorem to find the area of a hypocycloid by changing the parameterization to x=aθt and y=aθt, where 0≤t≤2. It recommends using the formula that involves both x and y, giving the equations dx=-3aθdt and dy=3aθdt to substitute into the Green's theorem area formula and solve the resulting integration.
This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
Rolle's theorem and the mean value theorem have some similarities and differences:
- Both theorems deal with continuous functions on closed intervals, but Rolle's theorem requires the function be differentiable on the open interval while mean value theorem does not.
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1. It provides examples of determining whether a transformation T is a linear combination or not based on checking if T(u+v)=T(u)+T(v) and T(ku)=kT(u).
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Eigenvalues and eigenvectors were found for several matrices. For a 3x3 matrix with eigenvalues 2, 2, 5, bases for the eigenspaces were determined to be the vectors [-5/2, 1, 2], [0, 1, 0], and [-8/3, 1, 3]. Another matrix was shown to be diagonalizable with eigenvalues 1, -1, 2 and change of basis matrix P.
The document provides information about linear algebra tutorial 7. It discusses properties of orthogonal matrices including:
- Matrix A is an orthogonal matrix since AT = A-1
- The rows and columns of A form orthonormal sets, confirming A is orthogonal
- Methods to find the inverse and transition matrices between different bases are presented.
Properties of orthogonality and procedures for working with orthogonal matrices are examined through examples.
1. The document discusses the Gram-Schmidt process for generating an orthogonal basis from a set of vectors.
2. The Gram-Schmidt process works by taking the first vector as is, then subtracting the projection of each subsequent vector onto the previous vectors to make it orthogonal.
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This presentation is version 3 of the strategic plan for Real Bedford Football Club.
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Indian Premier League (IPL) ---2024.pptxrathinikunj60
The Indian Premier League (IPL) is one of the most prominent and lucrative Twenty20 (T20) cricket leagues in the world. Since its inception in 2008, the IPL has revolutionized the landscape of cricket by blending sports, entertainment, and commerce. This summary provides an overview of the IPL's history, structure, notable performances, controversies, and its impact on cricket and beyond.
History and Formation
The IPL was launched by the Board of Control for Cricket in India (BCCI) in 2008, inspired by the success of domestic T20 leagues like the English T20 Cup and the now-defunct Indian Cricket League (ICL). Lalit Modi, the then Vice-President of BCCI, played a crucial role in conceptualizing and launching the league. The inaugural season kicked off in April 2008 with eight franchises representing different cities in India.
Structure and Format
The IPL follows a franchise-based model, where teams are owned by a mix of corporations, Bollywood stars, and other high-profile individuals. The league originally started with eight teams, although the number has fluctuated over the years due to various reasons including expansions and terminations. As of the latest seasons, the IPL features ten teams.
The tournament format includes a double round-robin stage, where each team plays the others twice, followed by playoffs. The top four teams from the round-robin stage qualify for the playoffs, which consist of two qualifiers, an eliminator, and the final. This format ensures a highly competitive and engaging tournament, culminating in a grand finale to crown the champion.
Teams and Their Evolution
The founding teams of the IPL were:
Chennai Super Kings (CSK)
Delhi Daredevils (now Delhi Capitals)
Kings XI Punjab (now Punjab Kings)
Kolkata Knight Riders (KKR)
Mumbai Indians (MI)
Rajasthan Royals (RR)
Royal Challengers Bangalore (RCB)
Deccan Chargers (now defunct, replaced by Sunrisers Hyderabad)
Over the years, the league has seen new teams such as Pune Warriors India, Kochi Tuskers Kerala, Gujarat Lions, and Rising Pune Supergiant. The most recent additions are the Gujarat Titans and Lucknow Super Giants, introduced in the 2022 season.
Iconic Players and Performances
The IPL has attracted the best talent from around the world, with numerous iconic players making significant contributions. Some of the standout performers include:
Sachin Tendulkar (MI): The "Little Master" brought his legendary status to the IPL, winning the Orange Cap (top run-scorer) in 2010.
Chris Gayle (RCB, KXIP): Known for his explosive batting, Gayle holds the record for the highest individual score in an IPL match (175*).
MS Dhoni (CSK): Dhoni's leadership has been instrumental in CSK's success, leading them to multiple titles.
AB de Villiers (RCB): Renowned for his innovative stroke play, de Villiers has been a consistent match-winner.
Virat Kohli (RCB): The highest run-scorer in IPL history, Kohli's batting prowess is unmatched.
La
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3. 152 CHAPTER 3. APPLICATIONS OF DIFFERENTIATION
f (x0 ) 30
(a) x1 = x0 −
f (x0 )
3 1
= −1 − =− 20
−6 2
f (x1 ) y
x2 = x1 −
f (x1 ) 10
1 0.375
=− − = −0.4117647
2 −4.25
0
−5.0 −2.5 0.0 2.5 5.0
(b) The root is x ≈ −0.4064206546. x
−10
Start with x0 = −5 to find the root near −5:
15. f (x) = x4 − 3x2 + 1 = 0, x0 = 1 x1 = −4.718750, x2 = −4.686202,
f (x) = 4x3 − 6x x3 = −4.6857796, x4 = −4.6857795
f (x0 )
(a) x1 = x0 − 18. From the graph, we see two roots:
f (x0 )
14 − 3 · 12 + 1 1
=1− = 15
4 · 13 − 6 · 1 2
10
f (x1 )
x2 = x1 −
f (x1 ) 5
-1 0 1 2 3 4
1 4 1 2 0
1 2 −3 2 +1
= −
2 1 3 1 -5
4 2 −6 2
-10
5
=
8 -15
-20
(b) 0.61803
16. f (x) = x4 − 3x2 + 1, x0 = −1 f (xi )
Use xi+1 = xi − with
f (x) = 4x3 − 6x f (xi )
f (x) = x4 − 4x3 + x2 − 1, and
f (x) = 4x3 − 12x2 + 2x
f (x0 ) Start with x0 = 4 to find the root below 4:
(a) x1 = x0 − x1 = 3.791666667, x2 = 3.753630030, x3 =
f (x0 )
−1 1 3.7524339, x4 = 3.752432297
= −1 − =− Start with x = −1 to find the root just above
2 2
f (x1 ) −1:
x2 = x1 − x1 = −0.7222222222,
f (x1 )
x2 = −0.5810217936,
1 0.3125 x3 = −0.5416512863,
=− − = −0.625
2 2.5 x4 = −0.5387668233,
x5 = −0.5387519962
(b) The root is x ≈ −0.6180339887.
