Tangent Lines and Linear Approximations Solutions

Size: px
Start display at page:

Download "Tangent Lines and Linear Approximations Solutions"

Transcription

1 Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment, based on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice, free response, calculator, and non-calculator) in the time allotted. Multiple Choice Answers:. C (003 AB6/BC6) Using the point-slope form of the equation of the tangent line and the point of tangency: y7 ( f())( x ) 7 ( f ())( ) 9 3 f(); f() 3.. C (997 AB0) y sin( x) When x, y cos 0 ; therefore, the equation of the tangent line is y0x. 3. E (003 AB89) g() f() 3 6 g( x) xf( x) f( x) (product rule); g() f() f() ( 5) 3 7. y67( x ) The only answer choice with slope of 7 is answer E.. A (008 AB) Using the product rule and factoring: 3 f ( x) 6 x( x) ( x) = ( x) ( 6x x) When x, f () 7 giving y 7( x ) The slope-intercept form of the equation of the tangent line is answer A. 5. B (993 AB7) (3x )() (x3)(3) Using the quotient rule, y. (3x ) When x, y() 3 so the point-slope form of the tangent line equation is y53( x ) which is equivalent to the standard form in answer B. Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

2 6. C (997 AB) The tangent line is y5( x 3). Approximate the zero when x 3. 5( x 3) 3 x B (997 AB) The slope of the line, y 3 x, is m. Given y x, the derivative is y x. Set x (slope of the parallel line y x 3 ). Substituting for x in the original equation, y. 8 Therefore, answer B gives the correct point, B (969 AB36/BC36) When x 0, y(0) sin(0) ( sin ) y x (cos x ) y(0) The equation of the tangent line is y ( x 0). Use this tangent line to approximate: y (0.) B (003 AB6) Use implicit differentiation: 6yy x(xy y). Evaluate at (3, ) : 6() y(3) ((3) y ()) 8y 8 y 9 Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

3 0. A (993 BC7) Using implicit differentiation: ln( xy) x ln x ln y x y x y Evaluate the original equation when x, ln( y) yields y e. Therefore, y e which yields the slope of the tangent line, y 0.. A (997 AB80) x f( x) 8 x( e ) The slope of the tangent line is equal to 3 when f( x) 3. Use the graphing calculator to x determine the x value when 8 x( e ) 3. Student may also use the derivative functions to graph f without calculating the derivative by hand.. D (998 AB87) Using the graphing calculator, determine the x value when f( x) and store the value in A Evaluate f( A) and store in B. The equation of the tangent line will be y B ( x A) or y x 0. *storing the values in this question is not necessary to obtain the correct answer, but is a good practice for students. 3. C (998 AB77) The functions f and g have parallel tangent lines when f ( x) g( x). x Students can take the derivatives by hand, 6e 8x, or use the calculator to solve. x Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

4 Free Response. 00 AB6ab dy (a) f 8 dx, : f : answer An equation of the tangent line is y 8 x. (b) f..8 Since y f x 0 on the interval x., d y 3 y 3x y 0 on this interval. dx Therefore on the interval x., the line tangent to the graph of y f x at x lies below the curve and the approximation f...8 is less than : approximation : conclusion with explanation Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

5 5. 00 AB6b y 5 x (b) f The approximation is less than f. because the graph of f is concave up on the interval x.. 3 : tangent line : computes y on the tangent line at x. : answer with reason Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

6 AB6 a (a) let Q bea, a 500 dy x dx 50 setting slopes equal: a a 0 a a a 00 or dy x dx 50 a equation for l: y x0 50 setting y-values equal: a a a0a a 00 3 (b) y x 0 5 (c) height of hill at x 50 : 50 y 50 5 feet 500 height of line at x 50 : y feet 5 Yes, the spotlight hits the tree since the height of the line is less than the height of the hill + tree which is 75 feet. : slope of tangent line from parabola : uses the condition that a a, a is on linel 500 : uses the condition that slopes are equal atq : answer 0 if student is solving an irrelevant equation : slope 0 if m 0 : equation : height of hill : height of line 3 0 if height 0 : answer with analysis Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

