The document provides an overview of the structure and content covered on the AP Calculus AB exam, including:
- The exam is 3 hours 15 minutes long and divided into multiple choice and free response sections testing limits, derivatives, integrals, and applications of calculus.
- Content topics covered include limits of functions, continuity, derivatives and their applications (related rates, max/min problems), integrals, and differential equations.
- Formulas and strategies are provided for evaluating limits, finding derivatives using various rules, applying derivatives to sketch curves, solve optimization problems, and solve motion problems using related rates.
This document outlines the 7 steps for sketching the curve of a function: 1) Determine the domain, 2) Find critical points, 3) Determine graph direction and max/min, 4) Use the second derivative to find concavity and points of inflection, 5) Find asymptotes, 6) Find intercepts and important points, 7) Combine evidence to graph the function. Key tests are outlined for max/min, concavity, and points of inflection using the first and second derivatives.
BCA_Semester-II-Discrete Mathematics_unit-iv Graph theoryRai University
This document defines key concepts in graph theory, including:
- A graph is defined as a pair (V,E) where V is the set of vertices and E is the set of edges.
- Examples of graph terminology include vertices, edges, walks, paths, circuits, connectivity, and components.
- Different types of graphs are discussed such as simple graphs, complete graphs, subgraphs, and induced subgraphs.
The document discusses brute force algorithms. It provides examples of problems that can be solved using brute force, including sorting algorithms like selection sort and bubble sort. It then summarizes two geometric problems - the closest pair problem and the convex hull problem - and provides pseudocode for brute force algorithms to solve each problem. The time complexity of these brute force algorithms is O(n^3).
Functions play a crucial role in mathematics by describing how one quantity depends on others. A function assigns exactly one output value to each possible input value. Functions can represent real-world phenomena through mathematical models. There are four common ways to represent functions: verbally through descriptions, numerically in tables, visually with graphs, and algebraically with formulas.
1) The document discusses concepts in differential calculus including the product rule, quotient rule, derivatives of trigonometric functions, higher-order derivatives, and polar curves.
2) Polar curves are represented using polar coordinates (r, θ) where r is the distance from the origin and θ is the angle. Conversions between polar and Cartesian coordinates are covered.
3) Methods for calculating the area inside a polar graph, length of a polar curve, and surface area obtained by rotating a polar curve about the x-axis are presented using integral calculus.
The document discusses key concepts in calculus including continuity, differentiation, integration, and their applications. It defines continuity as being able to draw a function's graph without lifting the pen, and differentiation as computing the rate of change of a dependent variable with respect to changes in the independent variable. The document also covers differentiation rules and techniques for implicit, inverse, exponential, logarithmic, and parametric functions.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
This document outlines the 7 steps for sketching the curve of a function: 1) Determine the domain, 2) Find critical points, 3) Determine graph direction and max/min, 4) Use the second derivative to find concavity and points of inflection, 5) Find asymptotes, 6) Find intercepts and important points, 7) Combine evidence to graph the function. Key tests are outlined for max/min, concavity, and points of inflection using the first and second derivatives.
BCA_Semester-II-Discrete Mathematics_unit-iv Graph theoryRai University
This document defines key concepts in graph theory, including:
- A graph is defined as a pair (V,E) where V is the set of vertices and E is the set of edges.
- Examples of graph terminology include vertices, edges, walks, paths, circuits, connectivity, and components.
- Different types of graphs are discussed such as simple graphs, complete graphs, subgraphs, and induced subgraphs.
The document discusses brute force algorithms. It provides examples of problems that can be solved using brute force, including sorting algorithms like selection sort and bubble sort. It then summarizes two geometric problems - the closest pair problem and the convex hull problem - and provides pseudocode for brute force algorithms to solve each problem. The time complexity of these brute force algorithms is O(n^3).
Functions play a crucial role in mathematics by describing how one quantity depends on others. A function assigns exactly one output value to each possible input value. Functions can represent real-world phenomena through mathematical models. There are four common ways to represent functions: verbally through descriptions, numerically in tables, visually with graphs, and algebraically with formulas.
1) The document discusses concepts in differential calculus including the product rule, quotient rule, derivatives of trigonometric functions, higher-order derivatives, and polar curves.
2) Polar curves are represented using polar coordinates (r, θ) where r is the distance from the origin and θ is the angle. Conversions between polar and Cartesian coordinates are covered.
3) Methods for calculating the area inside a polar graph, length of a polar curve, and surface area obtained by rotating a polar curve about the x-axis are presented using integral calculus.
The document discusses key concepts in calculus including continuity, differentiation, integration, and their applications. It defines continuity as being able to draw a function's graph without lifting the pen, and differentiation as computing the rate of change of a dependent variable with respect to changes in the independent variable. The document also covers differentiation rules and techniques for implicit, inverse, exponential, logarithmic, and parametric functions.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
This document discusses a new basis for the algebra of non-commutative symmetric functions (NSym) called the immaculate basis. The immaculate basis was introduced in 2012 as a non-commutative analogue of the Schur basis for the algebra of symmetric functions (Sym). Like the Schur basis, the immaculate basis can be defined using non-commutative versions of the Jacobi-Trudi formula and the Pieri rule. It satisfies relations that are analogous to the classical properties of the Schur basis but in a non-commutative setting. The document explores how this basis relates to previous work and provides problems about finding natural non-commutative analogues of bases for Sym.
INVERSE TRIGONOMETRIC FUNCTIONS by Sadiq hussainficpsh
This document provides an overview of a lesson on inverse trigonometric functions. The lesson aims to teach students the inverse sine function y=sin-1(x). It begins with reviewing prerequisite concepts like trigonometric functions. Then it introduces the topic and develops the concept that y=sin-1(x) if x=siny. Examples are used to illustrate inverse functions and their graphs. Students complete practice problems finding inverse sines and their ranges. The lesson concludes with a summary of key points and homework assignments.
A complete and comprehensive lesson on concept delivery of Inverse Trigonometric Functions for HSSC level. This lesson is fully helpful for Pakistani and Foreigner.
