CHAPTER 3 DIFFERENTIATION

Similar documents
Review Guideline for Final

UNIT 3: DERIVATIVES STUDY GUIDE

Chapter 2: Differentiation

*Finding the tangent line at a point P boils down to finding the slope of the tangent line at point P.

Chapter 2: Differentiation

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)

Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

CW High School. Calculus/AP Calculus A

Define each term or concept.

AP CALCULUS AB Study Guide for Midterm Exam 2017

Implicit Differentiation and Inverse Trigonometric Functions

Announcements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.

2.2 The derivative as a Function

AP Calculus Summer Prep

f(x 0 + h) f(x 0 ) h slope of secant line = m sec

Learning Objectives for Math 165

CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg.

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts

Lecture Notes for Math 1000

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts

Chapter 3 Differentiation Rules

Calculus I Sample Exam #01

TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12

2.1 The derivative. Rates of change. m sec = y f (a + h) f (a)

A Library of Functions

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2

Hello Future Calculus Level One Student,

Blue Pelican Calculus First Semester

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts

SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)

Calculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA

Derivative of a Function

Review for Final. The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study:

Section 1.1: A Preview of Calculus When you finish your homework, you should be able to

function independent dependent domain range graph of the function The Vertical Line Test

sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations

Math 131. The Derivative and the Tangent Line Problem Larson Section 2.1

2.1 Limits, Rates of Change and Slopes of Tangent Lines

2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 2.6 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS

Welcome to AP Calculus!!!

THEOREM: THE CONSTANT RULE

Limit. Chapter Introduction

Using the Definitions of the Trigonometric Functions

MATH 1040 Objectives List

AP Calculus Summer Packet

DIFFERENTIATION RULES

AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm.

In this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)

UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS. Qu: What do you remember about exponential and logarithmic functions?

One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle.

Summer Work for Students Entering Calculus

DuVal High School Summer Review Packet AP Calculus

CALCULUS ASSESSMENT REVIEW

Chapter 3 Definitions and Theorems

Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam

Calculus I. When the following condition holds: if and only if

Preliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics MATHS 101: Calculus I

FUNCTIONS AND MODELS

AP Calculus I Summer Packet

Rules for Differentiation Finding the Derivative of a Product of Two Functions. What does this equation of f '(

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).

A.P. Calculus Summer Assignment

February 21 Math 1190 sec. 63 Spring 2017

Inverse Trig Functions

AP Calculus AB. Course Overview. Course Outline and Pacing Guide

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations

Unit IV Derivatives 20 Hours Finish by Christmas

Unit IV Derivatives 20 Hours Finish by Christmas

MAT137 Calculus! Lecture 6

Calculus AB Topics Limits Continuity, Asymptotes

Math 147 Exam II Practice Problems

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)

Technical Calculus I Homework. Instructions

f(x)

Section 2.1 The Derivative and the Tangent Line Problem

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Final Exam Review Exercise Set A, Math 1551, Fall 2017

Differentiation Rules and Formulas

2. Algebraic functions, power functions, exponential functions, trig functions

1 + x 2 d dx (sec 1 x) =

Pre-Calculus MATH 119 Fall Section 1.1. Section objectives. Section 1.3. Section objectives. Section A.10. Section objectives

CHAIN RULE: DAY 2 WITH TRIG FUNCTIONS. Section 2.4A Calculus AP/Dual, Revised /30/2018 1:44 AM 2.4A: Chain Rule Day 2 1

3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

Functions. Remark 1.2 The objective of our course Calculus is to study functions.

Summer 2017 Review For Students Entering AP Calculus AB/BC

CALCULUS II MATH Dr. Hyunju Ban

Math Section 4.3 Unit Circle Trigonometry

Calculus: Early Transcendental Functions Lecture Notes for Calculus 101. Feras Awad Mahmoud

p324 Section 5.2: The Natural Logarithmic Function: Integration

Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description

Chapter 3: Derivatives

Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)

f(x) f(a) Limit definition of the at a point in slope notation.

1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2

Chapter 5 Analytic Trigonometry

Transcription:

CHAPTER 3 DIFFERENTIATION

3.1 THE DERIVATIVE AND THE TANGENT LINE PROBLEM You will be able to: - Find the slope of the tangent line to a curve at a point - Use the limit definition to find the derivative of a function - Understand the relationship between differentiability and continuity

TANGENT LINE PROBLEM Find the tangent line to a curve Finding the slope of the tangent line at point P Approximate using a secant line through the point of tangency and a second point

DEFINITION OF TANGENT LINE WITH SLOPE M As the change in x approaches 0, the difference quotient gets closer and closer to the slope of the line at the point of tangency, hence the definition of the slope of the tangent line: This limit is actually one of the two fundamental operations in calculus - DIFFERENTIATION

DIFFERENTIATION NOTATION All of these symbols refer back to differentiation Most common are the first 3 this symbol is read as prime so f prime of x dy/dx is read the derivative of y with respect to x