f (xi ) f (xi )
17. Use xi+1 = xi − with 19. Use xi+1 = xi − with
f (xi ) f (xi )
f (x) = x3 + 4x2 − 3x + 1, and f (x) = x5 + 3x3 + x − 1, and
f (x) = 3x2 + 8x − 3 f (x) = 5x4 + 9x2 + 1
4. 3.1. LINEAR APPROXIMATIONS AND NEWTONS METHOD 153
10 x1 = −0.644108, x2 = −0.636751
x3 = −0.636733, x4 = −0.636733
Start with x0 = 1.5 to find the root near 1.5:
5
x1 = 1.413799, x2 = 1.409634
x3 = 1.409624, x4 = 1.409624
0
−1.0 −0.5 0.0 0.5 1.0
22. Use xi+1 = xi − f (xii)) with
f
(x
x
f (x) = cos x2 − x, and
y −5 f (x) = 2x sin x2 − 1
3
−10
2
Start with x0 = 0.5 to find the root near 0.5: y
x1 = 0.526316, x2 = 0.525262, 1
x3 = 0.525261, x4 = 0.525261
0
f (xi ) -2 -1 0 1 2
20. Use xi+1 = xi − with x
f (xi ) -1
f (x) = cos x − x, and
f (x) = − sin x − 1 -2
5.0
Start with x0 = 1 to find the root between 0
and 1:
2.5
x1 = 0.8286590991, x2 = 0.8016918647,
x3 = 0.8010710854, x4 = 0.8010707652
0.0
3
−5 −4 −3 −2 −1 0 1 2 3 4 5
x
2
y −2.5 y
1
−5.0
0
Start with x0 = 1 to find the root near 1: -2 -1 0 1
x
2
x1 = 0.750364, x2 = 0.739113, -1
x3 = 0.739085, x4 = 0.739085
-2
21. Use xi+1 = xi − f (xii)) with
f
(x
f (x) = sin x − x2 + 1, and f (xi )
f (x) = cos x − 2x 23. Use xi+1 = xi − with
f (xi )
5.0
f (x) = ex + x, and
f (x) = ex + 1
20
2.5
15
0.0
−5 −4 −3 −2 −1 0 1 2 3 4 5
y 10
x
y −2.5
5
−5.0
0
−3 −2 −1 0 1 2 3
x
Start with x0 = −0.5 to find the root near −5
−0.5: Start with x0 = −0.5 to find the root between
5. 154 CHAPTER 3. APPLICATIONS OF DIFFERENTIATION
0 and -1: zeros of f ), Newton’s method will succeed.
x1 = −0.566311, x2 = −0.567143 Which root is found depends on the starting
x3 = −0.567143, x4 = −0.567143 place.
f (xi ) 33. f (x) = x2 + 1, x0 = 0
24. Use xi+1 = xi − with
√ f (xi ) f (x) = 2x
f (x) = e−x − x, and f (x0 ) 1
1 x1 = x0 − =0−
f (x) = −e−x − √ f (x0 ) 0
2 x The method fails because f (x0 ) = 0. There
are no roots.
1
34. Newton’s method fails because the function
0.5
does not have a root!
4x2 − 8x + 1
0
0 0.5 1 1.5 2
35. f (x) = = 0, x0 = −1
4x2 − 3x − 7
Note: f (x0 ) = f (−1) is undefined, so New-
-0.5
ton’s Method fails because x0 is not in the do-
main of f . Notice that f (x) = 0 only when
-1
4x2 − 8x + 1 = 0. So using Newton’s Method
on g(x) = 4x2 − 8x + 1 with x0 = −1 leads to
x ≈ .1339. The other root is x ≈ 1.8660.
Start with x0 = 0.5 to find the root close to
36. The slope tends to infinity at the zero. For ev-
0.5:
ery starting point, the sequence does not con-
x1 = 0.4234369253, x2 = 0.4262982542,
verge.