7 7. 005B AB5/BC 5 (a) yy y xy yx y y y y y x : implicit differentiation : solves for y (b) y y x y y x x 0 y 0,, 0, y : y x : answer y (c) 0 y x y 0 The curve has no horizontal tangent since 0 x 0 for any x. : 0 y : explanation Copyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

c x y f() f (x) Determine the Determine the Approximate c : Replacin on the AP exam: under-approximation

c x y f() f (x) Determine the Determine the Approximate c : Replacin on the AP exam: under-approximation Tangent Lines and Linear Approximations Students should be able to: Determine the slope of tangent line to a curve at a point Determine the equations of tangent lines and normal lines Approximate a value

More information

Polar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative

Polar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole

More information

Euler s Method and Logistic Growth (BC Only)

Euler s Method and Logistic Growth (BC Only) Euler s Method Students should be able to: Approximate numerical solutions of differential equations using Euler s method without a calculator. Recognize the method as an recursion formula extension of

More information

Name: Date: 1. Match the equation with its graph. Page 1

Name: Date: 1. Match the equation with its graph. Page 1 Name: Date: 1. Match the equation with its graph. y 6x A) C) Page 1 D) E) Page . Match the equation with its graph. ( x3) ( y3) A) C) Page 3 D) E) Page 4 3. Match the equation with its graph. ( x ) y 1

More information

----- o Implicit Differentiation ID: A. dy r.---; d 2 Y 2. If- = '" 1-y- then - = dx 'dx 2. a c. -1 d. -2 e.

----- o Implicit Differentiation ID: A. dy r.---; d 2 Y 2. If- = ' 1-y- then - = dx 'dx 2. a c. -1 d. -2 e. Name: Class: Date: ----- ID: A Implicit Differentiation Multiple Choice Identify the choice that best completes the statement or answers the question.. The slope of the line tangent to the curve y + (xy

More information

The base of a solid is the region in the first quadrant bounded above by the line y = 2, below by

The base of a solid is the region in the first quadrant bounded above by the line y = 2, below by Chapter 7 1) (AB/BC, calculator) The base of a solid is the region in the first quadrant bounded above by the line y =, below by y sin 1 x, and to the right by the line x = 1. For this solid, each cross-section

More information

AP CALCULUS BC 2013 SCORING GUIDELINES

AP CALCULUS BC 2013 SCORING GUIDELINES AP CALCULUS BC 2013 SCORING GUIDELINES Question 4 The figure above shows the graph of f, the derivative of a twice-differentiable function f, on the closed interval 0 x 8. The graph of f has horizontal

More information

AP Calculus. Areas and Volumes. Student Handout

AP Calculus. Areas and Volumes. Student Handout AP Calculus Areas and Volumes Student Handout 016-017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss Copyright

More information

MATH 19520/51 Class 6

MATH 19520/51 Class 6 MATH 19520/51 Class 6 Minh-Tam Trinh University of Chicago 2017-10-06 1 Review partial derivatives. 2 Review equations of planes. 3 Review tangent lines in single-variable calculus. 4 Tangent planes to

More information

Direction Fields; Euler s Method

Direction Fields; Euler s Method Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this

More information

MATH 1020 WORKSHEET 10.1 Parametric Equations

MATH 1020 WORKSHEET 10.1 Parametric Equations MATH WORKSHEET. Parametric Equations If f and g are continuous functions on an interval I, then the equations x ft) and y gt) are called parametric equations. The parametric equations along with the graph

More information

First of all, we need to know what it means for a parameterize curve to be differentiable. FACT:

First of all, we need to know what it means for a parameterize curve to be differentiable. FACT: CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of

More information

Differentiation. The Derivative and the Tangent Line Problem 10/9/2014. Copyright Cengage Learning. All rights reserved.

