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

Size: px
Start display at page:

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

Transcription

1 Announcements Topics: - sections 4.5 and * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the SCHEDULE + HOMEWORK link)

2 Basic Differentiation Rules All rules are proved using the definition of the derivative: df dx = f '(x) = lim h 0 f (x + h) f (x) h The derivative exists (i.e. a function is differentiable) at all values of x for which this limit exists.

3 The Constant Function Rule If where is a constant, then f (x) = k, k f '(x) = 0. f (x) = 3 f '(x) = 0 y m=0 x

4 The Power Rule If where then f (x) = x n, n R, f '(x) = nx n 1. Differentiate the following. (a) g(x) = x 100 (b) f (x) = x (c) (d) h(x) = 1 x 6 s(t) = t

5 Let k Then The Constant Multiple Rule be a constant. d dx k f (x) [ ] = k d dx Find the derivative of each. (a) (b) [ f (x) ]. f (x) = 3x 7 g(x) = 5 x

6 The Sum/Difference Rule provided f and g are differentiable functions. Examples: Differentiate. (a) [ f (x) ± g(x)]'= f '(x) ± g'(x) f (x) = 3x 4 + 5x 2 10 (b) g(x) = 5x + x 5 π 2

7 Using the Derivative to Sketch the Graph of a Function Sketch the graph of F(x) = x + 1 x. y x

8 Derivative of the Natural Exponential Function Definition: The number e is the number for which y f (x) = e x lim h 0 e h 1 h =1 x Natural Exponential Function: f (x) = e x

9 Derivative of the Natural Exponential Function Note: This definition states that the slope of the tangent to the curve at (0,1) is exactly 1, i.e. f '(0) = lim h 0 e 0+h e 0 h y m =1 f (x) = e x x = lim h 0 e h 1 h =1

10 Derivative of the Natural Exponential Function If f (x) = e x, then ʹ f (x) = e x. In words: The slope of the tangent line to the curve f (x) = e x at the point P is equal to the value of the function at P. y f (x) = e x P (x, e x )... slope = e x (1, e)... slope = e (0, 1)... slope =1 x

11 The Product Rule [ f (x) g(x)]'= f '(x) g(x) + f (x) g'(x) provided f and g are differentiable functions. Find the critical numbers of f (x) = x 4 e x.

12 The Quotient Rule " $ # f (x) g(x) % ''= & f '(x) g(x) f (x) g'(x) [g(x)] 2 provided and are differentiable and f g g(x) 0. Determine where the graph of the function q(x) = x x has horizontal tangents.

13 Chain Rule [ f (g(x))]'= f '(g(x)) g'(x) derivative of the outer function evaluated at the inner function times the derivative of the inner function Differentiate the following. (a) (b) f (x) = (4x 3 +1) 10 g(x) = 1 x 2 + e 3x

14 Derivatives of General Exponential Functions If then f (x) = a x, f '(x) = a x ln a. Differentiate. (a) (b) (c) f (x) = e x g(x) = 7 x + x 7 h(x) = 2 5x 2 +1

15 If Derivatives of Logarithmic Functions f (x) = log a x, then f '(x) = 1 x ln a. Example 1: Differentiate. (a) f (x) = ln x (b) g(x) = log 4 (x 2 + 5x + 6) Example 2: Determine the equation of the tangent line to the curve f (x) = ln x at the point P(1,0). x

16 Derivatives of Trigonometric Functions 2 1 y = sin x y = sin x y = cos x

17 Derivatives of Trigonometric Functions (sin x)'= cos x (cos x)'= sin x (tan x)'= sec 2 x Find the derivative of each. (a) f (x) = cos(e 3x 2 ) (b) g(x) = csc x (c) h(x) = tan x 4 + tan 4 x

18 Derivatives of Inverse Trig Functions d dx (arcsin x) = 1 1 x 2 d dx (arctan x) = 1 1+ x 2 Differentiate. (a) (b) f (x) = arctan x 5 ( ) + arctan ( 5 x) g(x) = ln(arcsin(5x 3 +1))

19 Implicit Differentiation Using implicit differentiation, determine for y 3 + x 2 = e xy. y "

20 Logarithmic Differentation Using logarithmic differentiation, find the derivative of h(x) = (x +1)ex x 3 sin x

21 Related Rates The concentration of a pollutant (measured in grams per cubic metre) at a location x metres away from the source is given by c(x) = 0.28e 0.25x2 An observer is located 5m from the source. How does the concentration change as she runs away from the source at a speed of 4.8m/s?