x3 = 0.4263027510
√ 37. (a) With x0 = 1.2,
25. f (x) = x2 − 11; x0 = 3; 11 ≈ 3.316625
√ x1 = 0.800000000,
26. Newton’s method for x near x = 23 is xn+1 = x2 = 0.950000000,
1
2 (xn + 23/xn ). Start with x0 = 5 to get:
x3 = 0.995652174,
x1 = 4.8, x2 = 4.7958333, and x3 = 4.7958315. x4 = 0.999962680,
√ x5 = 0.999999997,
27. f (x) = x3 − 11; x0 = 2; 3 11 ≈ 2.22398 x6 = 1.000000000,
√ x7 = 1.000000000
28. Newton’s method for 3 x near x = 23 is
xn+1 = 1 (2xn + 23/x2 ). Start with x0 = 3
3 n (b) With x0 = 2.2,
to get: x0 = 2.200000, x1 = 2.107692,
x1 = 2.851851851, x2 = 2.843889316, and x2 = 2.056342, x3 = 2.028903,
x3 = 2.884386698 x4 = 2.014652, x5 = 2.007378,
√ x6 = 2.003703, x7 = 2.001855,
29. f (x) = x4.4 − 24; x0 = 2; 4.4 24 ≈ 2.059133
x8 = 2.000928, x9 = 2.000464,
√
30. Newton’s method for 4.6 x near x = 24 is x10 = 2.000232, x11 = 2.000116,
1
xn+1 = 4.6 (3.6xn +24/x3.6 ). Start with x0 = 2
n x12 = 2.000058, x13 = 2.000029,
to get: x14 = 2.000015, x15 = 2.000007,
x1 = 1.995417100, x2 = 1.995473305, and x16 = 2.000004, x17 = 2.000002,
x3 = 1.995473304 x18 = 2.000001, x19 = 2.000000,
x20 = 2.000000
31. f (x) = 4x3 − 7x2 + 1 = 0, x0 = 0 The convergence is much faster with x0 =
f (x) = 12x2 − 14x 1.2.
f (x0 ) 1
x1 = x0 − =0−
f (x0 ) 0 38. Starting with x0 = 0.2 we get a sequence that
The method fails because f (x0 ) = 0. Roots converges to 0 very slowly. (The 20th itera-
are 0.3454, 0.4362, 1.659. tion is still not accurate past 7 decimal places).
Starting with x0 = 3 we get a sequence that
32. Newton’s method fails because f = 0. As long
7 quickly converges to π. (The third iteration is
as the sequence avoids xn = 0 and xn = (the already accurate to 10 decimal places!)
6
6. 3.1. LINEAR APPROXIMATIONS AND NEWTONS METHOD 155
√
39. (a) With x0 = −1.1 43. f (x) = √ 4 + x
x1 = −1.0507937, f (0) = 4 + 0 = 2
x2 = −1.0256065, 1
f (x) = (4 + x)−1/2
x3 = −1.0128572, 2
1 1
x4 = −1.0064423, f (0) = (4 + 0)−1/2 =
x5 = −1.0032246, 2 4
1
x6 = −1.0016132, L(x) = f (0) + f (0)(x − 0) = 2 + x
4
x7 = −1.0008068, 1
x8 = −1.0004035, L(0.01) = 2 + (0.01) = 2.0025
√ 4
x9 = −1.0002017, f (0.01) = 4 + 0.01 ≈ 2.002498
x10 = −1.0001009, 1
L(0.1) = 2 + (0.1) = 2.025
x11 = −1.0000504, √ 4
x12 = −1.0000252, f (0.1) = 4 + 0.1 ≈ 2.0248
x13 = −1.0000126, 1
L(1) = 2 + (1) = 2.25
x14 = −1.0000063, √ 4
x15 = −1.0000032, f (1) = 4 + 1 ≈ 2.2361
x16 = −1.0000016,
x17 = −1.0000008,
x18 = −1.0000004,
x19 = −1.0000002,
x20 = −1.0000001, 44. The linear approximation for ex at x = 0 is
x21 = −1.0000000, L(x) = 1 + x. The error when x = 0.01 is
x22 = −1.0000000 0.0000502, when x = 0.1 is 0.00517, and when
(b) With x0 = 2.1 x = 1 is 0.718.
x0 = 2.100000000,
x1 = 2.006060606,
x2 = 2.000024340,
x3 = 2.000000000,
x4 = 2.000000000 45. (a) f (0) = g(0) = h(0) = 1, so all pass
The rate of convergence in (a) is slower through the point (0, 1).
than the rate of convergence in (b). f (0) = 2(0 + 1) = 2,
g (0) = 2 cos(2 · 0) = 2, and
40. From exercise 37, f (x) = (x − 1)(x − 2)2 . New-
h (0) = 2e2·0 = 2,
ton’s method converges slowly near the double
so all have slope 2 at x = 0.
root. From exercise 39, f (x) = (x − 2)(x + 1)2 .
The linear approximation at x = 0 for all
Newton’s method again converges slowly near
three functions is L(x) = 1 + 2x.
the double root. In exercise 38, Newton’s
method converges slowly near 0, which is a zero
of both x and sin x but converges quickly near
π, which is a zero only of sin x. (b) Graph of f (x) = (x + 1)2 :
5
41. f (x) = tan x, f (0) = tan 0 = 0
f (x) = sec2 x, f (0) = sec2 0 = 1 4
L(x) = f (0) + f (0)(x − 0) L(0.01) = 0.01
3
= 0 + 1(x − 0) = x y
f (0.01) = tan 0.01 ≈ 0.0100003 2
L(0.1) = 0.1 1
f (0.1) = tan(0.1) ≈ 0.1003
L(1) = 1 0
f (1) = tan 1 ≈ 1.557 −3 −2 −1 0 1 2 3
−1
√ x
42. The linear approximation for 1 + x at x = 0
1
is L(x) = 1 + 2 x. The error when x = 0.01 is
0.0000124, when x = 0.1 is 0.00119, and when
x = 1 is 0.0858. Graph of f (x) = 1 + sin(2x):
7. 156 CHAPTER 3. APPLICATIONS OF DIFFERENTIATION
5
2
4
1
3
y
2 0
-2 -1 0 1 2
x
1
-1
0
−3 −2 −1 0 1 2 3
-2
x −1
Graph of h(x) = sinh x:
Graph of f (x) = e2x :
3
5
2
4
1
3 0
-2 -1 0 1 2
y
-1 x
2
-2
1
-3
0
−3 −2 −1 0 1 2 3
x −1
sin x is the closest fit, but sinh x is close.