Differentiation. The Derivative and the Tangent Line Problem 10/9/2014. Copyright Cengage Learning. All rights reserved. Differentiation Copyright Cengage Learning. All rights reserved. The Derivative and the Tangent Line Problem Copyright Cengage Learning. All rights reserved. 1 Objectives Find the slope of the tangent

More information

MAT 1475 Final Exam Review Problems

MAT 1475 Final Exam Review Problems MAT1475 Final Review Spring 2016 Spring 2016 MAT 1475 Final Exam Review Problems Revised by Prof. Kostadinov, Fall 2015, Fall 2014, Fall 2013, Fall 2012, Fall 2011, Fall 2010 Revised by Prof. Africk and

More information

AP Calculus AB Unit 2 Assessment

AP Calculus AB Unit 2 Assessment Class: Date: 203-204 AP Calculus AB Unit 2 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.

More information

Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each.

Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. Math 106/108 Final Exam Page 1 Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. 1. Factor completely. Do not solve. a) 2x

More information

Chapter 10 Homework: Parametric Equations and Polar Coordinates

Chapter 10 Homework: Parametric Equations and Polar Coordinates Chapter 1 Homework: Parametric Equations and Polar Coordinates Name Homework 1.2 1. Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the

More information

The following information is for reviewing the material since Exam 3:

The following information is for reviewing the material since Exam 3: Outcomes List for Math 121 Calculus I Fall 2010-2011 General Information: The purpose of this Outcomes List is to give you a concrete summary of the material you should know, and the skills you should

More information

Exam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA:

Exam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA: MA 114 Exam 3 Spring 217 Exam 3 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test.

More information

3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS

3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS 3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )

More information

Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives

Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. By the definition of a derivative, we have Then we are really

More information

practice: quadratic functions [102 marks]

practice: quadratic functions [102 marks] practice: quadratic functions [102 marks] A quadratic function, f(x) = a x 2 + bx, is represented by the mapping diagram below. 1a. Use the mapping diagram to write down two equations in terms of a and

More information

The diagram above shows a sketch of the curve C with parametric equations

The diagram above shows a sketch of the curve C with parametric equations 1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate

More information

Curves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent

Curves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Planes, and Differentials ( 11.3-11.4) Feb. 26, 2012 (Sun) Signs of Partial Derivatives on Level Curves Level curves are shown for a function

More information

Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes

Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes 1 of 11 1) Give f(g(1)), given that Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes 2) Find the slope of the tangent line to the graph of f at x = 4, given that 3) Determine

More information

Final Exam Review Algebra Semester 1

Final Exam Review Algebra Semester 1 Final Exam Review Algebra 015-016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)

More information

Euler s Method for Approximating Solution Curves

Euler s Method for Approximating Solution Curves Euler s Method for Approximating Solution Curves As you may have begun to suspect at this point, time constraints will allow us to learn only a few of the many known methods for solving differential equations.

More information

13.1. Functions of Several Variables. Introduction to Functions of Several Variables. Functions of Several Variables. Objectives. Example 1 Solution

13.1. Functions of Several Variables. Introduction to Functions of Several Variables. Functions of Several Variables. Objectives. Example 1 Solution 13 Functions of Several Variables 13.1 Introduction to Functions of Several Variables Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives Understand

More information

Area and Volume. where x right and x left are written in terms of y.

Area and Volume. where x right and x left are written in terms of y. Area and Volume Area between two curves Sketch the region and determine the points of intersection. Draw a small strip either as dx or dy slicing. Use the following templates to set up a definite integral:

More information

Math 209, Fall 2009 Homework 3

Math 209, Fall 2009 Homework 3 Math 209, Fall 2009 Homework 3 () Find equations of the tangent plane and the normal line to the given surface at the specified point: x 2 + 2y 2 3z 2 = 3, P (2,, ). Solution Using implicit differentiation

More information

Jim Lambers MAT 169 Fall Semester Lecture 33 Notes

Jim Lambers MAT 169 Fall Semester Lecture 33 Notes Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 33 Notes These notes correspond to Section 9.3 in the text. Polar Coordinates Throughout this course, we have denoted a point in the plane by an ordered

More information

Multivariate Calculus: Review Problems for Examination Two

Multivariate Calculus: Review Problems for Examination Two Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double

More information

Differentiation. J. Gerlach November 2010

Differentiation. J. Gerlach November 2010 Differentiation J. Gerlach November 200 D and diff The limit definition of the derivative is covered in the Limit tutorial. Here we look for direct ways to calculate derivatives. Maple has two commands

More information

Exam 1 Review. MATH Intuitive Calculus Fall Name:. Show your reasoning. Use standard notation correctly.