DIFFERENTIABILITY AND CONTINUITY The derivative of f at c is This is provided the limit exists, which means that both sides of the limit agree, hence continuity If f is differentiable at x=c, then f is continuous at x=c

HOMEWORK

3.2 BASIC DIFFERENTIATION RULES AND RATES OF CHANGE You will be able to: - Find the derivative of a function using the Constant Rule, Power Rule, Constant Multiple Rule, Sum and Difference Rules - Find the derivative of the sine, cosine, and exponential functions - Use derivative to find rates of change

CONSTANT AND POWER The Constant Rule The derivative of a constant function is 0. If c is a real number then The Power Rule If n is a rational number, then the function differentiable and is

CONSTANT MULTIPLE, SUM, AND DIFFERENCE If f is a differentiable function and c is a real number, then cf is also differentiable and The sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f+g (or f-g) is the sum (or difference) of the derivatives of f and g

SINE, COSINE, AND E

RATES OF CHANGE Distance = Rate x Time, Rate = distance/time s is the position function of time t Average velocity is change in distance/change in time = Instantaneous velocity is the slope of the position function Position function of a free-falling object:

HOMEWORK

3.3 PRODUCT AND QUOTIENT RULES AND HIGHER ORDER You will be able to: - find the derivative of a function using the Product and Quotient Rules - find the derivative of a trigonometric function - find a higher order derivative of a function

THE PRODUCT RULE The product of two differentiable functions f and g is itself differentiable Words: The derivative fg is the first function times the derivative of the second function, plus the second function times the derivative of the first function Symbols: Mnemonic 1 d2 + 2 d1

THE QUOTIENT RULE The quotient f/g of two differentiable functions f and g is itself differentiable at all values of x for which g(x) is not 0. Words: The derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Symbols: Mnemonic: Low d-high minus high d-low over the square of whats below

DERIVATIVES OF TRIGONOMETRICS Something to remember: sec and tan go together, cot and csc go together and are negative

HIGHER ORDER DERIVATIVES The second derivative is simply the derivative of the derivative Mainly used since acceleration is the slope of the velocity or the second derivative of the position function: a(t)=v (t)=s (t)

HOMEWORK

3.4 THE CHAIN RULE You will be able to: - find derivatives using the Chain Rule and General Power Rule

THE CHAIN RULE If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and or equivalently: For the chain rule, identify the inside function and the outside function

THE GENERAL POWER RULE AND NATURAL LOG If, where u is a differentiable function of x and n is a rational number, then Let u be a differentiable function of x.

ABSOLUTE, DIFFERENT BASES If u is a differentiable function of x such that u is not 0, then Derivatives of Bases other than e If a is a positive real number (a is not 1) and x is any real number then If a is a positive real number (a is not 1) and x is any positive real number then

BOOKMARK P. 160 HOMEWORK

3.5 IMPLICIT DIFFERENTIATION You will be able to: - distinguish between functions written in implicit form and explicit form - use implicit differentiation to find the derivative of a function - find derivatives of functions using logarithmic differentiation

IMPLICIT VS EXPLICIT Implicit: xy = 1 Explicit: y=1/x Differentiation with respect to x (dy/dx) (dx/dx)

IMPLICIT DIFFERENTIATION Guidelines Differentiate both sides with respect to x Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation Factor dy/dx out of the left side of the equation Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx

LOGARITHMIC DIFFERENTIATION On occasion, it is convenient to use logarithms as aids in differentiation nonlogarithmic functions This means we take the ln of both sides and then derive

HOMEWORK

3.6 DERIVATIVES OF INVERSE FUNCTIONS You will be able to: - Find the derivative of an inverse functions - Differentiate an inverse trigonometric function - Review that basic differentiation formulas for elementary functions

INVERSE FUNCTIONS Let f be a function whose domains is an interval I. If f has an inverse function, then the following statements are true. If f is continuous on its domain, then the inverse of f is continuous on its domain If f is differentiable on an interval containing c and f (c) is not 0, then the inverse of f is differentiable at f(c) If f has an inverse function g then

DERIVATIVES OF INVERSE TRIG FUNCTIONS Let u be a differentiable function of x.

HOMEWORK

3.7 RELATED RATES You will be able to: - Find a related rate - Use related rates to solve real-life problems

GUIDELINES FOR SOLVING RELATED-RATE PROBLEMS Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. Write an equation involving the variables whose rates of change either are given or are to be determined. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.

HOMEWORK

3.8 NEWTON S METHOD You will be able to approximate a zero of a function using Newton s Method.

NEWTON S METHOD Basically Newton s Method is using tangent lines to approximate the graph of the function near its x-intercepts. Let f(c)=0, where f is differentiable on an open interval containing c. To approximate c, use the following steps: Make an initial estimate that is close to c (graph) Determine a new approximation: If is within the desired accuracy, let serve as the final approximation. Otherwise return to Step 2 and calculate again.

HOMEWORK

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback