This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand the derivative of constants, the constant multiple rule, the sum rule, the difference rule, and derivatives of sine and cosine. Examples are provided to find the derivatives of squaring, cubing, square root, and cube root functions using the definition of the derivative. Graphs and properties of these functions and their derivatives are also discussed.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
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.
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.
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
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.
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 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.
The document defines the derivative and discusses rules for computing derivatives. It introduces the derivative as describing the slope of a curve at a point. It then outlines several basic rules for determining derivatives, such as the power rule, sum rule, and rules for constants and combinations of functions. The document also discusses the product rule, chain rule, and applications of derivatives to motion and rates of change problems.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
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.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
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.
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
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 an introduction to integration, which is the inverse process of differentiation. It discusses the basic notation and concepts of integration, such as taking the integral of a function to find its anti-derivative and adding a constant C. Examples are provided of integrating common functions and determining the constant value based on initial conditions. Questions and answers about integration techniques are also presented.
Lesson 11 derivative of trigonometric functionsRnold Wilson
The document provides differentiation formulas for trigonometric functions. It defines trigonometric and inverse trigonometric functions as transcendental functions. It then presents formulas for deriving the derivatives of sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u). Examples are given of applying the formulas to find derivatives of trigonometric functions and expressions involving trigonometric functions. Practice problems are provided at the end to differentiate various trigonometric expressions.
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.
The document provides explanations and examples for concepts related to differentiation and integration including:
- The differentiation of various functions like 3x^2, e^x, x^5, etc. and explanations of the differentiation process
- The integration of functions like 3x^3 and what integration undoes from differentiation
- Examples of calculating permutations, combinations, and arrangements of different objects
The document discusses differentiation and its history. It was independently developed in the 17th century by Isaac Newton and Gottfried Leibniz. Differentiation allows the calculation of instantaneous rates of change and is used in many areas including mathematics, physics, engineering and more. Key concepts covered include calculating speed, estimating instantaneous rates of change, the rules for differentiation, and differentiation of expressions with multiple terms.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
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.
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.
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
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.
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 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.
The document defines the derivative and discusses rules for computing derivatives. It introduces the derivative as describing the slope of a curve at a point. It then outlines several basic rules for determining derivatives, such as the power rule, sum rule, and rules for constants and combinations of functions. The document also discusses the product rule, chain rule, and applications of derivatives to motion and rates of change problems.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
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.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
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.
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
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 an introduction to integration, which is the inverse process of differentiation. It discusses the basic notation and concepts of integration, such as taking the integral of a function to find its anti-derivative and adding a constant C. Examples are provided of integrating common functions and determining the constant value based on initial conditions. Questions and answers about integration techniques are also presented.
Lesson 11 derivative of trigonometric functionsRnold Wilson
The document provides differentiation formulas for trigonometric functions. It defines trigonometric and inverse trigonometric functions as transcendental functions. It then presents formulas for deriving the derivatives of sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u). Examples are given of applying the formulas to find derivatives of trigonometric functions and expressions involving trigonometric functions. Practice problems are provided at the end to differentiate various trigonometric expressions.
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.
The document provides explanations and examples for concepts related to differentiation and integration including:
- The differentiation of various functions like 3x^2, e^x, x^5, etc. and explanations of the differentiation process
- The integration of functions like 3x^3 and what integration undoes from differentiation
- Examples of calculating permutations, combinations, and arrangements of different objects
The document discusses differentiation and its history. It was independently developed in the 17th century by Isaac Newton and Gottfried Leibniz. Differentiation allows the calculation of instantaneous rates of change and is used in many areas including mathematics, physics, engineering and more. Key concepts covered include calculating speed, estimating instantaneous rates of change, the rules for differentiation, and differentiation of expressions with multiple terms.
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
The document discusses how personalization and dynamic content are becoming increasingly important on websites. It notes that 52% of marketers see content personalization as critical and 75% of consumers like it when brands personalize their content. However, personalization can create issues for search engine optimization as dynamic URLs and content are more difficult for search engines to index than static pages. The document provides tips for SEOs to help address these personalization and SEO challenges, such as using static URLs when possible and submitting accurate sitemaps.
Reuters: Pictures of the Year 2016 (Part 2)maditabalnco
This document contains 20 photos from news events around the world between January and November 2016. The photos show international events like the US presidential election, the conflict in Ukraine, the migrant crisis in Europe, the Rio Olympics, and more. They also depict human interest stories and natural phenomena from various countries.
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.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
This document is from a Calculus I class at New York University and covers basic differentiation rules. It includes announcements about homework and a quiz. The objectives are to understand and use rules for differentiating constant functions, constants multiplied by functions, sums and differences of functions, and sine and cosine functions. Examples are provided of finding the derivatives of squaring, cubing, and square root functions using the definition of the derivative. Graphs and properties of derivatives are also discussed.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Mel Anthony Pepito
This document provides an overview of the key points covered in a calculus lecture on derivatives of logarithmic and exponential functions:
1) It discusses the derivatives of exponential functions with any base, as well as the derivatives of logarithmic functions with any base.
2) It covers using the technique of logarithmic differentiation to find derivatives of functions involving products, quotients, and/or exponentials.
3) The document provides examples of finding derivatives of various logarithmic and exponential functions.
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.
The document summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
The document discusses key concepts related to limits, continuity, and differentiation. It defines what it means for a variable x to approach a finite number a or infinity, and provides the formal definitions of one-sided limits and two-sided limits. It also discusses indeterminate forms when limits take on forms like 0/0, infinity/infinity, or infinity - infinity. The document outlines several properties of limits, including limits of even and odd functions. It distinguishes between the limit of a function as x approaches a, denoted limx→af(x), versus the function value at that point, f(a). Finally, it states standard theorems about limits, such as the sum and product of two functions whose limits exist
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
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 explains the limit laws for addition, subtraction, multiplication, division and nth roots of functions. It uses the error-tolerance game framework to justify the limit laws.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document discusses partial derivatives, which are used to describe the rate of change of functions with multiple variables. It defines:
1) Partial derivatives as the rate of change of the dependent variable with respect to one independent variable, while holding other variables constant.
2) Functions of two variables have level curves where the function value is constant. Their graphs are surfaces in 3D space.
3) Higher order partial derivatives describe the rate of change of the first partial derivatives.
4) The chain rule extends differentiation to composite functions, allowing functions of variables that are themselves functions of other variables.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
This chapter discusses 4 concepts: (1) function relations involving domains and ranges, (2) absolute value functions, (3) composite functions using substitution, and (4) inverse functions by making "x" equal to "y" and solving for y. Examples are provided for each concept to demonstrate finding outputs and inverses of simple functions.
A function is continuous at a point if the limit of the function at the point equals the value of the function at that point. Another way to say it, f is continuous at a if values of f(x) are close to f(a) if x is close to a. This property has deep implications, such as this: right now there are two points on opposites sides of the world with the same temperature!
This document contains notes from a calculus class section on continuity. Key points include:
- The definition of continuity requires that the limit of a function as x approaches a value exists and is equal to the value of the function at that point.
- Many common functions like polynomials, rational functions, trigonometric functions, exponentials and logarithms are continuous based on properties of limits.
- Functions can fail to be continuous if the limit does not exist or the function is not defined at a point. An example function is given that is not continuous at x=1.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
Similar to Lesson 8: Basic Differentation Rules (slides) (20)
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
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The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
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- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
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Lesson 8: Basic Differentation Rules (slides)
1. Sec on 2.3
Basic Differenta on Rules
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University
. NYUMathematics
2. Announcements
Quiz 1 this week on
1.1–1.4
Quiz 2 March 3/4 on 1.5,
1.6, 2.1, 2.2, 2.3
Midterm Monday March
7 in class
3. Objectives
Understand and use
these differen a on
rules:
the deriva ve of a
constant func on (zero);
the Constant Mul ple
Rule;
the Sum Rule;
the Difference Rule;
the deriva ves of sine
and cosine.
4. Recall: the derivative
Defini on
Let f be a func on and a a point in the domain of f. If the limit
f(a + h) − f(a) f(x) − f(a)
f′ (a) = lim = lim
h→0 h x→a x−a
exists, the func on is said to be differen able at a and f′ (a) is the
deriva ve of f at a.