Exam 1 Review. MATH Intuitive Calculus Fall Name:. Show your reasoning. Use standard notation correctly. MATH 11012 Intuitive Calculus Fall 2012 Name:. Exam 1 Review Show your reasoning. Use standard notation correctly. 1. Consider the function f depicted below. y 1 1 x (a) Find each of the following (or

More information

Sec.4.1 Increasing and Decreasing Functions

Sec.4.1 Increasing and Decreasing Functions U4L1: Sec.4.1 Increasing and Decreasing Functions A function is increasing on a particular interval if for any, then. Ie: As x increases,. A function is decreasing on a particular interval if for any,

More information

Precalculus 2 Section 10.6 Parametric Equations

Precalculus 2 Section 10.6 Parametric Equations Precalculus 2 Section 10.6 Parametric Equations Parametric Equations Write parametric equations. Graph parametric equations. Determine an equivalent rectangular equation for parametric equations. Determine

More information

SHOW ALL NEEDED WORK IN YOUR NOTEBOOK.

SHOW ALL NEEDED WORK IN YOUR NOTEBOOK. DO NOW: 1 3: NO CALCULATORS 1. Consider the function f () x the value of f (4.1)? SHOW ALL NEEDED WORK IN YOUR NOTEBOOK. x. We all know that f (4), but without a calculator, what is . The approximate value

More information

NAME: Section # SSN: X X X X

NAME: Section # SSN: X X X X Math 155 FINAL EXAM A May 5, 2003 NAME: Section # SSN: X X X X Question Grade 1 5 (out of 25) 6 10 (out of 25) 11 (out of 20) 12 (out of 20) 13 (out of 10) 14 (out of 10) 15 (out of 16) 16 (out of 24)

More information

Math 104, Spring 2010 Course Log

Math 104, Spring 2010 Course Log Math 104, Spring 2010 Course Log Date: 1/11 Sections: 1.3, 1.4 Log: Lines in the plane. The point-slope and slope-intercept formulas. Functions. Domain and range. Compositions of functions. Inverse functions.

More information

(1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives.

(1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives. Section 11.3 Partial Derivatives (1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives. MATH 127 (Section 11.3) Partial Derivatives The University of Kansas

More information

Math 2260 Exam #1 Practice Problem Solutions

Math 2260 Exam #1 Practice Problem Solutions Math 6 Exam # Practice Problem Solutions. What is the area bounded by the curves y x and y x + 7? Answer: As we can see in the figure, the line y x + 7 lies above the parabola y x in the region we care

More information

Math 131. Implicit Differentiation Larson Section 2.5

Math 131. Implicit Differentiation Larson Section 2.5 Math 131. Implicit Differentiation Larson Section.5 So far we have ealt with ifferentiating explicitly efine functions, that is, we are given the expression efining the function, such as f(x) = 5 x. However,

More information

Tangents of Parametric Curves

Tangents of Parametric Curves Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 32 Notes These notes correspond to Section 92 in the text Tangents of Parametric Curves When a curve is described by an equation of the form y = f(x),

More information

f sin the slope of the tangent line is given by f sin f cos cos sin , but it s also given by 2. So solve the DE with initial condition: sin cos

f sin the slope of the tangent line is given by f sin f cos cos sin , but it s also given by 2. So solve the DE with initial condition: sin cos Math 414 Activity 1 (Due by end of class August 1) 1 Four bugs are placed at the four corners of a square with side length a The bugs crawl counterclockwise at the same speed and each bug crawls directly

More information

True/False. MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY

True/False. MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY True/False 10 points: points each) For the problems below, circle T if the answer is true and circle F is the answer is false. After you ve chosen

More information

Look up partial Decomposition to use for problems #65-67 Do Not solve problems #78,79

Look up partial Decomposition to use for problems #65-67 Do Not solve problems #78,79 Franklin Township Summer Assignment 2017 AP calculus AB Summer assignment Students should use the Mathematics summer assignment to identify subject areas that need attention in preparation for the study

More information

Without fully opening the exam, check that you have pages 1 through 11.