Announcements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook!

Announcements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Announcements Topics: - sections 5.2 5.7, 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems

More information

Announcements. Topics: Homework:

Announcements. Topics: Homework: Announcements Topics: - sections 7.1 (differential equations), 7.2 (antiderivatives), and 7.3 (the definite integral +area) * Read these sections and study solved examples in your textbook! Homework: -

More information

UNIT 3: DERIVATIVES STUDY GUIDE

UNIT 3: DERIVATIVES STUDY GUIDE Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average

More information

Announcements. Topics: Homework: - sections 2.2, 2.3, 4.1, and 4.2 * Read these sections and study solved examples in your textbook!

Announcements. Topics: Homework: - sections 2.2, 2.3, 4.1, and 4.2 * Read these sections and study solved examples in your textbook! Announcements Topics: - sections 2.2, 2.3, 4.1, and 4.2 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the

More information

Test one Review Cal 2

Test one Review Cal 2 Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single

More information

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

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 NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus

More information

Announcements. Topics: Homework: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook!

Announcements. Topics: Homework: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook! Announcements Topics: - sections 1.4, 2.2, and 2.3 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook

More information

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

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever

More information

MAT137 Calculus! Lecture 6

MAT137 Calculus! Lecture 6 MAT137 Calculus! Lecture 6 Today: 3.2 Differentiation Rules; 3.3 Derivatives of higher order. 3.4 Related rates 3.5 Chain Rule 3.6 Derivative of Trig. Functions Next: 3.7 Implicit Differentiation 4.10

More information

Section 3.5: Implicit Differentiation

Section 3.5: Implicit Differentiation Section 3.5: Implicit Differentiation In the previous sections, we considered the problem of finding the slopes of the tangent line to a given function y = f(x). The idea of a tangent line however is not

More information

Chapter 3 Differentiation Rules

Chapter 3 Differentiation Rules Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. y For example,, or y = x sin x,

More information

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

Calculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you

More information

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

CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg. CALCULUS: Graphical,Numerical,Algebraic b Finne,Demana,Watts and Kenned Chapter : Derivatives.: Derivative of a function pg. 116-16 What ou'll Learn About How to find the derivative of: Functions with

More information

Announcements. Topics: Homework:

Announcements. Topics: Homework: Topics: Announcements - section 2.6 (limits at infinity [skip Precise Definitions (middle of pg. 134 end of section)]) - sections 2.1 and 2.7 (rates of change, the derivative) - section 2.8 (the derivative

More information

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

Preliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics   MATHS 101: Calculus I Preliminaries 2 1 2 Lectures Department of Mathematics http://www.abdullaeid.net/maths101 MATHS 101: Calculus I (University of Bahrain) Prelim 1 / 35 Pre Calculus MATHS 101: Calculus MATHS 101 is all about

More information

Week 1: need to know. November 14, / 20

Week 1: need to know. November 14, / 20 Week 1: need to know How to find domains and ranges, operations on functions (addition, subtraction, multiplication, division, composition), behaviors of functions (even/odd/ increasing/decreasing), library

More information

Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class.

Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class. Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180

More information

Math Practice Exam 3 - solutions

Math Practice Exam 3 - solutions Math 181 - Practice Exam 3 - solutions Problem 1 Consider the function h(x) = (9x 2 33x 25)e 3x+1. a) Find h (x). b) Find all values of x where h (x) is zero ( critical values ). c) Using the sign pattern

More information

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

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 3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)).

More information

Review Guideline for Final

Review Guideline for Final Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.

More information

2.2 The derivative as a Function

2.2 The derivative as a Function 2.2 The derivative as a Function Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x)

More information

CHAPTER 3 DIFFERENTIATION

CHAPTER 3 DIFFERENTIATION 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

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

More information

Chapter 3 Differentiation Rules (continued)

Chapter 3 Differentiation Rules (continued) Chapter 3 Differentiation Rules (continued) Sec 3.5: Implicit Differentiation (continued) Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph

More information

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

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 Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc

More information

Derivative of a Function

Derivative of a Function Derivative of a Function (x+δx,f(x+δx)) f ' (x) = (x,f(x)) provided the limit exists Can be interpreted as the slope of the tangent line to the curve at any point (x, f(x)) on the curve. This generalizes

More information

3.9 Derivatives of Exponential and Logarithmic Functions

3.9 Derivatives of Exponential and Logarithmic Functions 322 Chapter 3 Derivatives 3.9 Derivatives of Exponential and Logarithmic Functions Learning Objectives 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions.