This document discusses various applications of differentiation including finding extrema, Rolle's theorem, the mean value theorem, and determining whether a function is increasing or decreasing. It provides examples of using the derivative to find relative extrema, applying Rolle's theorem to show a horizontal tangent exists between two roots, using the mean value theorem to find a point where the tangent line is parallel to the secant line, and determining the intervals where a function is increasing or decreasing using the first derivative test.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
This document provides an introduction to calculus and functions. It discusses that calculus originated in the 17th century through the work of Newton and Leibniz. A function is defined as a relationship between variables where each value of one variable corresponds to a unique value of the other. Functions can relate multiple variables as well. Examples of functions include the area of a circle as a function of its radius and the perimeter of a rectangle as a function of its length. The document provides exercises on evaluating functions and relating composite functions.
The document discusses limits of functions and continuity. It defines a limit of a function as approaching a value A as the input x approaches a number a. A function is continuous at a point a if the limit exists and equals the function value at a. A function is discontinuous if it is not continuous at a point. Examples of continuous and discontinuous functions are provided to illustrate the concepts.
This document discusses autoregressive models for financial time series analysis. It introduces autoregressive (AR) and moving average (MA) processes. The autoregressive integrated moving average (ARIMA) model is presented as a way to fit time series data that accounts for correlation between observations. The document outlines the Box-Jenkins methodology for identifying and fitting an ARIMA model to time series data, including checking for stationarity, identifying orders using autocorrelation and partial autocorrelation functions, and selecting the best model. It applies this process to Shanghai Stock Exchange index data, finding that an ARIMA(48,1,0) model provided the best fit.
Here are the key steps to find the instantaneous rate of change using a graphing calculator:
1. Graph the function over the appropriate domain.
2. Use the arrow keys to move the cursor to the point where you want to find the instantaneous rate of change.
3. Press the TRACE button and select the tangent option.
4. The calculator will display the slope of the tangent line, which is the instantaneous rate of change at that point.
5. For example, if finding the IROC at x=1 for the function f(x) = x3, you would:
a) Graph f(x) = x3
b) Use arrows to move cursor
5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fifth part which is discussing singular value decomposition and principal component analysis.
Here are the slides of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Here are the slides of the fourth part which is discussing eigenvalues and eigenvectors.
https://www.slideshare.net/CeniBabaogluPhDinMat/4-linear-algebra-for-machine-learning-eigenvalues-eigenvectors-and-diagonalization
1. Write the function in standard form y=ax^2+bx+c.
2. Find the vertex by using the formula x=-b/2a.
3. Find the axis of symmetry by setting x=-b/2a.
4. Sketch the parabola using the vertex and axis of symmetry as guides.
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
The document discusses equations of lines. It separates lines into two cases - horizontal and vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines which use the point-slope formula y=m(x-x1)+y1 to find the equation given the slope m and a point (x1,y1). Examples are provided to demonstrate finding equations of lines from their descriptions.
General Mathematics - Representation and Types of FunctionsJuan Miguel Palero
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about the representation, definition, and types of functions.
This document summarizes key concepts about transformations and graphs in statistics:
1) Transformations like translations, scale changes, and symmetries can be used to adjust graphs into more useful forms for analysis. Translations move a graph along the x- or y-axis, while scale changes dilate a graph vertically or horizontally.
2) Symmetries include reflections like lines of symmetry and rotational symmetry about the origin. Even functions are symmetrical about the y-axis, while odd functions are symmetrical about the origin.
3) Inverse functions can be found by reflecting a function across the line y=x. The inverse of a function is a function itself if no horizontal line intersects the original function in more than
This document discusses Euler's method for numerically approximating solutions to first-order initial value problems. It begins by introducing Euler's method and its use of tangent lines to approximate the solution curve. Examples are provided to illustrate the application of the method and analyze errors compared to exact solutions. The discussion notes that Euler's method relies on a sequence of tangent lines to different solution curves, so accuracy depends on whether the family of solutions is converging or diverging. It emphasizes the importance of error bounds when exact solutions are unknown.
This document provides an introduction and outline for a discussion of orthonormal bases and eigenvectors. It begins with an overview of orthonormal bases, including definitions of the dot product, norm, orthogonal vectors and subspaces, and orthogonal complements. It also discusses the relationship between the null space and row space of a matrix. The document then provides an introduction to eigenvectors and outlines topics that will be covered, including what eigenvectors are useful for and how to find and use them.
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
This document discusses a new basis for the algebra of non-commutative symmetric functions (NSym) called the immaculate basis. The immaculate basis was introduced in 2012 as a non-commutative analogue of the Schur basis for the algebra of symmetric functions (Sym). Like the Schur basis, the immaculate basis can be defined using non-commutative versions of the Jacobi-Trudi formula and the Pieri rule. It satisfies relations that are analogous to the classical properties of the Schur basis but in a non-commutative setting. The document explores how this basis relates to previous work and provides problems about finding natural non-commutative analogues of bases for Sym.
INVERSE TRIGONOMETRIC FUNCTIONS by Sadiq hussainficpsh
This document provides an overview of a lesson on inverse trigonometric functions. The lesson aims to teach students the inverse sine function y=sin-1(x). It begins with reviewing prerequisite concepts like trigonometric functions. Then it introduces the topic and develops the concept that y=sin-1(x) if x=siny. Examples are used to illustrate inverse functions and their graphs. Students complete practice problems finding inverse sines and their ranges. The lesson concludes with a summary of key points and homework assignments.
A complete and comprehensive lesson on concept delivery of Inverse Trigonometric Functions for HSSC level. This lesson is fully helpful for Pakistani and Foreigner.
This document discusses various applications of differentiation including finding extrema, Rolle's theorem, the mean value theorem, and determining whether a function is increasing or decreasing. It provides examples of using the derivative to find relative extrema, applying Rolle's theorem to show a horizontal tangent exists between two roots, using the mean value theorem to find a point where the tangent line is parallel to the secant line, and determining the intervals where a function is increasing or decreasing using the first derivative test.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
This document provides an introduction to calculus and functions. It discusses that calculus originated in the 17th century through the work of Newton and Leibniz. A function is defined as a relationship between variables where each value of one variable corresponds to a unique value of the other. Functions can relate multiple variables as well. Examples of functions include the area of a circle as a function of its radius and the perimeter of a rectangle as a function of its length. The document provides exercises on evaluating functions and relating composite functions.
The document discusses limits of functions and continuity. It defines a limit of a function as approaching a value A as the input x approaches a number a. A function is continuous at a point a if the limit exists and equals the function value at a. A function is discontinuous if it is not continuous at a point. Examples of continuous and discontinuous functions are provided to illustrate the concepts.
This document discusses autoregressive models for financial time series analysis. It introduces autoregressive (AR) and moving average (MA) processes. The autoregressive integrated moving average (ARIMA) model is presented as a way to fit time series data that accounts for correlation between observations. The document outlines the Box-Jenkins methodology for identifying and fitting an ARIMA model to time series data, including checking for stationarity, identifying orders using autocorrelation and partial autocorrelation functions, and selecting the best model. It applies this process to Shanghai Stock Exchange index data, finding that an ARIMA(48,1,0) model provided the best fit.
Here are the key steps to find the instantaneous rate of change using a graphing calculator:
1. Graph the function over the appropriate domain.
2. Use the arrow keys to move the cursor to the point where you want to find the instantaneous rate of change.
3. Press the TRACE button and select the tangent option.
4. The calculator will display the slope of the tangent line, which is the instantaneous rate of change at that point.
5. For example, if finding the IROC at x=1 for the function f(x) = x3, you would:
a) Graph f(x) = x3
b) Use arrows to move cursor
5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fifth part which is discussing singular value decomposition and principal component analysis.
Here are the slides of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Here are the slides of the fourth part which is discussing eigenvalues and eigenvectors.
https://www.slideshare.net/CeniBabaogluPhDinMat/4-linear-algebra-for-machine-learning-eigenvalues-eigenvectors-and-diagonalization
1. Write the function in standard form y=ax^2+bx+c.
2. Find the vertex by using the formula x=-b/2a.
3. Find the axis of symmetry by setting x=-b/2a.
4. Sketch the parabola using the vertex and axis of symmetry as guides.
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
The document discusses equations of lines. It separates lines into two cases - horizontal and vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines which use the point-slope formula y=m(x-x1)+y1 to find the equation given the slope m and a point (x1,y1). Examples are provided to demonstrate finding equations of lines from their descriptions.
General Mathematics - Representation and Types of FunctionsJuan Miguel Palero
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about the representation, definition, and types of functions.
This document summarizes key concepts about transformations and graphs in statistics:
1) Transformations like translations, scale changes, and symmetries can be used to adjust graphs into more useful forms for analysis. Translations move a graph along the x- or y-axis, while scale changes dilate a graph vertically or horizontally.
2) Symmetries include reflections like lines of symmetry and rotational symmetry about the origin. Even functions are symmetrical about the y-axis, while odd functions are symmetrical about the origin.
3) Inverse functions can be found by reflecting a function across the line y=x. The inverse of a function is a function itself if no horizontal line intersects the original function in more than
This document discusses Euler's method for numerically approximating solutions to first-order initial value problems. It begins by introducing Euler's method and its use of tangent lines to approximate the solution curve. Examples are provided to illustrate the application of the method and analyze errors compared to exact solutions. The discussion notes that Euler's method relies on a sequence of tangent lines to different solution curves, so accuracy depends on whether the family of solutions is converging or diverging. It emphasizes the importance of error bounds when exact solutions are unknown.
This document provides an introduction and outline for a discussion of orthonormal bases and eigenvectors. It begins with an overview of orthonormal bases, including definitions of the dot product, norm, orthogonal vectors and subspaces, and orthogonal complements. It also discusses the relationship between the null space and row space of a matrix. The document then provides an introduction to eigenvectors and outlines topics that will be covered, including what eigenvectors are useful for and how to find and use them.
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
This document provides an introduction to the concept of derivatives. It begins by defining slope for straight lines and curves, and introduces the idea of average slope over small intervals of a curve. It then defines the derivative as the limit of the average slope as the interval approaches 0. This represents the instantaneous slope or slope at a single point. Examples are used to illustrate calculating derivatives using limits. The power rule is introduced, stating that the derivative of x^n is nx^(n-1). Finally, basic differentiation rules are covered and examples of calculating derivatives of simple functions are provided.
- The document discusses revising and sketching parabolic functions of the form y=ax+b, including how the parameters a and b affect the graph shape and position.
- It introduces parabolic equations in standard form y=a(x-p)+q and turning point form, identifying characteristics like the turning point, axes of symmetry, intercepts, and asymptotes.
- Examples are provided to demonstrate how to determine these characteristics, sketch the graph, and state the domain and range for parabolic functions given in equation form.
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Transform as a vector? Tying functional parity with rotation angle of coordin...SayakBhattacharjee4
Oral presentation for Undergraduate Session VII of the American Physical Society (APS) March Meeting 2020: {http://meetings.aps.org/Meeting/MAR20/Session/F12.12}.
This document discusses various topics related to piecewise functions and rational functions:
- It defines piecewise functions and provides examples of evaluating piecewise functions at given values.
- It introduces rational functions as functions of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not equal to zero. It discusses representing rational functions in different forms.
- It explains how to identify restrictions or extraneous roots of rational functions by setting the denominator equal to zero. It also discusses how to determine the domain of a rational function based on its restrictions.
- Finally, it defines vertical and horizontal asymptotes of rational functions. It provides
Grade 10_Math-Lesson 2-3 Graphs of Polynomial Functions .pptxErlenaMirador1
The document discusses how to graph polynomial functions by determining:
1) The end behavior using the leading coefficient test
2) The maximum number of turning points from the degree of the polynomial
3) The x-intercepts by finding the zeros of the polynomial
4) The y-intercept by evaluating the polynomial at x=0
It provides examples of using these steps to graph various polynomial functions of degrees 1-5.
Grade 10_Math-Lesson 2-3 Graphs of Polynomial Functions .pptxErlenaMirador1
The document discusses how to graph polynomial functions by determining:
1) The end behavior using the leading coefficient test
2) The maximum number of turning points from the degree of the polynomial
3) The x-intercepts by finding the zeros of the polynomial
4) The y-intercept by evaluating the polynomial at x=0
It provides examples of using these steps to graph various polynomial functions of degrees 1-5.
Probability theory provides a framework for quantifying and manipulating uncertainty. It allows optimal predictions given incomplete information. The document outlines key probability concepts like sample spaces, events, axioms of probability, joint/conditional probabilities, and Bayes' rule. It also covers important probability distributions like binomial, Gaussian, and multivariate Gaussian. Finally, it discusses optimization concepts for machine learning like functions, derivatives, and using derivatives to find optima like maxima and minima.