Without fully opening the exam, check that you have pages 1 through 11. Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard

More information

302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES. 4. Function of several variables, their domain. 6. Limit of a function of several variables

302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES. 4. Function of several variables, their domain. 6. Limit of a function of several variables 302 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 3.8 Chapter Review 3.8.1 Concepts to Know You should have an understanding of, and be able to explain the concepts listed below. 1. Boundary and interior points

More information

Math 21a Tangent Lines and Planes Fall, What do we know about the gradient f? Tangent Lines to Curves in the Plane.

Math 21a Tangent Lines and Planes Fall, What do we know about the gradient f? Tangent Lines to Curves in the Plane. Math 21a Tangent Lines and Planes Fall, 2016 What do we know about the gradient f? Tangent Lines to Curves in the Plane. 1. For each of the following curves, find the tangent line to the curve at the point

More information

2. Solve for x when x < 22. Write your answer in interval notation. 3. Find the distance between the points ( 1, 5) and (4, 3).

2. Solve for x when x < 22. Write your answer in interval notation. 3. Find the distance between the points ( 1, 5) and (4, 3). Math 6 Practice Problems for Final. Find all real solutions x such that 7 3 x = 5 x 3.. Solve for x when 0 4 3x

More information

Multivariate Calculus Review Problems for Examination Two

Multivariate Calculus Review Problems for Examination Two Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,

More information

Daily WeBWorK, #1. This means the two planes normal vectors must be multiples of each other.

Daily WeBWorK, #1. This means the two planes normal vectors must be multiples of each other. Daily WeBWorK, #1 Consider the ellipsoid x 2 + 3y 2 + z 2 = 11. Find all the points where the tangent plane to this ellipsoid is parallel to the plane 2x + 3y + 2z = 0. In order for the plane tangent to

More information

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #3 - FALL DR. DAVID BRIDGE

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #3 - FALL DR. DAVID BRIDGE MATH 2053 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #3 - FALL 2007 - DR. DAVID BRIDGE MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a calculator

More information

Derivatives. Day 8 - Tangents and Linearizations

Derivatives. Day 8 - Tangents and Linearizations Derivatives Day 8 - Tangents and Linearizations Learning Objectives Write an equation for the tangent line to a graph Write an equation for the normal line to a graph Find the locations of horizontal and

More information

f xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is

f xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is 1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,

More information

Social Science/Commerce Calculus I: Assignment #6 - Solutions Page 1/14

Social Science/Commerce Calculus I: Assignment #6 - Solutions Page 1/14 Social Science/Commerce Calculus I: Assignment #6 - Solutions Page 1/14 3 1. Let f (x) = -2x - 5x + 1. Use the rules of differentiation to compute: 2 The first derivative of f (x): -6x - 5 The second derivative

More information

minutes/question 26 minutes

minutes/question 26 minutes st Set Section I (Multiple Choice) Part A (No Graphing Calculator) 3 problems @.96 minutes/question 6 minutes. What is 3 3 cos cos lim? h hh (D) - The limit does not exist.. At which of the five points

More information

Application 2.4 Implementing Euler's Method

Application 2.4 Implementing Euler's Method Application 2.4 Implementing Euler's Method One's understanding of a numerical algorithm is sharpened by considering its implementation in the form of a calculator or computer program. Figure 2.4.13 in

More information

Loki s Practice Sets for PUBP555: Math Camp Spring 2014

Loki s Practice Sets for PUBP555: Math Camp Spring 2014 Loki s Practice Sets for PUBP555: Math Camp Spring 2014 Contents Module 1... 3 Rounding Numbers... 3 Square Roots... 3 Working with Fractions... 4 Percentages... 5 Order of Operations... 6 Module 2...