More information

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

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics

More information

AP Calculus Summer Packet

AP Calculus Summer Packet AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept

More information

AP Calculus Summer Prep

AP Calculus Summer Prep AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have

More information

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

2.1 The derivative. Rates of change. m sec = y f (a + h) f (a) 2.1 The derivative Rates of change 1 The slope of a secant line is m sec = y f (b) f (a) = x b a and represents the average rate of change over [a, b]. Letting b = a + h, we can express the slope of the

More information

Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. AB Fall Final Exam Review 200-20 Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) The position of a particle

More information

Final Exam Review Problems

Final Exam Review Problems Final Exam Review Problems Name: Date: June 23, 2013 P 1.4. 33. Determine whether the line x = 4 represens y as a function of x. P 1.5. 37. Graph f(x) = 3x 1 x 6. Find the x and y-intercepts and asymptotes

More information

7.1. Calculus of inverse functions. Text Section 7.1 Exercise:

7.1. Calculus of inverse functions. Text Section 7.1 Exercise: Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential

More information

Announcements. Topics: Homework:

Announcements. Topics: Homework: Announcements Topics: - sections 7.5 (additional techniques of integration), 7.6 (applications of integration), * Read these sections and study solved examples in your textbook! Homework: - review lecture

More information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - Math 101 Lecture Note - Dr. Said Algarni 2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that

More information

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

Final Exam Review Exercise Set A, Math 1551, Fall 2017 Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete

More information

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

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Fall 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Fall 01/01! This problems in this packet are designed to help you review the topics from Algebra

More information

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

Functions. Remark 1.2 The objective of our course Calculus is to study functions. Functions 1.1 Functions and their Graphs Definition 1.1 A function f is a rule assigning a number to each of the numbers. The number assigned to the number x via the rule f is usually denoted by f(x).

More information

February 21 Math 1190 sec. 63 Spring 2017

February 21 Math 1190 sec. 63 Spring 2017 February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative

More information

Unit #3 : Differentiability, Computing Derivatives, Trig Review

Unit #3 : Differentiability, Computing Derivatives, Trig Review Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute

More information

Calculus I Announcements

Calculus I Announcements Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.

More information

Implicit Differentiation and Inverse Trigonometric Functions

Implicit Differentiation and Inverse Trigonometric Functions Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5

More information

Calculus & Analytic Geometry I

Calculus & Analytic Geometry I TQS 124 Autumn 2008 Quinn Calculus & Analytic Geometry I The Derivative: Analytic Viewpoint Derivative of a Constant Function. For c a constant, the derivative of f(x) = c equals f (x) = Derivative of

More information

FUNCTIONS AND MODELS

FUNCTIONS AND MODELS 1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.6 Inverse Functions and Logarithms In this section, we will learn about: Inverse functions and logarithms. INVERSE FUNCTIONS The table gives data from an experiment

More information

Review for the Final Exam

Review for the Final Exam Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x

More information

Calculus I Sample Exam #01

Calculus I Sample Exam #01 Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6

More information

Workbook for Calculus I

Workbook for Calculus I Workbook for Calculus I By Hüseyin Yüce New York 2007 1 Functions 1.1 Four Ways to Represent a Function 1. Find the domain and range of the function f(x) = 1 + x + 1 and sketch its graph. y 3 2 1-3 -2-1

More information

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

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 Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form,

More information

f(x)

f(x) Calculus m y_0e1^y jkdudtdaw ZS[oifntCwxaCrJej ilhl[cq.k i qatlplm mrpiyg^hztbsz YrmePsqeWrNvxeEdG. Calculus Ch. 3 Review Given the graph of f '(x), sketch a possible graph of f (x). 1) f '(x) f(x) 8 8

More information

Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative

More information

Topics and Concepts. 1. Limits

Topics and Concepts. 1. Limits Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits

More information

, find the value(s) of a and b which make f differentiable at bx 2 + x if x 2 x = 2 or explain why no such values exist.

, find the value(s) of a and b which make f differentiable at bx 2 + x if x 2 x = 2 or explain why no such values exist. Math 171 Exam II Summary Sheet and Sample Stuff (NOTE: The questions posed here are not necessarily a guarantee of the type of questions which will be on Exam II. This is a sampling of questions I have

More information

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

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 Differentiation CHAPTER 2 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 25 2.5 THE CHAIN RULE 2.6 DERIVATIVES OF TRIGONOMETRIC

More information

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

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Winter 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Winter 01/01! This problems in this packet are designed to help you review the topics from

More information

Chapter 12: Differentiation. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M.