The document discusses curve fitting and the principle of least squares. It describes curve fitting as constructing a mathematical function that best fits a series of data points. The principle of least squares states that the curve with the minimum sum of squared residuals from the data points provides the best fit. Specifically, it covers fitting a straight line to data by using the method of least squares to compute the constants a and b in the equation y = a + bx. Normal equations are derived to solve for these constants by minimizing the error between the observed and predicted y-values.
The document discusses curve fitting and the principle of least squares. It describes curve fitting as constructing a mathematical function that best fits a series of data points. The principle of least squares states that the curve with the minimum sum of squared residuals from the data points provides the best fit. Specifically, it covers fitting a straight line to data by using the method of least squares to compute the constants a and b in the equation y = a + bx. Normal equations are derived to solve for these constants by minimizing the error between the observed and predicted y-values.
Optimum Engineering Design - Day 2b. Classical Optimization methodsSantiagoGarridoBulln
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
Taller grupal parcial ii nrc 3246 sebastian fueltala_kevin sánchezkevinct2001
The document is a report in Spanish for a Calculus course discussing applications of derivatives in mechanical engineering. It contains an introduction stating that calculus was developed in the 17th century to solve geometry and physics problems. It then discusses how derivatives are used in mechanical engineering, specifically for analyzing signals with amplitude and frequency using sine and cosine functions. The report has objectives of developing skills for manipulating algebraic functions and their relationship to mechanical engineering problems. It provides theoretical foundations for the definition and calculation of derivatives.
This document provides a summary of topics related to algebra, functions, and calculus including: linear and quadratic expressions, simultaneous equations, completing the square, trigonometric ratios, differentiation, tangents, normals, and finding stationary points through higher derivatives. It outlines key steps and methods for solving various types of problems within these topics.
A polynomial function is a function where all exponents are non-negative integers. It is of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a non-negative integer and the coefficients ai are real numbers. The degree of a polynomial is the highest exponent present, and the leading coefficient is the coefficient of the term with the highest degree. The end behavior of a polynomial function can be determined by looking at the sign of the leading coefficient and whether the highest degree is even or odd. A polynomial function has at most n real zeros, where n is its degree, and at most n-1 turning points.
This document discusses functions and their graphs. It defines what a function is and introduces function notation. It discusses the domain and range of a function and how to graph functions on a coordinate plane. It also covers even and odd functions, operations on functions like addition and composition, and finding the inverse of a function. Finally, it discusses elementary functions like linear, quadratic, cubic, exponential and logarithmic functions and how their graphs can be transformed through horizontal and vertical shifts, stretches, flips, and scaling.
This document discusses the normal distribution and its key properties. It also discusses sampling distributions and the central limit theorem. Some key points:
- The normal distribution is bell-shaped and symmetric. It is defined by its mean and standard deviation. Approximately 68% of values fall within 1 standard deviation of the mean.
- Sample statistics like the sample mean follow sampling distributions. When samples are large and random, the sampling distributions are often normally distributed according to the central limit theorem.
- Correlation and regression analyze the relationship between two variables. Correlation measures the strength and direction of association, while regression finds the best-fitting linear relationship to predict one variable from the other.
This document discusses the behavior and key characteristics of quadratic functions and their graphs (parabolas). It begins by defining quadratic functions, expressions, and equations. It then explains that a quadratic function has two variables, x and y, where x is the independent variable and y is the dependent variable.
The document goes on to discuss several important aspects of parabolas including the vertex (point of minimum or maximum), axis of symmetry, zeros (where the graph crosses the x-axis), and point of origin. It then analyzes the graphs of various quadratic functions of the form y=ax^2, observing that when a is positive the graph opens upward and the opening gets narrower as a increases, and when a
The document outlines the structure and content of a 5-week SAT workshop. Week 1 covers an introduction, familiarizing students with the test format and structure, and assigning practice problems. Future weeks will focus on specific strategies for the Reading, Writing and Language, and Math sections, including reviewing practice tests and assigning homework. The goal is for students to improve their scores by 50-150 points by establishing a target score and focusing study on their weaker sections.
The document provides information about an upcoming AP Biology exam, including:
- The exam will take place on May 8 from 8:00-11:00 AM. It will consist of multiple choice and free response questions.
- The exam will cover 4 big ideas: evolution, biological systems utilize energy/matter, living systems store/transmit information, and complex interactions in biological systems.
- The document then provides a detailed review of content including chemistry of life, cellular structure/function, cellular energetics, and molecular genetics.
The document provides guidance on obtaining letters of recommendation for medical school applications, noting that most schools require at least 3 letters - 2 from science professors and 1 from a non-science professor. It outlines the process for choosing letter writers, providing them with necessary forms and background information, maintaining communication, and ensuring the letters are submitted through the appropriate channels by the applicable deadlines. Obtaining strong letters of recommendation that are submitted on time can substantially support an applicant's medical school application.
Class Scheduling & Academic Planning Workshop
Presented at UC Berkeley for Sigma Mu Delta's "PreMed Survival Guide" 4/15/17
Originally created for Sigma Mu Delta's "How To Get Into Medical School" Symposium 3/16
The document provides information about an upcoming AP Biology exam, including exam structure, content topics, and a review of key biology concepts. The exam will be on May 8 from 8:00-11:00 AM and consist of multiple choice and free response questions covering four big ideas: evolution, biological systems utilize energy/matter, living systems store/transmit information, and complex interactions in biological systems. The review covers chemistry of life, cells, cellular energetics, photosynthesis, molecular genetics, genetics/heredity.
The document outlines the structure and content of a 5-week SAT preparation workshop. Week 1 covers an introduction, test structure, target score planning, and homework assignment. Students are instructed to take a practice test, establish study goals, and join a group messaging platform. Future weeks will focus on specific strategies for the Evidence-Based Reading, Evidence-Based Writing, Math, and Essay sections through practice questions and homework.
The document outlines the structure and content of a 5-week SAT preparation workshop. Week 1 covers an introduction, test structure, target score planning, and homework assignment. Future weeks will focus on specific strategies for the Reading, Writing, and Math sections through practice tests, review of techniques, and homework. The goal is for students to improve their scores by 50-150 points by the end of the workshop through targeted preparation and practice.
This document provides information and recommendations for navigating pre-med requirements at UC Berkeley. It outlines the typical course requirements, including one year each of calculus, chemistry, organic chemistry, physics, biology, and English. It recommends strategies for scheduling courses, choosing a major, and blocking out time. The document also shares the author's own academic history and provides additional tips and resources for students.