More information

Math 113 Calculus III Final Exam Practice Problems Spring 2003

Math 113 Calculus III Final Exam Practice Problems Spring 2003 Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross

More information

During the timed portion for Part A, you may work only on the problems in Part A.

During the timed portion for Part A, you may work only on the problems in Part A. SECTION II Time: hour and 30 minutes Percent of total grade: 50 Part A: 45 minutes, 3 problems (A graphing calculator is required for some problems or parts of problems.) During the timed portion for Part

More information

Graphing Functions. 0, < x < 0 1, 0 x < is defined everywhere on R but has a jump discontinuity at x = 0. h(x) =

Graphing Functions. 0, < x < 0 1, 0 x < is defined everywhere on R but has a jump discontinuity at x = 0. h(x) = Graphing Functions Section. of your tetbook is devoted to reviewing a series of steps that you can use to develop a reasonable graph of a function. Here is my version of a list of things to check. You

More information

Slope of the Tangent Line. Estimating with a Secant Line

Slope of the Tangent Line. Estimating with a Secant Line Slope of the Tangent Line Given a function f find the slope of the line tangent to the graph of f, that is, to the curve, at the point P(a, f (a)). The graph of a function f and the tangent line at a point

More information

Math Exam 2a. 1) Take the derivatives of the following. DO NOT SIMPLIFY! 2 c) y = tan(sec2 x) ) b) y= , for x 2.

Math Exam 2a. 1) Take the derivatives of the following. DO NOT SIMPLIFY! 2 c) y = tan(sec2 x) ) b) y= , for x 2. Math 111 - Exam 2a 1) Take the derivatives of the following. DO NOT SIMPLIFY! a) y = ( + 1 2 x ) (sin(2x) - x- x 1 ) b) y= 2 x + 1 c) y = tan(sec2 x) 2) Find the following derivatives a) Find dy given

More information

Section 1: Section 2: Section 3: Section 4:

Section 1: Section 2: Section 3: Section 4: Announcements Topics: In the Functions of Several Variables module: - Section 1: Introduction to Functions of Several Variables (Basic Definitions and Notation) - Section 2: Graphs, Level Curves + Contour

More information

Directional Derivatives as Vectors

Directional Derivatives as Vectors Directional Derivatives as Vectors John Ganci 1 Al Lehnen 2 1 Richland College Dallas, TX jganci@dcccd.edu 2 Madison Area Technical College Madison, WI alehnen@matcmadison.edu Statement of problem We are

More information

To sketch the graph we need to evaluate the parameter t within the given interval to create our x and y values.

To sketch the graph we need to evaluate the parameter t within the given interval to create our x and y values. Module 10 lesson 6 Parametric Equations. When modeling the path of an object, it is useful to use equations called Parametric equations. Instead of using one equation with two variables, we will use two

More information

GRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.

GRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),

More information

Differentiability and Tangent Planes October 2013

Differentiability and Tangent Planes October 2013 Differentiability and Tangent Planes 14.4 04 October 2013 Differentiability in one variable. Recall for a function of one variable, f is differentiable at a f (a + h) f (a) lim exists and = f (a) h 0 h

More information

Tangent line problems

Tangent line problems You will find lots of practice problems and homework problems that simply ask you to differentiate. The following examples are to illustrate some of the types of tangent line problems that you may come

More information

5/27/12. Objectives. Plane Curves and Parametric Equations. Sketch the graph of a curve given by a set of parametric equations.

5/27/12. Objectives. Plane Curves and Parametric Equations. Sketch the graph of a curve given by a set of parametric equations. Objectives Sketch the graph of a curve given by a set of parametric equations. Eliminate the parameter in a set of parametric equations. Find a set of parametric equations to represent a curve. Understand

More information

MA FINAL EXAM INSTRUCTIONS VERSION 01 DECEMBER 9, Section # and recitation time

MA FINAL EXAM INSTRUCTIONS VERSION 01 DECEMBER 9, Section # and recitation time MA 6500 FINAL EXAM INSTRUCTIONS VERSION 0 DECEMBER 9, 03 Your name Student ID # Your TA s name Section # and recitation time. You must use a # pencil on the scantron sheet (answer sheet).. Check that the

More information

11.3 The Tangent Line Problem

11.3 The Tangent Line Problem 11.3 The Tangent Line Problem Copyright Cengage Learning. All rights reserved. What You Should Learn Understand the tangent line problem. Use a tangent line to approximate the slope of a graph at a point.