Chapter 12: Differentiation. SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Chapter 12: Differentiation SSMth2: Basic Calculus Science and Technology, Engineering and Mathematics (STEM) Strands Mr. Migo M. Mendoza Chapter 12: Differentiation Lecture 12.1: The Derivative Lecture

More information

Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 1 Motivation Goal: We want to derive rules to find the derivative of

More information

Math Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8

Math Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree

More information

Announcements. Topics: Homework:

Announcements. Topics: Homework: Announcements Topics: - sections 7.4 (FTC), 7.5 (additional techniques of integration), 7.6 (applications of integration) * Read these sections and study solved examples in your textbook! Homework: - review

More information

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

UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS. Qu: What do you remember about exponential and logarithmic functions? UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS 5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS, y = e X Qu: What do you remember about exponential and logarithmic functions? e, called Euler s

More information

( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f

( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,

More information

Hello Future Calculus Level One Student,

Hello Future Calculus Level One Student, Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will

More information

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y. 90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y

More information

The Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over,

The Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over, The Derivative of a Function Measuring Rates of Change of a function y f(x) f(x 0 ) P Q Secant line x 0 x x Average rate of change of with respect to over, " " " " - Slope of secant line through, and,

More information

Fall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?

Fall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x? . What are the domain and range of the function Fall 9 Math 3 Final Exam Solutions f(x) = + ex e x? Answer: The function is well-defined everywhere except when the denominator is zero, which happens when

More information

1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x

1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x 1.5 Topics: The Derivative lutions 1. Use the limit definition of derivative (the one with x in it) to find f x given f x 4x 5x 6 4 x x 5 x x 6 4x 5x 6 f x x f x f x x0 x x0 x xx x x x x x 4 5 6 4 5 6

More information

(ii) y = ln 1 ] t 3 t x x2 9

(ii) y = ln 1 ] t 3 t x x2 9 Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside

More information

Find the rectangular coordinates for each of the following polar coordinates:

Find the rectangular coordinates for each of the following polar coordinates: WORKSHEET 13.1 1. Plot the following: 7 3 A. 6, B. 3, 6 4 5 8 D. 6, 3 C., 11 2 E. 5, F. 4, 6 3 Find the rectangular coordinates for each of the following polar coordinates: 5 2 2. 4, 3. 8, 6 3 Given the

More information

Unit #3 : Differentiability, Computing Derivatives

Unit #3 : Differentiability, Computing Derivatives Unit #3 : Differentiability, Computing Derivatives Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute derivative

More information

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

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 CHAIN RULE: DAY WITH TRIG FUNCTIONS Section.4A Calculus AP/Dual, Revised 018 viet.dang@humbleisd.net 7/30/018 1:44 AM.4A: Chain Rule Day 1 THE CHAIN RULE A. d dx f g x = f g x g x B. If f(x) is a differentiable

More information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - Math 101 Lecture Note - Dr. Said Algarni 3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.

More information

Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).

Practice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics). Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental

More information

Chapter 2 THE DERIVATIVE

Chapter 2 THE DERIVATIVE Chapter 2 THE DERIVATIVE 2.1 Two Problems with One Theme Tangent Line (Euclid) A tangent is a line touching a curve at just one point. - Euclid (323 285 BC) Tangent Line (Archimedes) A tangent to a curve

More information

Formulas that must be memorized:

Formulas that must be memorized: Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits

More information

Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives

Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives Topic 4 Outline 1 Derivative Rules Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives D. Kalajdzievska (University of Manitoba) Math 1500 Fall 2015 1 / 32 Topic

More information

Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam

Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam This outcomes list summarizes what skills and knowledge you should have reviewed and/or acquired during this entire quarter in Math 121, and what

More information

Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 611b Assignment #6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a formula for the function graphed. 1) 1) A) f(x) = 5 + x, x < -

More information

CW High School. Calculus/AP Calculus A

CW High School. Calculus/AP Calculus A 1. Algebra Essentials (25.00%) 1.1 I can apply the point-slope, slope-intercept, and general equations of lines to graph and write equations for linear functions. 4 Pro cient I can apply the point-slope,