5. The deriva ve …
…measures the slope of the line through (a, f(a)) tangent to
the curve y = f(x);
…represents the instantaneous rate of change of f at a
…produces the best possible linear approxima on to f near a.
6. Notation
Newtonian nota on Leibnizian nota on
dy d df
f′ (x) y′ (x) y′ f(x)
dx dx dx
7. Link between the notations
f(x + ∆x) − f(x) ∆y dy
f′ (x) = lim = lim =
∆x→0 ∆x ∆x→0 ∆x dx
dy
Leibniz thought of as a quo ent of “infinitesimals”
dx
dy
We think of as represen ng a limit of (finite) difference
dx
quo ents, not as an actual frac on itself.
The nota on suggests things which are true even though they
don’t follow from the nota on per se
8. Outline
Deriva ves so far
Deriva ves of power func ons by hand
The Power Rule
Deriva ves of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Mul ple Rule
Deriva ves of sine and cosine
9. Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
10. Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
Solu on
f(x + h) − f(x)
f′ (x) = lim
h→0 h
11. Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)2 − x2
f (x) = lim = lim
h→0 h h→0 h
12. Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)2 − x2
f (x) = lim = lim
h→0 h h→0 h
+ 2xh + h −
2
x2
x2
= lim
h→0 h
13. Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)2 − x2
f (x) = lim = lim
h→0 h h→0 h
+ 2xh + h −
x2
2
x2
2x + h2
h
¡
= lim = lim
h→0 h h→0 h
14. Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)2 − x2
f (x) = lim = lim
h→0 h h→0 h
+ 2xh + h −
x2
2
x2
2x + h2
h
¡
= lim = lim
h→0 h h→0 h
= lim (2x + h) = 2x.
h→0
15. The second derivative
If f is a func on, so is f′ , and we can seek its deriva ve.
f′′ = (f′ )′
It measures the rate of change of the rate of change!
16. The second derivative
If f is a func on, so is f′ , and we can seek its deriva ve.
f′′ = (f′ )′
It measures the rate of change of the rate of change! Leibnizian
nota on:
d2 y d2 d2 f
f(x)
dx2 dx2 dx2
19. The squaring function and its derivatives
y
f′
f increasing =⇒ f′ ≥ 0
f decreasing =⇒ f′ ≤ 0
. f x horizontal tangent at 0
=⇒ f′ (0) = 0
20. The squaring function and its derivatives
y
f′
f′′ f increasing =⇒ f′ ≥ 0
f decreasing =⇒ f′ ≤ 0
. f x horizontal tangent at 0
=⇒ f′ (0) = 0
21. Derivative of the cubing function
Example
Suppose f(x) = x3 . Use the defini on of deriva ve to find f′ (x).
22. Derivative of the cubing function
Example
Suppose f(x) = x3 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)3 − x3
f (x) = lim = lim
h→0 h h→0 h
23. Derivative of the cubing function
Example
Suppose f(x) = x3 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)3 − x3
f (x) = lim = lim
h→0 h h→0 h
x3
+ 3x2 h + 3xh2 + h3 − 3
x
= lim
h→0 h
24. Derivative of the cubing function
Example
Suppose f(x) = x3 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)3 − x3
f (x) = lim = lim
h→0 h h→0 h
1 2
!
¡ !
¡
x3
+
2 2
3x h + 3xh + h 3
− 3
x 3x2 +
h
¡
3xh2 + ¡
h3
= lim = lim
h→0 h h→0 h
25. Derivative of the cubing function
Example
Suppose f(x) = x3 . Use the defini on of deriva ve to find f′ (x).
Solu on
′ f(x + h) − f(x) (x + h)3 − x3
f (x) = lim = lim
h→0 h h→0 h
1 2
!
¡ !