This document provides information and recommendations for pre-med students at UC Berkeley regarding course requirements, recommended course combinations, major selection, scheduling blocks of classes, and resources. It outlines the typical pre-med requirements of 1 year each of calculus, chemistry, organic chemistry, physics, biology, English, and a social science. It recommends against taking more than 2 technical courses in a semester and provides examples of course schedules. It also stresses exploring interests through major selection and engaging in extracurricular activities to stand out in medical school applications.
The document is a study guide for a chemistry exam covering various organic chemistry topics including allylic and conjugated systems, aromaticity, electrophilic aromatic substitution, carbonyl chemistry, amino acids, and peptide sequencing. It provides definitions, reaction mechanisms, and practice problems for key concepts that will be tested. The study guide emphasizes memorizing fundamental steps and rules for different reaction types as well as clearly indicating hybridizations and understanding how underlying concepts link various topics together. It concludes by recommending getting sufficient rest before the exam and trusting one's conceptual understanding of material to answer problems, even those involving unfamiliar reactions.
The document is a study guide for a chemistry exam covering various organic chemistry topics including allylic and conjugated systems, aromaticity, electrophilic aromatic substitution, carbonyl chemistry, amino acids, and peptide sequencing. It provides definitions, reaction mechanisms, and practice problems for students to review key concepts that will be tested like identifying hybridizations and drawing Frost diagrams for aromatic compounds, outlining the steps of electrophilic aromatic substitution and Friedel-Crafts reactions, interconverting functional groups like carbonyls, hemiacetals, and acetals, and sequencing peptides after cleavage by specific proteases. The study guide also offers general exam preparation advice and reminds students to trust their conceptual understanding of material to answer problems.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
AP Advantage: AP Calculus
1. AP Advantage: AP Calculus AB
Classroom Matters
Instructor: Shashank Patil
May 6-7, 2017
Exam Day:
Tuesday, May 9 @ 8:00AM
Exam Structure:
Time: 3 hours 15 minutes
Section I: Multiple Choice | 45 Questions | 1 hour and 45 minutes | 50% of Final Exam Score
Part A — 30 questions | 60 minutes (calculator not permitted)
Part B — 15 questions | 45 minutes (graphing calculator required)
Section II: Free-Response | 6 Questions | 1 hour and 30 minutes | 50% of Final Exam Score
Part A — 2 problems | 30 minutes (graphing calculator required)
Part B — 4 problems | 60 minutes (calculator not permitted)
Note: You may not take both the Calculus AB and Calculus BC exams within the same year.
Content Review
Limits
• Properties of Limits:
o lim
𝑥→𝑐
(𝑓(𝑥) + 𝑔(𝑥)) = lim
𝑥→𝑎
𝑓(𝑥) + lim
𝑥→𝑎
𝑔(𝑥)
o lim
𝑥→𝑐
(𝑓(𝑥) − 𝑔(𝑥)) = lim
𝑥→𝑎
𝑓(𝑥) − lim
𝑥→𝑎
𝑔(𝑥)
o lim
𝑥→𝑐
(𝑓(𝑥)𝑔(𝑥)) = lim
𝑥→𝑎
𝑓(𝑥) × lim
𝑥→𝑎
𝑔(𝑥)
o lim
𝑥→𝑎
[𝑐𝑓(𝑥)] = c lim
𝑥→𝑎
𝑓(𝑥)
o lim
𝑥→𝑎
𝑓(𝑥)
𝑔(𝑥)
=
lim
𝑥→𝑎
𝑓(𝑥)
lim
𝑥→𝑎
𝑔(𝑥)
Finding Limits of Equations
- General: If lim
𝑥→𝑎+
𝑓(𝑥) = L and lim
𝑥→𝑎−
𝑓(𝑥) = L, then the limit lim
𝑥→𝑎
𝑓(𝑥) exists.
o If not, the limit does not exist.
- Limits by direct substitution
o F is continuous at x= a if lim
𝑥→𝑎
𝑓(𝑥) = f(a)
- Finding limits by factoring
o Find limit as x approaches 2 of f(x) = (x2
+ x – 6)/(x-2)
▪ Factor numerator and simplify to f(x) = x + 3
▪ Solution: 5
- If k and n are constants, |x| > 1, and n >0, then lim
𝑥→
𝑘
𝑥 𝑛
= 0, and lim
𝑥→ −
𝑘
𝑥 𝑛
= 0
2. - Rational function points of discontinuity
o Strategy
▪ Functions are undefined for inputs that make the denominator equal to
zero
▪ Inputs where the function is defined and the numerator is equal to zero are
zeros of the function. These are simply the x-intercepts of the graph of
the function.
▪ Simplify the function expression by canceling out common factors. Any
undefined input that no longer makes the denominator equal to zero is a
removable discontinuity. The remaining undefined inputs are vertical
asymptotes.
o Rationalizing functions with square roots
▪ Multiply by conjugate if you get the indeterminate form 0/0 when you
plug in the value x approaches into the function
• Ex. Conjugate of √4𝑥 + 4 + 4 is √4𝑥 + 4 − 4
o Polynomial Limits
▪ If the highest power of x in a rational expression is in the numerator, then
the limit as x approaches infinity is infinity.
▪ If the highest power of x in a rational expression is in the denominator,
then the limit as x approaches infinity is zero.
o Limits of Trigonometric Functions
▪ There are four standard limits you can memorize – with these, you can
evaluate all of the trigonometric limits that appear on the test:
1. lim
𝑥→0
sin(𝑥)
𝑥
= lim
𝑥→0
𝑥
sin(𝑥)
= 1 (where x is in radians)
a. Remember that both sin(x) and x have approximately the same
slope near the origin as x gets closer to zero
2. lim
𝑥→0
cos(𝑥)−1
𝑥
= 0
3. lim
𝑥→0
sin(𝑎𝑥)
𝑥
= 𝑎
4. lim
𝑥→0
sin(𝑎𝑥)
sin(𝑏𝑥)
=
𝑎
𝑏
Continuity
A function f is continuous at x = a if and only if:
1. f(a) exists
2. lim
𝑥→𝑎
𝑓(𝑥) exists
3. lim
𝑥→𝑎
𝑓(𝑥) = 𝑓(𝑎)
Types of Discontinuity (Draw an example of each of these!)
• Jump Discontinuity
o Occurs when the curve “breaks” at a particular place and starts somewhere else.
