Quiz 4A Solutions. Math 150 (62493) Spring Name: Instructor: C. Panza
|
|
- Oswald Hopkins
- 5 years ago
- Views:
Transcription
1 Math 150 (62493) Spring 2019 Quiz 4A Solutions Instructor: C. Panza Quiz 4A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). Answer with a fraction unless instructed to round. (5 pts ) 1. Use the limit definition to compute f (a) for f(x) = x + 3 at a = 2. f f(a + h) f(a) (a) h 0 h f f( 2 + h) f( 2) ( 2) h 0 h h h 0 h h h 0 h ( h ) h h 0 h ( h ) h 0 1 h = 1 2 h h (5 pts ) 2. Use the limit definition to compute f (a) and find an equation of the tangent line at x = a. f(x) = x 2 2x, a = 3 f (a) x a f(x) f(a) x a f (3) x 3 f(x) f(3) x 3 x 2 2x 3 x 3 x 3 (x 3)(x + 1) x 3 x 3 (x + 1) x 3 = 4 Equation of tangent line: y f(3) = f (3)(x 3) y 3 = 4(x 3) y = 4x 9
2 Math 150 (62493)/Quiz 4A Solutions Page 2 of 2 (5 pts ) 3. Use the power rule to compute the derivative of f(x) = 2x 2 6x + 9 and find the equation of the tangent line at x = 1. Derivative: f (x) = 4x 6 Tangent line: f(1) = 5 f (1) = 2 y 5 = 2(x 1) y = 2x + 7 (5 pts ) 4. Use the power rule to compute the derivative. g(x) = 2x5 x 3 + 5x 1 x 2 g(x) = 2x 3 x + 5x 3 g (x) = 6x x 4
3 Math 150 (62493) Spring 2019 Quiz 4B Solutions Instructor: C. Panza Quiz 4B Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). Answer with a fraction unless instructed to round. (5 pts ) 1. Use the limit definition to compute f (a) for f(x) = x 2 at a = 3. f f(a + h) f(a) (a) h 0 h f f(3 + h) f(3) ( 2) h 0 h 1 + h 1 h 0 h 1 + h 1 h 0 h ( 1 + h + 1 ) h h 0 h ( h ) h 0 1 h = h h + 1 (5 pts ) 2. Use the limit definition to compute f (a) and find an equation of the tangent line at x = a. f(x) = x 2 + 3x, a = 1 f (a) x a f(x) f(a) x a f (1) x 1 f(x) f(1) x 1 x 2 + 3x 4 x 1 x 1 (x 1)(x + 4) x 1 x 1 (x + 4) x 1 = 5 Equation of tangent line: y f(1) = f (21)(x 1) y 4 = 5(x 1) y = 5x 1
4 Math 150 (62493)/Quiz 4B Solutions Page 2 of 2 (5 pts ) 3. Use the power rule to compute the derivative of f(x) = 2x 2 + 8x 2 and find the equation of the tangent line at x = 1. Derivative: f (x) = 4x + 8 Tangent line: f( 1) = 8 f ( 1) = 4 y ( 8) = 4(x ( 1)) y = 4x 4 (5 pts ) 4. Use the power rule to compute the derivative. g(x) = x6 5x 4 2x 1 x 3 g(x) = x 3 5x 2x 4 g (x) = 3x x 5
5 Math 150 (62493) Spring 2019 Quiz 5A Solutions Instructor: C. Panza Quiz 5A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). Answer word problems with a sentence. (5 pts ) 1. Use the product rule to compute the derivative of f(x) = x 4 e x. f (x) = e x d dx x4 + x 4 d dx ex = 4x 3 e x + x 4 e x = e x x 3 (x + 4) (5 pts ) 2. Use the quotient rule to compute the derivative of f(t) = t t 2 3. f (t) = (t2 3) d dt t t d dt (t2 3) (t 2 3) 2 = (t2 3) 1 t (2t) (t 2 3) 2 = t2 3 (t 2 3) 2
6 Math 150 (62493)/Quiz 5A Solutions Page 2 of 2 (5 pts ) 3. A ball tossed in the air vertically from ground level returns to the earth 4 seconds later. Find the maximum height (in meters) and the initial velocity (in m/sec) of the ball. [Use g = 9.8 m/s 2.] At 2 seconds the ball is at its maximum height with a velocity of 0 m/sec. v(2) = v 0 9.8(2) 0 = v = v 0 s(t) = t 4.9t 2 s(2) = 19.6(2) 4.9(2) 2 = 19.6 The ball had a maximum height of 19.6 m an initial velocity of 19.6 m/sec. (5 pts ) 4. Given f(x) = x 3 e x, find f (x) and evaluate f (1). Round your answer to two decimal places. f (x) = 3x 2 e x + x 3 e x f (x) = 6xe x + 3x 2 e x + 3x 2 e x + x 3 e x = 6xe x + 6x 2 e x + x 3 e x f (1) = 6(1)e 1 + 6(1) 2 e e 1 = 13e 35.34