√
4
47. (a) 16.04 = 2.0012488
L(0.04) = 2.00125
|2.0012488 − 2.00125| = .00000117
46. (a) f (0) = g(0) = h(0) = 0, so all pass √
4
through the point (0, 0). (b) 16.08 = 2.0024953
f (0) = cos 0 = 1, L(.08) = 2.0025
1 |2.0024953 − 2.0025| = .00000467
g (0) = = 1, and
1 + 02 √
h (0) = cosh 0 = 1, (c) 4
16.16 = 2.0049814
so all have slope 1 at x = 0. L(.16) = 2.005
The linear approximation at x = 0 for all |2.0049814 − 2.005| = .0000186
three functions is L(x) = x.
(b) Graph of f (x) = sin x: 48. If you graph | tan x − x|, you see that the dif-
ference is less than .01 on the interval −.306 <
2 x < .306 (In fact, a slightly larger interval
would work as well).
1
49. The first tangent line intersects the x-axis at a
0
-2 -1 0 1 2 point a little to the right of 1. So x1 is about
x
1.25 (very roughly). The second tangent line
-1
intersects the x-axis at a point between 1 and
x1 , so x2 is about 1.1 (very roughly). Newton’s
-2 Method will converge to the zero at x = 1.
Starting with x0 = −2, Newton’s method con-
Graph of g(x) = tan−1 x: verges to x = −1.
8. 3.1. LINEAR APPROXIMATIONS AND NEWTONS METHOD 157
f (x) = 2x − 1
3
3
At x0 =
2
2
2
y 3 3 1
f (x0 ) = − −1=−
1 2 2 4
and
3
-2 -1
0
0 1 2
f (x0 ) = 2 −1=2
x
2
-1 By Newton’s formula,
f (x0 ) 3 −1 13
x1 = x0 − = − 4 =
-2 f (x0 ) 2 2 8
Starting with x0 = 0.4, Newton’s method con- (b) f (x) = x2 − x − 1
verges to x = 1. f (x) = 2x − 1
5
At x0 = 3
3 2
5 5 1
f (x0 ) = − −1=
2 3 3 9
y
and
5 7
1 f (x0 ) = 2 −1=
3 3
0
By Newton’s formula,
-2 -1 0 1 2
f (x0 )
x x1 = x0 −
-1 f (x0 )
1
5 9 5 1 34
-2 = − 7 = − =
3 3
3 21 21
50. It wouldn’t work because f (0) = 0. x0 = 0.2 (c) f (x) = x2 − x − 1
works better as an initial guess. After jumping f (x) = 2x − 1
8
to x1 = 2.55, the sequence rapidly decreases At x0 = 5
2
toward x = 1. Starting with x0 = 10, it takes 8 8 1
f (x0 ) = − −1=−
several steps to get to 2.5, on the way to x = 1. 5 5 25
and
f (xn ) 8 11
51. xn+1 = xn − f (x0 ) = 2 −1=
f (xn ) 5 5
x2 − c
n By Newton’s formula,
= xn − f (x0 )
2xn x1 = x0 −
x2 c f (x0 )
= xn − n + 8 − 25 1
8 1 89
2xn 2xn = − 11 = + =
xn c 5 5 55 55
= + 5
2 2xn
1 c (d) From part (a),
= xn + F4 F7
2 xn sincex0 = , hence x1 = .
√ √ √ F3 F6
If x0 < a, then a/x0 > a, so x0 < a < From part (b),
a/x0 . F5 F9
√ since x0 = hence x1 = .
52. The root of xn − c is n c, so Newton’s method F4 F8
From part (c),
approximates this number. F6 F11
Newton’s method gives since x0 = hence x1 = .
f (xi ) xn − c F5 F10
xi+1 = xi − = xi − i n−1 Fn+1
f (xi ) nxi Thus in general if x0 = , then x1 =
Fn
1 F2n+1
= (nxi − xi + cx1−n ),
i implies m = 2n + 1 and k = 2n
n F2n
as desired.
3 Fn+1
53. (a) f (x) = x2 − x − 1 (e) Given x0 = , then lim will be
2 n→∞ Fn
9. 158 CHAPTER 3. APPLICATIONS OF DIFFERENTIATION
the zero of the function f (x) = x2 − 2P x
L(x) = 120 − .01(120) = P −
x − 1 which is 1.618034. Therefore, R
Fn+1 2 · 120x
lim = 1.618034 = 120 −
n→∞ Fn R
2x
.01 =
54. The general form of functionf (x) is, R
1 n+2 1 1 x = .005R = .005(20,900,000)
fn (x) = 2 x − 3 for n < x < n−1 .
5 2 2 = 104,500 ft
Hence
2n+2 1 1 58. If m = m0 (1 − v 2 /c2 )1/2 , then
f (x) = fn (x) = for n < x < n−1 .
5 2 2 m = (m0 /2)(1 − v 2 /c2 )−1/2 (−2v/c2 ), and
By Newton’s method,
m = 0 when v = 0. The linear approxima-
3 f 34 3 f1 34
x1 = − = − tion is the constant function m = m0 for small
4 f 3 4
4 f1 4 3
values v.