More information

Functions of Two variables.

Functions of Two variables. Functions of Two variables. Ferdinánd Filip filip.ferdinand@bgk.uni-obuda.hu siva.banki.hu/jegyzetek 27 February 217 Ferdinánd Filip 27 February 217 Functions of Two variables. 1 / 36 Table of contents

More information

University of California, Berkeley

University of California, Berkeley University of California, Berkeley FINAL EXAMINATION, Fall 2012 DURATION: 3 hours Department of Mathematics MATH 53 Multivariable Calculus Examiner: Sean Fitzpatrick Total: 100 points Family Name: Given

More information

4) Simplify 5( 6) Simplify. 8) Solve 1 x 2 4

4) Simplify 5( 6) Simplify. 8) Solve 1 x 2 4 Algebra Summer Assignment 1) Simplify x 4y 10 x ) Simplify 4y 6x 3( x y) 3) Simplify 1 3 ( x 3) 4) Simplify 5( x 9x) (3x 4) 7 5) Simplify ( x 3)( x ) 6) Simplify ( x 4) 3 7) Simplify ( 5x 8)(4x 1) 8) Solve

More information

Worksheet 2.2: Partial Derivatives

Worksheet 2.2: Partial Derivatives Boise State Math 275 (Ultman) Worksheet 2.2: Partial Derivatives From the Toolbox (what you need from previous classes) Be familiar with the definition of a derivative as the slope of a tangent line (the

More information

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total

More information

Lincoln County Schools Patriot Day Instructional Expectations Patriot Day 5 School: LCHS Course/Subject: AP Calculus Teacher: Susan Harris

Lincoln County Schools Patriot Day Instructional Expectations Patriot Day 5 School: LCHS Course/Subject: AP Calculus Teacher: Susan Harris Lincoln County Schools Patriot Day Instructional Expectations Patriot Day 5 School: LCHS Course/Subject: AP Calculus Teacher: Susan Harris Learning Target: 4..1 I can use implicit differentiation to find

More information

3 D DIFFERENTIATION RULES

3 D DIFFERENTIATION RULES 3 D DIFFERENTIATION RULES 3.1 Derivatives ofpolnomials and Eponential Functions h 1. (a) e is the number such that lim e h- 1 = 1. h----+o (b).7 X - 1-0.001 0.998-0.0001 0.993 0.001 0.9937 0.0001 0.9933.8

More information

Solved Examples. Parabola with vertex as origin and symmetrical about x-axis. We will find the area above the x-axis and double the area.

Solved Examples. Parabola with vertex as origin and symmetrical about x-axis. We will find the area above the x-axis and double the area. Solved Examples Example 1: Find the area common to the curves x 2 + y 2 = 4x and y 2 = x. x 2 + y 2 = 4x (i) (x 2) 2 + y 2 = 4 This is a circle with centre at (2, 0) and radius 2. y = (4x-x 2 ) y 2 = x

More information

Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal

Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to

More information

Math 205 Test 3 Grading Guidelines Problem 1 Part a: 1 point for figuring out r, 2 points for setting up the equation P = ln 2 P and 1 point for the initial condition. Part b: All or nothing. This is really

More information

27. Tangent Planes & Approximations

27. Tangent Planes & Approximations 27. Tangent Planes & Approximations If z = f(x, y) is a differentiable surface in R 3 and (x 0, y 0, z 0 ) is a point on this surface, then it is possible to construct a plane passing through this point,

More information

Practice problems from old exams for math 233 William H. Meeks III December 21, 2009

Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These

More information

MA 114 Worksheet #17: Average value of a function

MA 114 Worksheet #17: Average value of a function Spring 2019 MA 114 Worksheet 17 Thursday, 7 March 2019 MA 114 Worksheet #17: Average value of a function 1. Write down the equation for the average value of an integrable function f(x) on [a, b]. 2. Find

More information

Tangent Planes/Critical Points

Tangent Planes/Critical Points Tangent Planes/Critical Points Christopher Croke University of Pennsylvania Math 115 UPenn, Fall 2011 Problem: Find the tangent line to the curve of intersection of the surfaces xyz = 1 and x 2 + 2y 2

More information

REVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections

REVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections REVIEW I MATH 254 Calculus IV Exam I (Friday, April 29 will cover sections 14.1-8. 1. Functions of multivariables The definition of multivariable functions is similar to that of functions of one variable.