7 Math 150 (62493) Spring 2019 Quiz 5B Solutions Instructor: C. Panza Quiz 5B Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). Answer word problems with a sentence. (5 pts ) 1. Use the product rule to compute the derivative of f(x) = x 3 e x. f (x) = e x d dx x3 + x 3 d dx ex = 3x 2 e x + x 3 e x = e x x 2 (x + 3) (5 pts ) 2. Use the quotient rule to compute the derivative of f(t) = t t f (t) = (t2 + 2) d dt t t d dt (t2 + 2) (t 2 + 2) 2 = (t2 + 2) 1 t (2t) (t 2 + 2) 2 = 2 t2 (t 2 + 2) 2
8 Math 150 (62493)/Quiz 5B Solutions Page 2 of 2 (5 pts ) 3. A ball tossed in the air vertically from ground level returns to the earth 6 seconds later. Find the maximum height (in meters) and the initial velocity (in m/sec) of the ball. [Use g = 9.8 m/s 2.] At 3 seconds the ball is at its maximum height with a velocity of 0 m/sec. v(3) = v 0 9.8(3) 0 = v = v 0 s(t) = t 4.9t 2 s(3) = 29.4(3) 4.9(3) 2 = 44.1 The ball had a maximum height of 44.1 m and an initial velocity of 29.4 m/sec. (5 pts ) 4. Given f(x) = x 1 e x, find f (x) and evaluate f (1). Round your answer to two decimal places. f (x) = x 2 e x + x 1 e x f (x) = 2x 3 e x + ( x 2 e x) + ( x 2 e x) + x 1 e x = 2x 3 e x 2x 2 e x + x 1 e x f (1) = 2(1) 3 e 1 2(1) 2 e e 1 = e 2.72
9 Math 150 (62493) Spring 2019 Quiz 6A Solutions Instructor: C. Panza Quiz 6A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). (5 pts ) 1. Compute the derivative using the Chain rule. f(x) = e x4 5x f (x) = ( ) e x4 5x (4x 3 5 ) = e x4 5x ( 4x 3 5 ) (5 pts ) 2. Compute the derivative using the Chain Rule. y = ( ) 3 sin 1 (4x) 1 ( ) ( ) 2 y = 3 sin 1 1 (4x) (4x) 2 = ( ) 2 12 sin 1 (4x) x 2
10 Math 150 (62493)/Quiz 6A Solutions Page 2 of 2 (5 pts ) 3. Compute the higher derivative y. y = sec x y = d ( ) 1 dx cos x 0 ( sin x) = cos 2 x = sec x tan x y = tan x d d sec x + sec x dx dx tan x = tan x sec x tan x + sec x sec 2 x = sec x tan 2 x + sec 3 x = sec x ( tan 2 x + sec 2 x ) (5 pts ) 4. Find the derivative with respect to x using implicit differentiation. Solve for y. 2x 3 + cos y = sin y d ( 2x 3 + cos y ) = d dx dx sin y 6x 2 sin yy = cos yy 6x 2 cos y + sin y = y 6x 2 = cos yy + sin yy 6x 2 = y (cos y + sin y)
11 Math 150 (62493) Spring 2019 Quiz 6B Solutions Instructor: C. Panza Quiz 6B Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). (5 pts ) 1. Compute the derivative using the Chain rule. f(x) = e 2x4 +3x f (x) = ( ) e 2x4 +3x (8x ) = ( 8x ) e 2x4 +3x (5 pts ) 2. Compute the derivative using the Chain Rule. y = ( ) sin 1 (3x) ( ) ( ) 4 y = sin 1 1 (3x) 3 1 (3x) 2 = ( sin 1 (3x) 1 9x 2 ) 4
12 Math 150 (62493)/Quiz 6B Solutions Page 2 of 2 (5 pts ) 3. Compute the higher derivative y. y = sec x y = d ( ) 1 dx cos x 0 ( sin x) = cos 2 x = sec x tan x y = tan x d d sec x + sec x dx dx tan x = tan x sec x tan x + sec x sec 2 x = sec x tan 2 x + sec 3 x = sec x ( tan 2 x + sec 2 x ) (5 pts ) 4. Find the derivative with respect to x using implicit differentiation. Solve for y. sin y = 4x 2 cos y d dx sin y = d ( 4x 2 cos y ) dx cos yy = 8x + sin yy cos yy sin yy = 8x y (cos y sin y) = 8x y 8x = cos y sin y