3 (3/5 ) 3 x0
= − = = 59. The only positive solution is 0.6407.
4 (8/5 ) 8 2
x1 x0 x0
Similarly, x2 = = 2 and x3 = 3 60. The smallest positive solution of the first equa-
2 2 2 tion is 0.132782, and for the second equa-
x0
Continuing this, we get, xn−1 = n−1 It may tion the smallest positive solution is 1, so the
2
also be observed that, for each fn (x) species modeled by the second equation is cer-
(1/2n ) + 1/2n+1 3 tain to go extinct. This is consistent with the
x0 = = n+1 ,
2 2 models, since the expected number of offspring
x0 3 3 for the population modeled by the first equa-
xn = n = 2n+1 ⇒ xn+1 = 2n+2 which
2 2 2 tion is 2.2, while for the second equation it is
is the zero of F . Therefore Newton’s method
only 1.3
converges to zero of F .
61. The linear approximation for the inverse tan-
55. For small x we approximate ex by x + 1 gent function at x = 0 is
(see exercise 44) f (x) ≈ f (0) + f (0)(x − 0)
Le2πd/L − e−2πd/L tan−1 (x) ≈ tan−1 (0) + 1+02 (x − 0)
1
e2πd/L + e−2πd/L tan−1 (x) ≈ x
L 1 + 2πd − 1 − 2πd
L L
Using this approximation,
≈ 3[1 − d/D] − w/2
1 + 2πd + 1 − 2πd
L L φ = tan−1
4πd
D−d
L L
≈ = 2πd 3[1 − d/D] − w/2
2 φ≈
4.9 D−d
f (d) ≈ · 2πd = 9.8d If d = 0, then φ ≈ 3−w/2 . Thus, if w or D
π D
increase, then φ decreases.
8πhcx−5
56. If f (x) = , then using the linear 62. d (θ) = D(w/6 sin θ)
ehc/(kT x) − 1 d(0) = D(1 − w/6) so
approximation we see that
8πhcx−5 d(θ) ≈ d(0) + d (0)(θ − 0)
f (x) ≈ hc
= 8πkT x−4 = D(1 − w/6) + 0(θ) = D(1 − w/6),
(1 + kT x ) − 1
as desired. as desired.
P R2
57. W (x) =
(R + x)2
, x0 = 0 3.2 Indeterminate Forms and
W (x) =
−2P R2 L’Hˆpital’s Rule
o
(R + x)3
L(x) = W (x0 ) + W (x0 )(x − x0 ) x+2
1. lim
x→−2 x2 − 4
P R2 −2P R2 x+2
= + (x − 0) = lim
(R + 0)2 (R + 0)3 x→−2 (x + 2)(x − 2)
2P x 1 1
=P− = lim =−
R x→−2 x − 2 4
10. ˆ
3.2. INDETERMINATE FORMS AND L’HOPITAL’S RULE 159
x2 − 4 sin x − x 0
2. lim 11. lim 3
is type ;
x→2 x2− 3x + 2 x→0 x 0
(x − 2)(x + 2) we apply L’Hˆpital’s Rule thrice to get
o
= lim cos x − 1 − sin x
x→2 (x − 2)(x − 1) = lim = lim
x+2 x→0 3x2 x→0 6x
= lim =4 − cos x 1
x→2 x − 1 = lim =− .
x→0 6 6
3x2 + 2
3. lim tan x − x 0
x→∞ x2 − 4 12. lim is type ;
2
3 + x2
x→0 x3 0
= lim we apply L’Hˆpital’s Rule to get
o
x→∞ 1 − 4 sec2 x − 1
x2
3 lim .
= =3 x→0 3x2
1
Apply L’Hˆpital’s Rule twice more to get
o
x+1 ∞ 2 sec2 x tan x
4. lim is type ; lim
x→−∞ x2 + 4x + 3 ∞ x→0 6x
4 sec2 x tan2 x + 2 sec4 x 1
we apply L’Hˆpital’s Rule to get
o = lim = .
1 x→0 6 3
lim = 0. √ √ √
x→−∞ 2x + 4
t−1 t−1 t+1
13. lim = lim · √
e2t − 1 0 t→1 t − 1 t→1 t − 1 t+1
5. lim is type ;
t→0 t 0 (t − 1)
we apply L’Hˆpital’s Rule to get
o = lim √
t→1 (t − 1) t + 1
d
e2t − 1
lim dt d 1 1
= lim √ =
dt t
t→0
t→1 t+1 2
2e2t 2
lim = =2 ln t 0
t→0 1 1 14. lim is type ;
sin t 0
t→1 t −1 0
6. lim is type ;
t→0 e3t−1 0 we apply L’Hˆpital’s Rule to get
o
we apply L’Hˆpital’s Rule to get
o d 1
dt (ln t)
d
(sin t) cos t 1 lim d = lim t = 1
t→1 1
dt (t − 1)
t→1
lim ddt 3t = lim 3t =
t→0 3e 3
dt (e − 1)
t→0
x3 ∞
tan−1 t 0 15. lim x is type ;
7. lim is type ; x→∞ e ∞
t→0 sin t 0 we apply L’Hˆpital’s Rule thrice to get
o
we apply L’Hˆpital’s Rule to get
o
d
tan−1 t 1/(1 + t2 ) 3x2 6x
lim dt d = lim =1 lim = lim x
t→0
dt (sin t)
t→0 cos t x→∞ ex x→∞ e
6
sin t 0 = lim x = 0.