In other words, lim
𝑥→𝑎+
𝑓(𝑥) ≠ lim
𝑥→𝑎−
𝑓(𝑥).
3. • Point Discontinuity
o Occurs when the curve has a “hole” in it from a missing point because the
function has a value at that point that is “off the curve.” In other words,
lim
𝑥→𝑎
𝑓(𝑥) ≠ 𝑓(𝑎) .
• Removable Discontinuity
o Occurs when you have a rational expression with common factors in the
numerator and denominator.
▪ Since the factors can be canceled, the discontinuity is “removable.”
• Essential Discontinuity (Vertical Asymptotes)
o Occurs when the curve has a vertical asymptote.
Derivatives: Definition & Formulas
• Derivatives = slope of the tangent line to the graph of f(x) at some point
o Tangent line: straight line that touches a function at only one point = represents
instantaneous rate of change of a function at one point
o Secant line: straight line joining two points on a function, which gives us the
average rate of change (= slope between the two points)
o Limit definition of a derivative
▪ “Use slope of secant line between two nearby points to approximate slope
of tangent line at a certain point”
▪ f’(x) = lim
ℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
• Power Rule
o If y = 𝑦 = 𝑥 𝑛
, 𝑡ℎ𝑒𝑛
𝑑𝑦
𝑑𝑥
= 𝑛𝑥 𝑛−1
a. If y = x, then
𝑑𝑦
𝑑𝑥
= 1
b. If y = kx, then
𝑑𝑦
𝑑𝑥
= 𝑘 (where k is a constant)
c. If y = k, then
𝑑𝑦
𝑑𝑥
= 0 (where k is a constant)
• Addition Rule
o If 𝑦 = 𝑎𝑥 𝑛
+ 𝑏𝑥 𝑛
, where a and b are constants, then
𝑑𝑦
𝑑𝑥
= 𝑎(𝑛𝑥 𝑛−1) +
𝑏(𝑛𝑥 𝑛−1
).
• Product Rule
o If f(x) = uv, then 𝑢𝑣′ + 𝑣𝑢′
• Quotient Rule
o If f(x) = u/v, then 𝑓′(𝑥) =
𝑣𝑢′− 𝑢𝑣′
𝑣2
• How to Remember: “Low De-High minus High De-Low, Cross the line
and Square the Below”
• Chain Rule
o If y = f(g(x)), then y’ = g’(x) • f’(g(x))
• How to Remember: “Derivative of the inside times the derivative of the
outside of the inside”
• Trig Functions:
o
𝑑
𝑑𝑥
sin( 𝑥) = cos( 𝑥)
o
𝑑
𝑑𝑥
cos(𝑥) = − sin(𝑥)
4. o
𝑑
𝑑𝑥
tan(𝑥) = sec2
(𝑥)
o
𝑑
𝑑𝑥
sec(𝑥) = sec(x)tan(x)
o
𝑑
𝑑𝑥
csc(𝑥) = -csc(x)cot(x)
o
𝑑
𝑑𝑥
cot(𝑥) = -csc2
(x)
Exponential and Logarithmic Derivatives
o If y = ln(x), then
𝑑𝑦
𝑑𝑥
=
1
𝑥
• Similarly, if y = ln(u), then
𝑑𝑦
𝑑𝑥
=
1
𝑢
(
𝑑𝑢
𝑑𝑥
)
o If y = ex
, then
𝑑𝑦
𝑑𝑥
= 𝑒 𝑥
• Similarly, if y = eu
, then
𝑑𝑦
𝑑𝑥
= 𝑒 𝑢
(
𝑑𝑢
𝑑𝑥
)
o If y= log 𝑎 𝑥, then
𝑑𝑦
𝑑𝑥
=
1
𝑥𝑙𝑛(𝑎)
• Similarly, if y = log 𝑎 𝑢, then
𝑑𝑦
𝑑𝑥
=
1
𝑢𝑙𝑛(𝑎)
(
𝑑𝑢
𝑑𝑥
)
o If y = ax
, then
𝑑𝑦
𝑑𝑥
= e(xlna)
(ln a) = (ax
)(ln a)
• Similarly, if y = au
, then
𝑑𝑦
𝑑𝑥
= au
(ln a) (
𝑑𝑢
𝑑𝑥
)
Implicit Differentiation
o Use when you cannot isolate y in terms of x, so you derive in terms of y AND x
o Ex. Find
𝑑𝑦
𝑑𝑥
if y3
– 4y2
= x5
+ 3x4
• 3𝑦2
(
𝑑𝑦
𝑑𝑥
) − 8𝑦 (
𝑑𝑦
𝑑𝑥
) = 5𝑥4
(
𝑑𝑥
𝑑𝑥
) + 12𝑥3
(
𝑑𝑥
𝑑𝑥
)
•
𝑑𝑥
𝑑𝑥
= 1, 𝑠𝑜
𝑑𝑦
𝑑𝑥
(3𝑦2
− 8𝑦) = 5𝑥4
+ 12𝑥3
• Thus:
𝑑𝑦
𝑑𝑥
=
5𝑥4+12𝑥3
3𝑦2−8𝑦
Derivative of an Inverse Function
o Strategy
• 1. Solve the equation with respect to x (e.g. x = something)
• 2. Switch the x and y. You now have the inverse function
• 3. Take derivative of the function.
• 4. Plug in point value into the equation and solve.
• Remember: You will be plugging in the given y value for x
(Because this is the inverse function!)
Derivative Applications
o Finding the equations for the tangent and normal line to the curve f(x) = y at a
point (x1, y1) as
• Finding the slope: Take the derivative of the function and then plug in the
value of x into the derivative function. The resulting value is the slope
(m). To find slope of the normal (perpendicular) line, take the negative
reciprocal of m.
• Finally, plug in values of m, x1, and y1 into the point-slope equation:
• y- y1 = m(x-x1)
5. o Mean Value Theorem for Derivatives
• If y = f(x) is continuous on the interval [a,b] and is differentiable
everywhere on the interval (a,b), then there is at least one number c
between a and b such that,
• 𝑓′(𝑐) =
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
• “There’s some point in the interval where the slope of the tangent
line equals the slope of the secant line that connects the endpoints
of the interval”
• Rolle’s Theorem (Special Case of MVT)
• Same as Mean Value Theorem but in this case f(a) = f(b) = 0, so
the formula simplifies to finding the value of c where f’(c) = 0
o Maxima and Minima
• A maximum or minimum of a function occurs at a point where the
derivative of the function is zero, or where the derivative fails to exist.