13 Math 150 (62493) Spring 2019 Quiz 7A Solutions Instructor: C. Panza Quiz 7A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). (5 pts ) 1. Find the derivative using logarithmic differentiation. (2x + 5)3 y = (x 3 8) 2 [ ] (2x + 5) 3 ln y = ln (x 3 8) 2 ln y = 3 ln (2x + 5) 2 ln ( x 3 8 ) y y = 6 2x + 5 y = (2x + 5)3 (x 3 8) 2 6x2 x 3 8 ( 6 2x + 5 ) 6x2 x 3 8 (5 pts ) 2. Calculate the derivative of f(x) = x 2x4. y = x 2x4 ln y = ln x 2x4 ln y = 2x 4 ln x y y = 8x3 ln x + 2x 4 1 x y y = 8x3 ln x + 2x 3 y = y ( 8x 3 ln x + 2x 3) f (x) = x 2x4 ( 8x 3 ln x + 2x 3) f (x) = 2x 3 x 2x4 (4 ln x + 1) f (x) = 2x 2x4 +3 (4 ln x + 1)
14 Math 150 (62493)/Quiz 7A Solutions Page 2 of 2 (5 pts ) 3. A sailboat 4 kilometers from shore passes by SBCC traveling parallel to the beach at 3 kilometers per hour. How fast is the distance between SBCC and the sailboat changing 30 minutes later? x = the sailboat s horizontal distance from SBCC z = the distance from SBCC to the sailboat 4 km x z z = x 2 dz dt = 2x x dx 2 dt At t = 30 minutes, ( ) 30 x = 3 = km dx dt = 3 km/hr dz dt = 3/ (3/2) The distance between SBCC and the sailboat is changing at about 1.05 km/hr. (5 pts ) 4. A man of height 1.8 m walks away from a 4-meter lamppost. After 2 seconds, his shadow is increasing at a rate of 1 meter per second. How fast is the man walking? x = the distance between the man and lamppost y = the length of the man s shadow 4 m x The man is walking about 1.2 meters per second. 1.8 y 4 x + y = 1.8 y 4y = 1.8(x + y) 4 dy ( dx dt = 1.8 dt + dy ) dt 2.2 dy dt = 1.8dx dt At t = 2 seconds, dy dt = 1 m/s 2.2(1) = 1.8 dx dt dx dt
15 Math 150 (62493) Spring 2019 Quiz 7B Solutions Instructor: C. Panza Quiz 7B Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality, correctly using the equals sign (=). (5 pts ) 1. Find the derivative using logarithmic differentiation. (3x 7)4 y = (x 2 8) 3 [ ] (3x 7) 4 ln y = ln (x 2 8) 3 ln y = 4 ln (3x 7) 3 ln ( x 2 8 ) y y = 12 3x 7 y = (3x 7)4 (x 2 8) 3 6x x 2 8 ( 12 3x 7 6x ) x 2 8 (5 pts ) 2. Calculate the derivative of f(x) = x 4x3. y = x 4x3 ln y = ln x 4x3 ln y = 4x 3 ln x y y = 12x2 ln x + 4x 3 1 x y y = 12x2 ln x + 4x 2 y = y ( 12x 2 ln x + 4x 2) f (x) = x 4x3 ( 12x 2 ln x + 4x 2) f (x) = 4x 2 x 4x3 (3 ln x + 1) f (x) = 4x 4x3 +2 (3 ln x + 1)
16 Math 150 (62493)/Quiz 7B Solutions Page 2 of 2 (5 pts ) 3. A plane passes directly overhead SBCC at 800 kilometers per hour at a height of 3 kilometers above ground. How fast is the distance between SBCC and the plane changing one minute later? x = the plane s horizontal distance from SBCC z = the distance from SBCC to the plane 3 km x z z = x 2 dz dt = 2x x dx 2 dt At t = 1 minute, ( ) 1 x = 800 = km dx = 800 km/hr dt dz dt = 40/ (40/3) The distance between SBCC and the plane is changing at about km/hr. (5 pts ) 4. A woman of height 1.6 m walks away from a 5-meter lamppost. After 2 seconds, her shadow is increasing at a rate of 1 meter per second. How fast is the woman walking? x = the distance between the woman and lamppost y = the length of the woman s shadow 5 m x The woman is walking about 2.2 meters per second. 1.6 y 5 x + y = 1.6 y 5y = 1.6(x + y) 5 dy ( dx dt = 1.6 dt + dy ) dt 3.4 dy dt = 1.6dx dt At t = 2 seconds, dy dt = 1 m/s 3.4(1) = 1.6 dx dt = dx dt