8. lim is type ; x→∞ e
t→0 sin−1 t 0
ex ∞
we apply L’Hˆpital’s Rule to get
o 16. lim is type ;
d x→∞ x4 ∞
dt (sin t) cos t
lim = lim √ =1 we apply L’Hˆpital’s Rule four times to get
o
t→0 d sin−1 t t→0 1/( 1 − t2 )
dt ex ex
lim 3
= lim
sin 2x 0 x→∞ 4x x→∞ 12x2
9. lim is type ; ex ex
x→π sin x 0 = lim = lim = ∞.
x→∞ 24x x→∞ 24
we apply L’Hˆpital’s Rule to get
o
x cos x − sin x ∞
2 cos 2x 2(1) 17. limx→0 2 is type ;
lim = = −2. x sin x ∞
x→π cos x −1 we apply L’Hˆpital’s Rule twice to get
o
cos x − x sin x − cos x
cos−1 x limx→0
10. lim is undefined (numerator goes to sin2 x + 2x sin x cos x
x→−1 x2 − 1 −x sin x
π, denominator goes to 0). = lim
x→0 sin x (sin x + 2x cos x)
11. 160 CHAPTER 3. APPLICATIONS OF DIFFERENTIATION
−x x − π cos x
= lim = lim 2
= 0
x→0 sin x + 2x cos x
−1
π
x→ 2 cos x − x − π sin x
2
= lim
x→0 cos x + 2 cos x − 2x sin x ln x ∞
1 21. lim 2 is type
=− . x→∞ x ∞
3 we apply L’Hˆpital’s Rule to get
o
1/x 1
18. Rewrite as one fraction, we have lim = lim = 0.
x→∞ 2x x→∞ 2x2
1 x cos x − sin x
lim cot x − = lim ln x ∞
x→0 x x→0 x sin x 22. lim √ is type ;
0 x→∞ x ∞
which is of type we apply L’Hˆpital’s Rule to get
o
0 1
we apply L’Hˆpital’s Rule to get
o 2
cos x − x sin x − cos x lim x = lim √ = 0.
1
= lim
x→∞ √
2 x
x→∞ x
x→0 sin x + x cos x
t ∞
d
(−x sin x) 23. lim t is type
= lim dx t→∞ e ∞
x→0 d
(sin x + x cos x) we apply L’Hˆpital’s Rule to get
o
dx d
(t) 1
− sin x − x cos x lim dt = lim t = 0.
= lim =0 t→∞ d (et ) t→∞ e
x→0 cos x + cos x − x sin x dt
sin 1
t 0
19. Rewrite as one fraction, we have 24. lim 1 is type
t→∞
t
0
x+1 2 we apply L’Hˆpital’s Rule to get
o
lim −
x→0 x sin 2x - 1 cos 1
2 1
(x + 1) sin 2x − 2x 0 = lim t 1 t = lim cos = 1.
= lim is type ; t→∞ − t2 t→∞ t
x→0 x sin 2x 0
we apply L’Hˆpital’s Rule four times to get
o ln (ln t)
d 25. lim
(x + 1) sin 2x − 2x t→1 ln t
lim dx d As t approaches ln from below, ln t is a small
dx (x sin 2x)
x→0
negative number. Hence ln (ln t) is undefined,
sin 2x + 2(x + 1) cos 2x − 2 so the limit is undefined.
= lim
x→0 sin 2x + 2x cos 2x
d
sin (sin t) 0
(sin 2x + 2(x + 1) cos 2x − 2) 26. lim is type
= lim dx d t→0 sin t 0
dx (sin 2x + 2x cos 2x)
x→0 we apply L’Hˆpital’s Rule to get
o
2 cos 2x + 2 cos 2x − 4(x + 1) sin 2x cos (sin t) cos t
= lim lim = 1.
x→0 2 cos 2x + 2 cos 2x − 4x sin 2x t→0 cos t
4
= =1 sin (sinh x) 0
4 27. lim is type
x→0 sinh (sin x) 0
we apply L’Hˆpital’s Rule to get
o
1 cos (sinh x) cosh x
20. lim tan x + lim =1
π
x→ 2 x− π 2
x→0 cosh (sin x) cos x
In this case the limit has the form (∞ - ∞). sin x − sinh x
sin x 28. lim
Rewrite tan x as and then as one frac- x→0 cos x − cosh x
cos x
tion, we get 2 sin x − ex + e−x
= lim
1 x→0 2 cos x − ex − e−x
lim tan x +
x→ 2π
x− π 2
2ex sin x − e2x + 1 0
= lim is type
sin x 1 x→0 2ex cos x − e2x − 1 0
= lim + we apply L’Hˆpital’s Rule twice to get
o
x→ π
2 cos x x − π 2
2ex cos x + 2ex sin x − 2e2x
x − π sin x + cos x
2 0 lim
= lim is type x→0 −2ex sin x + 2ex cos x − 2e2x
x→ π
2 x − π cos x
2
0
cos x + sin x − 1 0
we apply L’Hˆpital’s Rule to get
o = lim is type
x→0 cos x − sin x − 1 0
sin x + x − π cos x − sin x
2 − sin x + cos x
= lim = lim = −1
x→ π
2 cos x − x − π sin x
2 x→0 − sin x − cos x
12. ˆ
3.2. INDETERMINATE FORMS AND L’HOPITAL’S RULE 161
ln x ∞ x+1
29. lim is type
x→0 + cot x ∞ ln x−2
we apply L’Hˆpital’s Rule to get
o = lim
x→∞ √ 1
1/x x2 −4
lim
x→0+ − csc2 x 0
sin x This last limit has indeterminate form , so
= lim+ − sin x · = (0)(1) = 0. 0
x→0 x we can apply L’Hˆpital’s Rule and simplify to
o
find that the above is equal to
√
x −3(x2 − 4)3/2
30. lim+ = 0 (numerator goes to 0 and de- lim and this is equal to 3. So
x→0 ln x x→∞ −x3 + x2 + 2x
nominator goes to −∞). lim ln y = 3.
x→∞
Thus lim y = lim eln y = e3 ≈ 20.086.
x→∞ x→∞
31. lim x2 + 1 − x √
x→∞
√ 1 x
x2 + 1 + x 35. lim+ √ −√
= lim x 2+1−x √ x→0 x√ x+1 √
x→∞ x2 + 1 + x x + 1 − ( x)2
2
x +1−x 2 = lim+ √ √
= lim √ x→0
√ x x+1
x→∞ x2 + 1 + x x+1−x
1 = lim √ √
= lim √ =0 x→0+ x x+1
x→∞ x 2+1+x = ∞.
√
5−x−2 0
ln x
−1 36. lim √ is type
32. lim ln x − x = lim x
= −∞ since the x→1 10 − x − 3 0
1
x→∞ x→∞
x
we apply L’Hˆpital’s Rule to get
o
numerator goes to −1 and the denominator 1 −1/2
2 (5 − x) (−1)
goes to 0+ . (Recall Example 2.8 which shows lim 1
x→1 (10 − x)−1/2 (−1)
2 √
ln x
lim = 0.) 10 − x 3
x→∞ x = lim √ = .
x→1 5−x 2
x x
1 37. Let y = (1/x) . Then ln y = x ln(1/x). Then
33. Let y = 1+
x lim+ ln y = lim+ x ln(1/x) = 0, by Exercise
x→0 x→0
1
⇒ ln y = x ln 1 + . Then 27. Thus lim+ y = lim+ eln y = 1.
x x→0 x→0
1
lim ln y = lim x ln 1 + 38. Let y = lim+ (cos x)1/x . Then
x→∞ x→∞ x x→0
ln 1 + x 1 1
= lim ln y = lim+ ln cos x
x→0 x
x→∞ 1/x
1 1 ln(cos x) 0
1+ x1 − x2 = lim is type
= lim x→0 + x 0
x→∞ −1/x2 so apply L’Hˆpital’s Rule to get
o
1 − tan x
= lim = 1. lim+ = 0.
x→∞ 1 + 1 x→0 1
x
Hence lim y = lim eln y = e. Therefore the limit is y = e0 = 1.
x→∞ x→∞
t t
t−3 (t − 3)
39. lim = lim
34. Notice that the limit in question has the inde- t→∞ t+2 t→∞ (t + 2)t
terminate form 1∞ . Also, note that as x gets 3 t lim 1 − 3
t
x+1 x+1 1− t t→∞ t
large, = . = lim =
t→∞ 2 t 2 t
x−2 x −√2 1+ t lim 1 + t
t→∞
x2 −4 −3 t
x+1 lim 1 +
Define y = . Then t→∞ t e−3
x−2 = = = e−5
2 t e2
√ x+1 lim 1 + t
t→∞
ln y = x2 − 4 ln and
x−2 t t
3
x+1 t−3 1− t
lim ln y = lim x2 − 4 ln 40. lim = lim 1
x→∞ x→∞ x−2 t→∞ 2t + 1 t→∞ 2+ t
13. 162 CHAPTER 3. APPLICATIONS OF DIFFERENTIATION
3 t
1− t e−3 we apply L’Hˆpital’s Rule to get
o
= lim t = lim =0
t→∞ 1/2 t→∞ 2t e1/2
2t 1 + n cos nx n
t lim = .
x→0 m cos mx m
41. L’Hˆpital’s rule does not apply. As x → 0, the
o sin x2 2x cos x2
numerator gets close to 1 and the denominator 50. (a) lim 2
= lim
x→0 x x→0 2x
is small and positive. Hence the limit is ∞.
= lim cos x2 = 1,
ex − 1 0 ex x→0
42. lim is type , but lim is not, so sin x
x→0 x 2 0 x→0 2x which is the same as lim .
L’Hˆpital’s Rule does not apply to this limit.
o x→0 x
1 − cos x2
43. L’Hˆpital’s rule does not apply. As x → 0, the
o (b) lim
x→0 x4
numerator is small and positive while the de- 2x sin x2 sin x2
nominator goes to −∞. Hence the limit is 0. = lim 3
= lim
x→0 4x x→0 2x2
2x 1 sin x 2
1
Also lim , which equals lim x2 , is not of = lim = (by part (a)),
x→0 2/x x→0
2 x→0 x2 2
0
the form so L’Hˆpital’s rule doesn’t apply
o
0 while
here either.
sin x 0 cos x 1 − cos x sin x 1 1
44. lim is type , but lim is not, so lim 2
= lim = (1) =
x→0 x2 0 x→0 2x x→0 x x→0 2x 2 2
L’Hˆpital’s rule does not apply. This limit is
o so both of these limits are the same.
undefined because the numerator goes to 1 and (c) Based on the patterns found in exercise
the denominator goes to 0. 45, we should guess
csc x sin x3 1 − cos x3 1
45. lim+ √ lim = 1 and lim = .
x→0 x x→0 x3 x→0 x6 2
∞
In this case limit has the form , L’Hˆspital’s
o
0 (x + 1)(2 + sin x)
Rule should not be used. 51. (a)
x(2 + cos x)
x−3/2 ∞ x
46. lim+ is type . In this case (b) x
x→0 ln x −∞ e
L’Hˆspital’s Rule should be used.
o 3x + 1
(c)
x2 − 3x + 1 x−7
47. lim = ∞. In this case limit has 3 − 8x
x→∞ tan−1 x (d)
the form ∞. So L’Hˆspital’s Rule should not
o 1 + 2x
be used.