• Relative (local) max/min: means that the curve has a horizontal
line at that point, but it is not the highest or lowest value that the
function attains.
• Absolute (global) max/min: occurs at an end point or an x-value
where there is a vertical asymptote
• Second Derivative Test: If a function has a critical value at x = c, then that
value is relative maximum if f’’(x) < 0 and it is a relative minimum if
f’’(x) > 0
• Intuition: Let’s look at the slope of the graph of f’(x)
• Application: Knowing the maxima or minima will help us to optimize
functions
o Curve Sketching
• Find Intercepts
• Set f(x) = 0, then solve for x. This tells you the function’s x-
intercepts (or roots)
• Set x= 0 to find y-intercept(s)
• Find Asymptotes
• Horizontal Asymptotes: Find limits of f(x) as x approaches + and
-
o If they give you an interval, evaluate f(x) at the endpoints
• Vertical Asymptotes: Find values of x that make f(x) undefined
• Test the First Derivative
• Find where f’(x) = 0. This tells you the critical points
(maxima/minima).
• When f’(x) >0, the curve is rising; when f’(x) < 0, the curve is
falling.
• Test the Second Derivative
• Find where f’’(x) = 0. This tells you the inflection points.
• When f’’(x) > 0, the curve is concave up; when f’’(x) < 0, the
curve is concave down
6. o Motion
• Related rates
• You’ll be given an equation relating two or more variables. These
variables will change with respect to time, and you’ll use
derivatives to determine how the rates of change are related.
• Position, Velocity, & Acceleration (Rectilinear Motion)
• Position Function x(t)
• Velocity Function v(t) = x’(t)
• Acceleration Function a(t) = v’(t) = x’’(t)
• When the velocity is negative, the particle is moving to the left.
o When the velocity is positive, the particle is moving to the
right.
• When the velocity and acceleration of the particle have the same
signs, the particle’s speed is increasing.
o When the velocity and acceleration of a particle have
opposite signs, the particle’s speed is decreasing (or
slowing down).
o When the velocity is zero and the acceleration is not zero,
the particle is momentarily stopped and changing direction.
o L’Hopital’s Rule
• Used to find the limit of an expression if it results in an indeterminate
form (
0
0
𝑜𝑟
∞
∞
)
• If f(c) = g(c)= 0, and if f’(c) and g’(c) exist, and if g’(c) ≠ 0, then
• lim
𝑥→𝑐
𝑓(𝑥)
𝑔(𝑥)
=
𝑓′(𝑥)
𝑔′(𝑥)
• Similarly, if f(c) = g(c)= ∞, and if f’(c) and g’(c) exist, and if g’(c) ≠ 0,
then:
• lim
𝑥→𝑐
𝑓(𝑥)
𝑔(𝑥)
=
𝑓′(𝑥)
𝑔′(𝑥)
• Translation: If the limit of an expression is undefined, take the derivative
of the top and bottom until you get a determinate expression. Then, take
the limit.
o Differentials/Linearization
o Differential = very small quantity that represents a change in a number (∆x)
o Used to approximate the value of a function
o Take the definition of a derivative, replace h with ∆x, and get rid of the
limit:
▪ 𝑓′(𝑥) ≈
𝑓(𝑥+∆𝑥) – 𝑓(𝑥)
∆𝑥
▪ Rearrange to get:
• 𝒇(𝒙 + ∆𝒙) ≈ 𝒇(𝒙) + 𝒇′(𝒙)∆𝒙
• This equation only works with radians (No Degrees!)
o A similar function used to estimate the error or change in a
measurement:
▪ dy = f’(x)dx
7. ______________________________________________________________________________
Integrals (Antiderivative)
• Used to help find area under curve or evaluate volume enclosed by a function
• Power Rule: If f(x) = xn
, then ∫ 𝑓(𝑥)𝑑𝑥 =
𝑥 𝑛+1
𝑛+1
+ 𝐶 (except when n = -1)
o It follows that:
▪ ∫ 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥
▪ ∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
▪ ∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶
• Integrals of Trig Functions
o ∫ sin(ax)𝑑𝑥 = −
cos(ax)
a
+ C
o ∫ cos(ax)𝑑𝑥 =
sin(ax)
a
+ C
o ∫ sec(ax) tan(𝑎𝑥) 𝑑𝑥 =
sec(ax)
a
+ C
o ∫ sec2
𝑎𝑥 𝑑𝑥 =
tan(ax)
a
+ C
o ∫ csc(ax) cot(𝑎𝑥) 𝑑𝑥 = −
csc(ax)
a
+ C
o ∫ csc2
𝑎𝑥 𝑑𝑥 = −
cot(ax)
a
+ C
o ∫
du
u
= ln |u| + C
o ∫ tan(x)𝑑𝑥 = −ln|cos 𝑥| + C
o ∫ cot(x)𝑑𝑥 = ln|sin 𝑥| + C
o ∫ sec(𝑥) = ln|sec(x) + tan(𝑥)| + C
o ∫ csc(x)𝑑𝑥 = −ln|csc(x) + cot(𝑥)| + C
o ∫ ex
dx =
1
k
ekx
+ C
• Integration technique: U-substitution
o ∫ un
du =
un+1
n+1
+ C
o Pick one expression to equal u, derive u, and then plug in values of u and du into
original equation.