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)
Final Exam Review AP Calculus AB 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) Use the graph to evaluate the limit. 2) lim x
More informationMultiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationMath 131 Exam 2 Spring 2016
Math 3 Exam Spring 06 Name: ID: 7 multiple choice questions worth 4.7 points each. hand graded questions worth 0 points each. 0. free points (so the total will be 00). Exam covers sections.7 through 3.0
More informationName 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 informationFind all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) =
Math 90 Final Review Find all points where the function is discontinuous. ) Find all vertical asymptotes of the given function. x(x - ) 2) f(x) = x3 + 4x Provide an appropriate response. 3) If x 3 f(x)
More informationFind 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 informationMath 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 informationMath 2413 General Review for Calculus Last Updated 02/23/2016
Math 243 General Review for Calculus Last Updated 02/23/206 Find the average velocity of the function over the given interval.. y = 6x 3-5x 2-8, [-8, ] Find the slope of the curve for the given value of
More information1 + 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 informationMATH 151, SPRING 2018
MATH 151, SPRING 2018 COMMON EXAM II - VERSIONBKEY LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. TURN OFF
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2014 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2015 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6 C) - 12 (6x - 7)3
Part B- Pre-Test 2 for Cal (2.4, 2.5, 2.6) Test 2 will be on Oct 4th, chapter 2 (except 2.6) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationMATH 1070 Test 1 Spring 2014 Version A Calc Student s Printed Name: Key & Grading Guidelines CUID:
Student s Printed Name: Key & Grading Guidelines CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell
More informationMATH 1207 R02 MIDTERM EXAM 2 SOLUTION
MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find
More information1. 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 informationSpring 2015 Sample Final Exam
Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than
More informationMath Exam 02 Review
Math 10350 Exam 02 Review 1. A differentiable function g(t) is such that g(2) = 2, g (2) = 1, g (2) = 1/2. (a) If p(t) = g(t)e t2 find p (2) and p (2). (Ans: p (2) = 7e 4 ; p (2) = 28.5e 4 ) (b) If f(t)
More informationTest 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 informationDRAFT - 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 information1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim
Spring 10/MAT 250/Exam 1 Name: Show all your work. 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 1 +f(x) = lim x 3 f(x) = lim x
More informationdollars for a week of sales t weeks after January 1. What is the total revenue (to the nearest hundred dollars) earned from t = 10 to t = 16?
MATH 7 RIOHONDO SPRING 7 TEST (TAKE HOME) DUE 5//7 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) A department store has revenue from the sale
More informationReview 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 informationMA 113 Calculus I Fall 2017 Exam 1 Tuesday, 19 September Multiple Choice Answers. Question
MA 113 Calculus I Fall 2017 Exam 1 Tuesday, 19 September 2017 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationThe Princeton Review AP Calculus BC Practice Test 2
0 The Princeton Review AP Calculus BC Practice Test CALCULUS BC SECTION I, Part A Time 55 Minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each
More informationFind the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis
Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais
More information(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
More informationMath 231 Final Exam Review
Math Final Eam Review Find the equation of the line tangent to the curve 4y y at the point (, ) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y 4 y 4 Find the eact
More informationMath 2413 t2rsu14. Name: 06/06/ Find the derivative of the following function using the limiting process.
Name: 06/06/014 Math 413 trsu14 1. Find the derivative of the following function using the limiting process. f( x) = 4x + 5x. Find the derivative of the following function using the limiting process. f(
More informationAP Calculus AB Chapter 2 Test Review #1
AP Calculus AB Chapter Test Review # Open-Ended Practice Problems:. Nicole just loves drinking chocolate milk out of her special cone cup which has a radius of inches and a height of 8 inches. Nicole pours
More informationMath 113/114 Lecture 22
Math 113/114 Lecture 22 Xi Chen 1 1 University of Alberta October 31, 2014 Outline 1 2 (Application of Implicit Differentiation) Given a word problem about related rates, we need to do: interpret the problem
More informationSolutions to Second Midterm(pineapple)
Math 125 Solutions to Second Midterm(pineapple) 1. Compute each of the derivatives below as indicated. 4 points (a) f(x) = 3x 8 5x 4 + 4x e 3. Solution: f (x) = 24x 7 20x + 4. Don t forget that e 3 is
More informationStudent s Printed Name:
MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this eam. If ou consider a question to
More informationCalculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016
Name: Class: Date: Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016 Short Answer 1. Decide whether the following problem can be solved using precalculus, or whether calculus is required.
More informationAP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm.
AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm. Name: Date: Period: I. Limits and Continuity Definition of Average
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
More informationStudent s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any
More informationReview: A Cross Section of the Midterm. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review: A Cross Section of the Midterm Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it eists. 4 + ) lim - - ) A) - B) -
More informationMATH 1242 FINAL EXAM Spring,
MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t
More information3.4 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 informationx+1 e 2t dt. h(x) := Find the equation of the tangent line to y = h(x) at x = 0.
Math Sample final problems Here are some problems that appeared on past Math exams. Note that you will be given a table of Z-scores for the standard normal distribution on the test. Don t forget to have
More informationKey- Math 231 Final Exam Review
Key- Math Final Eam Review Find the equation of the line tangent to the curve y y at the point (, ) y-=(-/)(-) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y y (ysiny+y)/(-siny-y^-^)
More informationPurdue University Study Guide for MA Credit Exam
Purdue University Study Guide for MA 16010 Credit Exam Students who pass the credit exam will gain credit in MA16010. The credit exam is a two-hour long exam with multiple choice questions. No books or
More informationMATH 151, Fall 2013, Week 10-2, Section 4.5, 4.6
MATH 151, Fall 2013, Week 10-2, Section 4.5, 4.6 Recall the derivative of logarithmic and exponential functions. Theorem 1 (ln x) = (ln f(x)) = (log a x) = (log a f(x)) = Theorem 2 (a x ) = (a f(x) ) =
More informationStudent s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, smart
More informationMATH 019: Final Review December 3, 2017
Name: MATH 019: Final Review December 3, 2017 1. Given f(x) = x 5, use the first or second derivative test to complete the following (a) Calculate f (x). If using the second derivative test, calculate
More informationIF you participate fully in this boot camp, you will get full credit for the summer packet.