52. (a) lim x − ln x = ∞ (see exercise 32).
ln x2 ∞ x→∞
48. lim is type . So L’Hˆspital’s Rule
o √
x→∞ ex/3 ∞ (b) lim x2 + 1 − x = 0 (see exercise 31).
should be used. x→∞
√
sin 3x (c) lim x2 + 4x − x
49. (a) Starting with lim , we cannot x→∞ √
sin 2x
x→0 = lim ( x2 + 4x − x)
3x x→∞
“cancel sin”to get lim . We can cancel 4x
x→0 2x = lim √
the x’s in the last limit to get the final an- x→∞ x2 + 4x + x
swser of 3/2. The first step is likely to give 1
4x x
a correct answer because the linear ap- = lim √
x→∞ 1
proximation of sin 3x is 3x, and the linear ( x2 + 4x + x)
x
approximation of sin 2x is 2x. The linear 4
= lim = 2,
approximations are better the closer x is x→∞ 4
1+ x +1
to zero, so the limits are likely to be the
where to get from the second to
same.
the third line, we have multiplied by
√
sin nx
(b) lim is type 0 ;
0
( x2 + 4x + x)
x→0 sin mx √ .
( x2 + 4x + x)
14. ˆ
3.2. INDETERMINATE FORMS AND L’HOPITAL’S RULE 163
53. lim ex = lim xn = ∞ In general,when the degree of exponential term
x→∞ x→∞
ex in the numerator and denominator are differ-
lim n = ∞. Since n applications of ln ekx + p(x)
x→∞ x ent, then the lim for polyno-
L’Hˆpital’s rule yields
o x→∞ ln (ecx + q(x))
ex mials p and q and positive numbers. k and c
lim = ∞.
x→∞ n! will be the fraction of degrees that is k .
c
Hence e dominates xn .
x
54. lim ln x = lim xp = ∞. 59. If x → 0, then x2 → 0, so if lim
f (x)
= L,
x→∞ x→∞
ln x ∞ x→0 g(x)
lim is of type f (x2 )
x→∞ xp ∞ then lim = L (but not conversely). If
we use L’Hˆpital’s Rule to get
o x→0 g(x2 )
1
x 1 f (x)
lim p−1
= lim = 0 (since p > 0). a = 0 or 1, then lim involves the be-
x→∞ px x→∞ pxp x→a g(x)
p
Therefore, x dominates ln x. f (x2 )
havior of the quotient near a, while lim
t t
x→a g(x2 )
55. lim e 2 − t3 Since e 2 dominates t3 . So involves the behavior of the quotient near the
t→∞
t different point a2 .
lim e − t3 = ∞
2
t→∞
60. Functions f (x) = |x| and g(x) = x work.
√ f (x)
x − ln x ∞ lim does not exist as it approaches −1
56. lim √ is type . x→0 g(x)
x→∞ x ∞
from the left and it approaches 1 from the
we apply L’Hˆpital’s Rule to get
o
√ − 1
1 √ f (x2 )
2 x x x−2 x right, but lim = 1.
lim = lim x→0 g(x2 )
x→∞ 1
√ x→∞ x
2 x
2 2.5(4ωt − sin 4ωt)
= lim 1 − √ = 1. 61. lim
x→∞ x ω→0 4ω 2
2.5(4t − 4t cos 4ωt)
= lim
ln x3 + 2x + 1
ω→0 8ω
57. lim 2.5(16t2 sin 4ωt)
x→∞ ln (x2 + x + 2) = lim =0
ω→0 8
we apply L’Hˆpital’s Rule
o
d
dx ln x3 + 2x + 1 π
lim 2.5 − 2.5 sin(4ωt + )
d 2 2 is type 0 ;
dx (ln (x + x + 2))
x→∞
62. lim 0
3x2 +2
ω→0 4ω 2
x3 +2x+1 we apply L’Hˆpital’s Rule to get
o
= lim 2x+1 −10t cos(4ωt + π )
x→∞ 2
x2 +x+2 lim
3x + 3x + 8x2 + 2x + 4
4 3
3 ω→0 8ω
= lim = 40t2 sin(4ωt + π )
x→∞ 2x 4 + x3 + 4x2 + 4x + 1 2 = lim 2
= 5t2 .
In general, for numerator and denominator the
ω→0 8
highest degee of polynomials p and q, such that
p(x) > 0 and q(x) > 0 for x > 0, 2
should be the lim ln(p(x)) .
ln(q(x))
x→∞
1.5
3x
ln e + x ∞
58. lim 2x + 4)
is ; 1
x→∞ ln (e ∞
we apply L’Hˆpital’s Rule
o 0.5
d 3x
dx ln e +x
lim d
x→∞
dx (ln (e2x + 4)) 0
0 0.1 0.2 0.3 0.4 0.5 0.6
3e3x +1 t
e3x +x
= lim 2e2x
x→∞
e2x +4
5x
3e + 12e3x + e2x + 4 3 63. The area of triangular region 1 is
= lim = (1/2)(base)(height)
x→∞ 2e5x + 2xe2x 2