¡
x3
2 2
3x h + 3xh + h
+
3
− 3
x 3x2 +
h
¡
3xh2 + ¡
h3
= lim = lim
h→0
( 2 h ) h→0 h
= lim 3x + 3xh + h2 = 3x2 .
h→0
28. The cubing function and its derivatives
No ce that f is increasing,
y and f′ 0 except
f′ f′ (0) = 0
f
. x
29. The cubing function and its derivatives
No ce that f is increasing,
y and f′ 0 except
f′′ f′ f′ (0) = 0
f
. x
30. The cubing function and its derivatives
No ce that f is increasing,
y and f′ 0 except
f′′ f′ f′ (0) = 0
No ce also that the
f tangent line to the graph
. x of f at (0, 0) crosses the
graph (contrary to a
popular “defini on” of
the tangent line)
31. Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
32. Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
Solu on
√ √
′ f(x + h) − f(x) x+h− x
f (x) = lim = lim
h→0 h h→0 h
33. Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
Solu on
√ √
′ f(x + h) − f(x) x+h− x
f (x) = lim = lim
h→0
√ h h→0 h
√ √ √
x+h− x x+h+ x
= lim ·√ √
h→0 h x+h+ x
34. Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
Solu on
√ √
′ f(x + h) − f(x) x+h− x
f (x) = lim = lim
h→0
√ h h→0 h
√ √ √
x+h− x x+h+ x
= lim ·√ √
h→0 h x+h+ x
(x + h) − x
¡ ¡
= lim (√ √ )
h→0 h x+h+ x
35. Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
Solu on
√ √
′ f(x + h) − f(x) x+h− x
f (x) = lim = lim
h→0
√ h h→0 h
√ √ √
x+h− x x+h+ x
= lim ·√ √
h→0 h x+h+ x
(x + h) − x
¡ ¡ h
= lim (√ √ ) = lim (√ √ )
h→0 h x+h+ x h x+h+ x
h→0
36. Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
Solu on
√ √
′ f(x + h) − f(x) x+h− x
f (x) = lim = lim
h→0
√ h h→0 h
√ √ √
x+h− x x+h+ x
= lim ·√ √
h→0 h x+h+ x
(x + h) − x
¡ ¡ h
1
= lim (√ √ ) = lim (√ √ )= √
h→0 h x+h+ x h x+h+ x
h→0 2 x
49. The cube root and its derivative
y
Here lim f′ (x) = ∞ and f
f x→0
is not differen able at 0
f′
. x
50. The cube root and its derivative
y
Here lim f′ (x) = ∞ and f
f x→0
is not differen able at 0
f′
. x No ce also
lim f′ (x) = 0
x→±∞
51. One more
Example
Suppose f(x) = x2/3 . Use the defini on of deriva ve to find f′ (x).
52. One more
Example
Suppose f(x) = x2/3 . Use the defini on of deriva ve to find f′ (x).
Solu on
f(x + h) − f(x) (x + h)2/3 − x2/3
f′ (x) = lim = lim
h→0 h h→0 h
53. One more
Example
Suppose f(x) = x2/3 . Use the defini on of deriva ve to find f′ (x).
Solu on
f(x + h) − f(x) (x + h)2/3 − x2/3
f′ (x) = lim = lim
h→0 h h→0 h
1/3 ( )
(x + h) − x
1/3
= lim · (x + h) + x
1/3 1/3
h→0 h
54. One more
Example
Suppose f(x) = x2/3 . Use the defini on of deriva ve to find f′ (x).
Solu on
f(x + h) − f(x) (x + h)2/3 − x2/3
f′ (x) = lim = lim
h→0 h h→0 h
1/3 ( )
(x + h) − x
1/3
= lim · (x + h) + x
1/3 1/3
h→0
( h)
1 −2/3
= 3x 2x1/3
55. One more
Example
Suppose f(x) = x2/3 . Use the defini on of deriva ve to find f′ (x).