18_19 AP Calculus BC Summer Packet NOTE - Please mark July on your calendars. We will have a boot camp in my room from 8am 11am on this day. We will work together on the summer packet. Time permitting,
More informationFINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed.
Math 150 Name: FINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed. 135 points: 45 problems, 3 pts. each. You
More informationAnnouncements. Topics: Homework: - sections 4.5 and * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook
More informationMath 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 informationMath 180, Exam 2, Spring 2013 Problem 1 Solution
Math 80, Eam, Spring 0 Problem Solution. Find the derivative of each function below. You do not need to simplify your answers. (a) tan ( + cos ) (b) / (logarithmic differentiation may be useful) (c) +
More information) # ( 281). Give units with your answer.
Math 120 Winter 2009 Handout 17: In-Class Review for Exam 2 The topics covered by Exam 2 in the course include the following: Implicit differentiation. Finding formulas for tangent lines using implicit
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More informationFinal Examination 201-NYA-05 May 18, 2018
. ( points) Evaluate each of the following limits. 3x x + (a) lim x x 3 8 x + sin(5x) (b) lim x sin(x) (c) lim x π/3 + sec x ( (d) x x + 5x ) (e) lim x 5 x lim x 5 + x 6. (3 points) What value of c makes
More informationSolutions to Math 41 First Exam October 15, 2013
Solutions to Math 41 First Exam October 15, 2013 1. (16 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether
More information1 The Derivative and Differrentiability
1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationUNIT 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 informationMath 1131Q Section 10
Math 1131Q Section 10 Section 3.9 and 3.10 Oct 19, 2010 Find the derivative of ln 3 5 e 2 ln 3 5 e 2 = ln 3 + ln 5/2 + ln e 2 = 3 ln + ( 5 ) ln + 2 2 (ln 3 5 e 2 ) = 3 + 5 2 + 2 Find the derivative of
More informationAP Calculus Testbank (Chapter 6) (Mr. Surowski)
AP Calculus Testbank (Chapter 6) (Mr. Surowski) Part I. Multiple-Choice Questions 1. Suppose that f is an odd differentiable function. Then (A) f(1); (B) f (1) (C) f(1) f( 1) (D) 0 (E). 1 1 xf (x) =. The
More information1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve
MAT 11 Solutions TH Eam 3 1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: Therefore, d 5 5 d d 5 5 d 1 5 1 3 51 5 5 and 5 5 5 ( ) 3 d 1 3 5 ( ) So the
More informationPractice 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 informationStudent s Printed Name:
Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, smart
More informationMA 113 Calculus I Fall 2013 Exam 3 Tuesday, 19 November Multiple Choice Answers. Question
MA 113 Calculus I Fall 2013 Exam 3 Tuesday, 19 November 2013 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions
More informationMath Practice Final - solutions
Math 151 - Practice Final - solutions 2 1-2 -1 0 1 2 3 Problem 1 Indicate the following from looking at the graph of f(x) above. All answers are small integers, ±, or DNE for does not exist. a) lim x 1
More informationSection 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 informationProblem Worth Score Total 14
MATH 241, Fall 14 Extra Credit Preparation for Final Name: INSTRUCTIONS: Write legibly. Indicate your answer clearly. Revise and clean up solutions. Do not cross anything out. Rewrite the page, I will
More informationSection 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 informationMATH 2203 Exam 3 Version 2 Solutions Instructions mathematical correctness clarity of presentation complete sentences
MATH 2203 Exam 3 (Version 2) Solutions March 6, 2015 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.