Evaluating Integrals using Geometry
o Using Riemann Sums to estimate the integral of a function
▪ Translation: Finding area under a curve by adding up areas of rectangles
▪ Formulas (Don’t memorize these!):
• Left-Handed Sum
o
𝑏−𝑎
𝑛
(y0+ y1 + y2 + y3 + y4 + …+yn-1)
• Right-Handed Sum
o
𝑏−𝑎
𝑛
(y1 + y2 + y3 + y4 +y5 …+yn)
• Midpoint Sums
o
𝑏−𝑎
𝑛
(y1/2 + y3/2 + y5/2 + y7/2 +y9/2 …+y[2n-1]/2)
o Note: Fractional subscript means to evaluate the function at
the number half-way between each integral pair of n-values
8. • Where a = left endpoint of the interval, b= right endpoint of the
interval, and n= number of triangles we’re using
o Trapezoid Rule = Better at estimating area than Riemann Sum method
▪ Finding area under a curve by adding up areas of trapezoids
▪ Formula
• Using formula for area of a trapezoid A= ½ (b1+ b2)(h) you can
derive the formula:
• (
1
2
) (
𝑏−𝑎
𝑛
)(y0+2y1+2y2+2y3+…+2yn-2+2yn-1+yn)
The First Fundamental Theorem of Calculus
o ∫ f(x)dx = F(b) − F(a); where F(x)is the antiderivative of f(x)
b
a
The Second Fundamental Theorem of Calculus
o If f(x) is continuous on [a,b], then the derivative of the function F(x) = ∫ 𝑓(𝑡)𝑑𝑡
𝑥
𝑎
is:
▪
dF
dx
=
𝑑
𝑑𝑥
∫ 𝑓(𝑡)𝑑𝑡 = 𝑓(𝑥)
𝑥
𝑎
• All we are concerned with is swapping the upper variable term (in
this case x) with the variable in the function (in this case t)
o The bottom term could be any constant value
• If the upper variable term is a function of x (e.g. x2
), we multiply
the answer by the derivative of that term (e.g. 2x)
Mean Value Theorem for Integrals
o Enables you to find the average value of a function
o If f(x) is continuous on a closed interval [a,b], then at some point c in the interval
[a,b] the following is true:
▪ ∫ 𝑓(𝑥)𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎)
𝑏
𝑎
• Translation: The area under the curve of f(x) on the interval [a,b] is
equal to the value of the function at some value c (between a and
b) times the length of the interval.
o This equation can be rearranged to find the average value of a function:
▪ 𝑓(𝑐) =
1
𝑏−𝑎
∫ 𝑓(𝑥)𝑑𝑥
𝑏
𝑎
Area Between Two Curves
o Vertical Slices
o If a region is bounded by f(x) above and g(x) below at all points of the interval
[a,b], then the area of the region is given by:
▪ ∫ [ 𝑓( 𝑥) − 𝑔( 𝑥)] 𝑑𝑥
𝑏
𝑎
o Horizontal Slices
o If a region is bounded by f(y) on the right and g(y) on the left at all points of the
interval [c,d], then the area of the region is given by
▪ ∫ [𝑓(𝑦) − 𝑔(𝑦)]𝑑𝑦
𝑑
𝑐
9. Volume of a Solid of Revolution
o Washers (Disk) Method: Finding the volume of a complex shape by adding up volumes
of many thin discs
o Think: “You’re adding up the volume of each thin pancake in a stack!”
o Disk: If you have a region whose area is bounded by the curve y = f(x) and the x-
axis on the interval [a,b], each disk has a radius of f(x), and the area of each disk
will be π[f(x)]2
(Just like the area of a circle of a circle is A = πr2
!)
▪ To find the volume, evaluate the integral:
• π ∫ [𝑓(𝑥)]2
𝑑𝑥
𝑏
𝑎
(where dx is some small depth along the x-axis)
o Washer: If you have a region whose area is bounded above the curve y = f(x) and
below by the curve y = g(x), on the interval [a,b], then each washer will have an
area of π[f(x)2
– g(x)2
] (We’re subtracting the area of one circle from another!)
▪ To find the volume, evaluate the integral: (rotated about the x-axis)
• π ∫ [𝑓( 𝑥)2
− 𝑔( 𝑥)2
]
𝑏
𝑎
dx
▪ Finding volume for washers when the region is rotated about y –axis:
• π ∫ [𝑓(𝑦)2
− 𝑔(𝑦)2
]
𝑑
𝑐
dy
o Remember: dy and dx just stand for really (infinitesimally) small depths along the
y or x axis respectively!
o Cylindrical Shells Method
o Think: “Finding the volume of each layer of an onion” (Shrek!)
o If you have a region whose area is bounded above by the curve y= f(x) and below
by the curve y = g(x), on the interval [a,b], then each cylinder will have a height
f(x) – g(x), a radius of x, and an area of 2 πx[f(x)-g(x)].
▪ Volume when the region is rotated around the y-axis:
• 2π∫ 𝑥[𝑓(𝑥) − 𝑔(𝑥)]𝑑𝑥
𝑏
𝑎
o If you have a region whose area is bounded on the right by the curve x =f(y) and
on the left by the curve x =g(y), on the interval [c,d], then each cylinder will have
a height of f(y) – g(y), a radius of y, and an area of 2 πy[f(y)-g(y)].
▪ Volume when the region is rotated around the x-axis:
• 2𝜋 ∫ 𝑦[𝑓(𝑦) − 𝑔(𝑦)]
𝑑
𝑐
dy
o Volumes of Solids with Known Cross-Sections
o If you’re given an object where you know 1) the shape of the base and 2) that the
perpendicular cross-sections are all the same regular, planar geometric shape
▪ You can integrate using the area of that shape!
▪ Strategy: Find the area of the regular (=all sides are equal) shape, multiply
by it some small depth dy or dx (depending on how the object is oriented),
and then integrate from the endpoints of an interval
Differential Equations
o Equations that relate a function with one or more of its derivatives
o How to solve differential problems:
o Separate the variables
o Integrate both sides
o Solve for the constant
10. o Applications:
o Position, Velocity, and Acceleration functions
▪ Derivative of Position Function Velocity Function
▪ Derivative of Velocity Function Acceleration Function
▪ We can go backwards as well:
• Integral of Acceleration function Velocity function
• Integral of Velocity function Position function
o Exponential Growths and Decay
Slope Fields (Direction Fields)
o Making a graphical representation of the slope of a function at various points in the plane.
o You will be given
𝑑𝑦
𝑑𝑥
. Plug-in different values for x and y and graph the slopes at the
respective coordinates.
Test-Taking Strategies
- Show all of your work (Free Response)
o Remember that the grader is more interested in seeing if you know HOW to solve
a problem rather than just checking to see if your answer is correct/incorrect
- Do not round partial answers
o Store them in your calculator so that you can use them unrounded in further
calculations
▪ Be sure you’re in the correct mode Most always radian mode
o Unless otherwise specified, your final answers should be accurate to three places
after the decimal point
- Use Process of Elimination and Bubble in for all problems
o There is No Guessing Penalty
- Pro-Tip: During the second timed portion of the free-response section (Part B), you are
permitted to continue work on problems in Part A, but you are not permitted to use a
calculator during this time.
- Breathe. You’ll do great! ☺