Solu on
f(x + h) − f(x) (x + h)2/3 − x2/3
f′ (x) = lim = lim
h→0 h h→0 h
1/3 ( )
(x + h) − x
1/3
= lim · (x + h) + x
1/3 1/3
h→0
( h)
1 −2/3
= 3x 2x1/3 = 2 x−1/3
3
66. Recap: The Tower of Power
y y′
x2 2x1 The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2
x1/2 2x
1 −2/3
x1/3 3x
2 −1/3
x2/3 3x
67. Recap: The Tower of Power
y y′
x2 2x The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2
x1/2 2x
1 −2/3
x1/3 3x
2 −1/3
x2/3 3x
68. Recap: The Tower of Power
y y′
x2 2x The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2
x1/2 2x
1 −2/3
x1/3 3x
2 −1/3
x2/3 3x
69. Recap: The Tower of Power
y y′
x2 2x The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2
x1/2 2x
1 −2/3
x1/3 3x
2 −1/3
x2/3 3x
70. Recap: The Tower of Power
y y′
x2 2x The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2
x1/2 2x
1 −2/3
x1/3 3x
2 −1/3
x2/3 3x
71. Recap: The Tower of Power
y y′
x2 2x The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2
x1/2 2x
1 −2/3
x1/3 3x
2 −1/3
x2/3 3x
72. Recap: The Tower of Power
y y′
x2 2x The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2 The coefficient in the
x1/2 2x deriva ve is the power of
1 −2/3
x1/3 3x
the original func on
2 −1/3
x2/3 3x
73. The Power Rule
There is moun ng evidence for
Theorem (The Power Rule)
Let r be a real number and f(x) = xr . Then
f′ (x) = rxr−1
as long as the expression on the right-hand side is defined.
Perhaps the most famous rule in calculus
We will assume it as of today
We will prove it many ways for many different r.
75. Outline
Deriva ves so far
Deriva ves of power func ons by hand
The Power Rule
Deriva ves of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Mul ple Rule
Deriva ves of sine and cosine
76. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
77. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
Proof.
We have
∑
n
(x + h) = (x + h) · (x + h) · (x + h) · · · (x + h) =
n
ck xk hn−k
n copies k=0
78. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
Proof.
We have
∑
n
(x + h) = (x + h) · (x + h) · (x + h) · · · (x + h) =
n
ck xk hn−k
n copies k=0
79. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
Proof.
We have
∑
n
(x + h) = (x + h) · (x + h) · (x + h) · · · (x + h) =
n
ck xk hn−k
n copies k=0
80. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
Proof.
We have
∑
n
(x + h) = (x + h) · (x + h) · (x + h) · · · (x + h) =
n
ck xk hn−k
n copies k=0
81. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
Proof.
We have
∑
n
(x + h) = (x + h) · (x + h) · (x + h) · · · (x + h) =
n
ck xk hn−k
n copies k=0
82. Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
Proof.
We have
∑
n
(x + h) = (x + h) · (x + h) · (x + h) · · · (x + h) =
n
ck xk hn−k
n copies k=0
87. Proving the Power Rule
Theorem (The Power Rule)
d n
Let n be a posi ve whole number. Then x = nxn−1 .
dx
88. Proving the Power Rule
Theorem (The Power Rule)
d n
Let n be a posi ve whole number. Then x = nxn−1 .
dx
Proof.
As we showed above,
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
(x + h)n − xn nxn−1 h + (stuff with at least two hs in it)
So =
h h
= nxn−1 + (stuff with at least one h in it)
and this tends to nxn−1 as h → 0.
89. The Power Rule for constants?
Theorem
d
Let c be a constant. Then c=0
dx
90. The Power Rule for constants?
d 0
Theorem like x = 0x−1
d dx
Let c be a constant. Then c = 0.
.
dx
91. The Power Rule for constants?
d 0
Theorem like x = 0x−1
d dx
Let c be a constant. Then c = 0.
.
dx
Proof.
Let f(x) = c. Then
f(x + h) − f(x) c − c
= =0
h h
So f′ (x) = lim 0 = 0.
h→0
93. Recall the Limit Laws
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M
x→a
2. lim [f(x) − g(x)] = L − M
x→a
3. lim [cf(x)] = cL
x→a
4. . . .
94. Adding functions
Theorem (The Sum Rule)
Let f and g be func ons and define
(f + g)(x) = f(x) + g(x)
Then if f and g are differen able at x, then so is f + g and
(f + g)′ (x) = f′ (x) + g′ (x).
Succinctly, (f + g)′ = f′ + g′ .
95. Proof of the Sum Rule
Proof.
Follow your nose:
(f + g)(x + h) − (f + g)(x)
(f + g)′ (x) = lim
h→0 h
96. Proof of the Sum Rule
Proof.