More information

AP * Calculus Review. Area and Volume

AP * Calculus Review. Area and Volume AP * Calculus Review Area and Volume Student Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production of,

More information

Use Derivatives to Sketch the Graph of a Polynomial Function.

Use Derivatives to Sketch the Graph of a Polynomial Function. Applications of Derivatives Curve Sketching (using derivatives): A) Polynomial Functions B) Rational Functions Lesson 5.2 Use Derivatives to Sketch the Graph of a Polynomial Function. Idea: 1) Identify

More information

(a) Find the equation of the plane that passes through the points P, Q, and R.

(a) Find the equation of the plane that passes through the points P, Q, and R. Math 040 Miterm Exam 1 Spring 014 S o l u t i o n s 1 For given points P (, 0, 1), Q(, 1, 0), R(3, 1, 0) an S(,, 0) (a) Fin the equation of the plane that passes through the points P, Q, an R P Q = 0,

More information

Properties of a Function s Graph

Properties of a Function s Graph Section 3.2 Properties of a Function s Graph Objective 1: Determining the Intercepts of a Function An intercept of a function is a point on the graph of a function where the graph either crosses or touches

More information

Math: Question 10

Math: Question 10 1 of 1 9/22/2016 7:55 PM Math: Question 10 A carpenter has $60 with which to buy supplies. The carpenter needs to buy both nails and screws. Nails cost $12.99 per box, and screws cost $14.99 per box. If

More information

SPRING 2015 Differentiation Practice (EXTRA PROBLEMS) 1

SPRING 2015 Differentiation Practice (EXTRA PROBLEMS) 1 SPRING 2015 Differentiation Practice (EXTRA PROBLEMS) 1 WARNING: These are EXTRA problems, which means you have to do all the homework, webassign and NYTI problems before doing this. Also You ll have to

More information

MAT1B01: Curves defined by parametric equations

MAT1B01: Curves defined by parametric equations MAT1B01: Curves defined by parametric equations Dr Craig 24 October 2016 My details: acraig@uj.ac.za Consulting hours: Thursday 11h20 12h55 Friday 11h30 13h00 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/

More information

Algebra 1 Notes Quarter

Algebra 1 Notes Quarter Algebra 1 Notes Quarter 3 2014 2015 Name: ~ 1 ~ Table of Contents Unit 9 Exponent Rules Exponent Rules for Multiplication page 6 Negative and Zero Exponents page 10 Exponent Rules Involving Quotients page

More information

AP Calculus BC. Find a formula for the area. B. The cross sections are squares with bases in the xy -plane.

AP Calculus BC. Find a formula for the area. B. The cross sections are squares with bases in the xy -plane. AP Calculus BC Find a formula for the area Homework Problems Section 7. Ax of the cross sections of the solid that are perpendicular to the x -axis. 1. The solid lies between the planes perpendicular to

More information

Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.

Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written. Math 2374 Spring 2006 Final May 8, 2006 Time Limit: 1 Hour Name (Print): Student ID: Section Number: Teaching Assistant: Signature: This exams contains 11 pages (including this cover page) and 10 problems.

More information

. The differential of y f (x)

. The differential of y f (x) Calculus I - Prof D Yuen Exam Review version 11/14/01 Please report any typos Derivative Rules Of course you have to remember all your derivative rules Implicit Differentiation Differentiate both sides

More information

MAT01B1: Curves defined by parametric equations

MAT01B1: Curves defined by parametric equations MAT01B1: Curves defined by parametric equations Dr Craig 10 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/

More information