More information

Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses

Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses Instructor: Sal Barone School of Mathematics Georgia Tech May 22, 2015 (updated May 22, 2015) Covered sections: 3.3 & 3.5 Exam 1 (Ch.1 - Ch.3) Thursday,

More information

Announcements. Topics: Homework:

Announcements. Topics: Homework: Announcements Topics: - sections 7.3 (the definite integral +area), 7.4 (FTC), 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Homework: - review

More information

Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point

Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the

More information

THE UNIVERSITY OF WESTERN ONTARIO

THE UNIVERSITY OF WESTERN ONTARIO Instructor s Name (Print) Student s Name (Print) Student s Signature THE UNIVERSITY OF WESTERN ONTARIO LONDON CANADA DEPARTMENTS OF APPLIED MATHEMATICS AND MATHEMATICS Calculus 1000A Midterm Examination

More information

Practice Differentiation Math 120 Calculus I Fall 2015

Practice Differentiation Math 120 Calculus I Fall 2015 . x. Hint.. (4x 9) 4x + 9. Hint. Practice Differentiation Math 0 Calculus I Fall 0 The rules of differentiation are straightforward, but knowing when to use them and in what order takes practice. Although

More information

Lecture Notes for Math 1000

Lecture Notes for Math 1000 Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: xswang@mun.ca Course

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

AP Calculus Summer Homework

AP Calculus Summer Homework Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.

More information

Chapter Product Rule and Quotient Rule for Derivatives

Chapter Product Rule and Quotient Rule for Derivatives Chapter 3.3 - Product Rule and Quotient Rule for Derivatives Theorem 3.6: The Product Rule If f(x) and g(x) are differentiable at any x then Example: The Product Rule. Find the derivatives: Example: The

More information

Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.

Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework. For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin

More information

Math 3B: Lecture 1. Noah White. September 23, 2016

Math 3B: Lecture 1. Noah White. September 23, 2016 Math 3B: Lecture 1 Noah White September 23, 2016 Syllabus Take a copy of the syllabus as you walk in or find it online at math.ucla.edu/~noah Class website There are a few places where you will find/receive

More information

Dr. Abdulla Eid. Section 3.8 Derivative of the inverse function and logarithms 3 Lecture. Dr. Abdulla Eid. MATHS 101: Calculus I. College of Science

Dr. Abdulla Eid. Section 3.8 Derivative of the inverse function and logarithms 3 Lecture. Dr. Abdulla Eid. MATHS 101: Calculus I. College of Science Section 3.8 Derivative of the inverse function and logarithms 3 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 19 Topics 1 Inverse Functions (1

More information

Math 1 Lecture 23. Dartmouth College. Wednesday

Math 1 Lecture 23. Dartmouth College. Wednesday Math 1 Lecture 23 Dartmouth College Wednesday 11-02-16 Contents Reminders/Announcements Last Time Derivatives of Logarithmic and Exponential Functions Examish Exercises Reminders/Announcements WebWork

More information

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

f(x) f(a) Limit definition of the at a point in slope notation. Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point

More information

Summer 2017 Review For Students Entering AP Calculus AB/BC

Summer 2017 Review For Students Entering AP Calculus AB/BC Summer 2017 Review For Students Entering AP Calculus AB/BC Holy Name High School AP Calculus Summer Homework 1 A.M.D.G. AP Calculus AB Summer Review Packet Holy Name High School Welcome to AP Calculus

More information

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

1 + x 2 d dx (sec 1 x) = Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating

More information

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS A2 level Mathematics Core 3 course workbook 2015-2016 Name: Welcome to Core 3 (C3) Mathematics. We hope that you will use this workbook to give you an organised set of notes for

More information

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 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). You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and

More information

Aim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)

Aim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x) Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x

More information

Limits at Infinity. Use algebraic techniques to help with indeterminate forms of ± Use substitutions to evaluate limits of compositions of functions.

Limits at Infinity. Use algebraic techniques to help with indeterminate forms of ± Use substitutions to evaluate limits of compositions of functions. SUGGESTED REFERENCE MATERIAL: Limits at Infinity As you work through the problems listed below, you should reference Chapter. of the recommended textbook (or the equivalent chapter in your alternative

More information

Chapter 6: Messy Integrals

Chapter 6: Messy Integrals Chapter 6: Messy Integrals Review: Solve the following integrals x 4 sec x tan x 0 0 Find the average value of 3 1 x 3 3 Evaluate 4 3 3 ( x 1), then find the area of ( x 1) 4 Section 6.1: Slope Fields

More information