Follow your nose:
(f + g)(x + h) − (f + g)(x)
(f + g)′ (x) = lim
h→0 h
f(x + h) + g(x + h) − [f(x) + g(x)]
= lim
h→0 h
97. Proof of the Sum Rule
Proof.
Follow your nose:
(f + g)(x + h) − (f + g)(x)
(f + g)′ (x) = lim
h→0 h
f(x + h) + g(x + h) − [f(x) + g(x)]
= lim
h→0 h
f(x + h) − f(x) g(x + h) − g(x)
= lim + lim
h→0 h h→0 h
98. Proof of the Sum Rule
Proof.
Follow your nose:
(f + g)(x + h) − (f + g)(x)
(f + g)′ (x) = lim
h→0 h
f(x + h) + g(x + h) − [f(x) + g(x)]
= lim
h→0 h
f(x + h) − f(x) g(x + h) − g(x)
= lim + lim
h→0 h h→0 h
′ ′
= f (x) + g (x)
99. Scaling functions
Theorem (The Constant Mul ple Rule)
Let f be a func on and c a constant. Define
(cf)(x) = cf(x)
Then if f is differen able at x, so is cf and
(cf)′ (x) = c · f′ (x)
Succinctly, (cf)′ = cf′ .
100. Proof of Constant Multiple Rule
Proof.
Again, follow your nose.
(cf)(x + h) − (cf)(x)
(cf)′ (x) = lim
h→0 h
101. Proof of Constant Multiple Rule
Proof.
Again, follow your nose.
(cf)(x + h) − (cf)(x) cf(x + h) − cf(x)
(cf)′ (x) = lim = lim
h→0 h h→0 h
102. Proof of Constant Multiple Rule
Proof.
Again, follow your nose.
(cf)(x + h) − (cf)(x) cf(x + h) − cf(x)
(cf)′ (x) = lim = lim
h→0 h h→0 h
f(x + h) − f(x)
= c lim
h→0 h
103. Proof of Constant Multiple Rule
Proof.
Again, follow your nose.
(cf)(x + h) − (cf)(x) cf(x + h) − cf(x)
(cf)′ (x) = lim = lim
h→0 h h→0 h
f(x + h) − f(x)
= c lim = c · f′ (x)
h→0 h
105. Derivatives of polynomials
Example
d ( 3 )
Find 2x + x4 − 17x12 + 37
dx
Solu on
d ( 3 ) d ( 3) d d ( ) d
2x + x4 − 17x12 + 37 = 2x + x4 + −17x12 + (37)
dx dx dx dx dx
106. Derivatives of polynomials
Example
d ( 3 )
Find 2x + x4 − 17x12 + 37
dx
Solu on
d ( 3 ) d ( 3) d d ( ) d
2x + x4 − 17x12 + 37 = 2x + x4 + −17x12 + (37)
dx dx dx dx dx
d d d
= 2 x3 + x4 − 17 x12 + 0
dx dx dx
107. Derivatives of polynomials
Example
d ( 3 )
Find 2x + x4 − 17x12 + 37
dx
Solu on
d ( 3 ) d ( 3) d d ( ) d
2x + x4 − 17x12 + 37 = 2x + x4 + −17x12 + (37)
dx dx dx dx dx
d d d
= 2 x3 + x4 − 17 x12 + 0
dx dx dx
= 2 · 3x + 4x − 17 · 12x11
2 3
108. Derivatives of polynomials
Example
d ( 3 )
Find 2x + x4 − 17x12 + 37
dx
Solu on
d ( 3 ) d ( 3) d d ( ) d
2x + x4 − 17x12 + 37 = 2x + x4 + −17x12 + (37)
dx dx dx dx dx
d d d
= 2 x3 + x4 − 17 x12 + 0
dx dx dx
= 2 · 3x + 4x − 17 · 12x11
2 3
= 6x2 + 4x3 − 204x11
109. Outline
Deriva ves so far
Deriva ves of power func ons by hand
The Power Rule
Deriva ves of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Mul ple Rule
Deriva ves of sine and cosine
110. Derivatives of Sine and Cosine
Fact
d
sin x = ???
dx
Proof.
From the defini on:
111. Derivatives of Sine and Cosine
Fact
d
sin x = ???
dx
Proof.
From the defini on:
d sin(x + h) − sin x
sin x = lim
dx h→0 h
112. Angle addition formulas
See Appendix A
sin(A + B) = . A cos B + cos A sin B
sin
cos(A + B) = cos A cos B − sin A sin B
113. Derivatives of Sine and Cosine
Fact
d
sin x = ???
dx
Proof.
From the defini on:
d sin(x + h) − sin x ( sin x cos h + cos x sin h) − sin x
sin x = lim = lim
dx h→0 h h→0 h
114. Derivatives of Sine and Cosine
Fact
d
sin x = ???
dx
Proof.
From the defini on:
d sin(x + h) − sin x ( sin x cos h + cos x sin h) − sin x
sin x = lim = lim
dx h→0 h h→0 h
cos h − 1 sin h
= sin x · lim + cos x · lim
h→0 h h→0 h
116. Derivatives of Sine and Cosine
Fact
d
sin x = ???
dx
Proof.
From the defini on:
d sin(x + h) − sin x ( sin x cos h + cos x sin h) − sin x
sin x = lim = lim
dx h→0 h h→0 h
cos h − 1 sin h
= sin x · lim + cos x · lim
h→0 h h→0 h
= sin x · 0 + cos x · 1
117. Derivatives of Sine and Cosine
Fact
d
sin x = ???
dx
Proof.
From the defini on:
d sin(x + h) − sin x ( sin x cos h + cos x sin h) − sin x
sin x = lim = lim
dx h→0 h h→0 h
cos h − 1 sin h
= sin x · lim + cos x · lim
h→0 h h→0 h
= sin x · 0 + cos x · 1
118. Derivatives of Sine and Cosine
Fact
d
sin x = cos x
dx
Proof.
From the defini on:
d sin(x + h) − sin x ( sin x cos h + cos x sin h) − sin x
sin x = lim = lim
dx h→0 h h→0 h
cos h − 1 sin h
= sin x · lim + cos x · lim
h→0 h h→0 h
= sin x · 0 + cos x · 1 = cos x
120. Illustration of Sine and Cosine
y
. x
π −π 0 π π cos x
2 2
sin x
f(x) = sin x has horizontal tangents where f′ = cos(x) is zero.
121. Illustration of Sine and Cosine
y
. x
π −π 0 π π cos x
2 2
sin x
f(x) = sin x has horizontal tangents where f′ = cos(x) is zero.
what happens at the horizontal tangents of cos?
123. Derivative of Cosine
Fact
d
cos x = − sin x
dx
Proof.
We already did the first. The second is similar (muta s mutandis):
d cos(x + h) − cos x
cos x = lim
dx h→0 h
124. Derivative of Cosine
Fact
d
cos x = − sin x
dx
Proof.
We already did the first. The second is similar (muta s mutandis):
d cos(x + h) − cos x (cos x cos h − sin x sin h) − cos x
cos x = lim = lim
dx h→0 h h→0 h
125. Derivative of Cosine
Fact
d
cos x = − sin x
dx
Proof.
We already did the first. The second is similar (muta s mutandis):
d cos(x + h) − cos x (cos x cos h − sin x sin h) − cos x
cos x = lim = lim
dx h→0 h h→0 h
cos h − 1 sin h
= cos x · lim − sin x · lim
h→0 h h→0 h
126. Derivative of Cosine
Fact
d
cos x = − sin x
dx
Proof.
We already did the first. The second is similar (muta s mutandis):
d cos(x + h) − cos x (cos x cos h − sin x sin h) − cos x
cos x = lim = lim
dx h→0 h h→0 h
cos h − 1 sin h
= cos x · lim − sin x · lim
h→0 h h→0 h
= cos x · 0 − sin x · 1 = − sin x
128. Summary
What have we learned today?
The Power Rule
The deriva ve of a sum is the sum of the deriva ves
The deriva ve of a constant mul ple of a func on is that
constant mul ple of the deriva ve
129. Summary
What have we learned today?
The Power Rule
The deriva ve of a sum is the sum of the deriva ves
The deriva ve of a constant mul ple of a func on is that
constant mul ple of the deriva ve
The deriva ve of sine is cosine
The deriva ve of cosine is the opposite of sine.