More informationMATH 2053 Calculus I Review for the Final Exam
MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x
More informationAP Calculus AB Semester 1 Practice Final
Class: Date: AP Calculus AB Semester 1 Practice Final Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the limit (if it exists). lim x x + 4 x a. 6
More informationCalculating 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 informationMath 106 Answers to Exam 3a Fall 2015
Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical
More informationMath 112 (Calculus I) Final Exam
Name: Student ID: Section: Instructor: Math 112 (Calculus I) Final Exam Dec 18, 7:00 p.m. Instructions: Work on scratch paper will not be graded. For questions 11 to 19, show all your work in the space
More informationFinal Exam. Math 3 December 7, 2010
Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationCalculus is Cool. Math 160 Calculus for Physical Scientists I Exam 1 September 18, 2014, 5:00-6:50 pm. NAME: Instructor: Time your class meets:
NAME: Instructor: Time your class meets: Math 160 Calculus for Physical Scientists I Exam 1 September 18, 2014, 5:00-6:50 pm How can it be that mathematics, being after all a product of human thought independent
More information2. Find the midpoint of the segment that joins the points (5, 1) and (3, 5). 6. Find an equation of the line with slope 7 that passes through (4, 1).
Math 129: Pre-Calculus Spring 2018 Practice Problems for Final Exam Name (Print): 1. Find the distance between the points (6, 2) and ( 4, 5). 2. Find the midpoint of the segment that joins the points (5,
More informationMath 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
More informationStudent s Printed Name:
Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this exam. If you consider a question to be ambiguous, state your assumptions in the margin and do the best you
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationMath 190 Chapter 3 Lecture Notes. Professor Miguel Ornelas
Math 190 Chapter 3 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 190 Lecture Notes Section 3.1 Section 3.1 Derivatives of Polynomials an Exponential Functions Derivative of a Constant Function
More informationChapter 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 informationSample Questions Exam II, FS2009 Paulette Saab Calculators are neither needed nor allowed.
Sample Questions Exam II, FS2009 Paulette Saab Calculators are neither needed nor allowed. Part A: (SHORT ANSWER QUESTIONS) Do the following problems. Write the answer in the space provided. Only the answers
More informationMath 251 Lecture Notes
Lecture Notes 2.1: Average Rates of Change Example 1. A population of bees was happily residing in someone s backyard a few years ago. Unfortunately, the population died off before the year s end. Let
More informationThe above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.
Math 7A Practice Midterm III Solutions Ch. 6-8 (Ebersole,.7-.4 (Stewart DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You
More informationMA4001 Engineering Mathematics 1 Lecture 15 Mean Value Theorem Increasing and Decreasing Functions Higher Order Derivatives Implicit Differentiation
MA4001 Engineering Mathematics 1 Lecture 15 Mean Value Theorem Increasing and Decreasing Functions Higher Order Derivatives Implicit Differentiation Dr. Sarah Mitchell Autumn 2014 Rolle s Theorem Theorem
More informationYour exam contains 5 problems. The entire exam is worth 70 points. Your exam should contain 6 pages; please make sure you have a complete exam.
MATH 124 (PEZZOLI) WINTER 2017 MIDTERM #2 NAME TA:. Section: Instructions: Your exam contains 5 problems. The entire exam is worth 70 points. Your exam should contain 6 pages; please make sure you have
More informationPart A: Short Answer Questions
Math 111 Practice Exam Your Grade: Fall 2015 Total Marks: 160 Instructor: Telyn Kusalik Time: 180 minutes Name: Part A: Short Answer Questions Answer each question in the blank provided. 1. If a city grows
More informationMath 1501 Calc I Fall 2013 Lesson 9 - Lesson 20
Math 1501 Calc I Fall 2013 Lesson 9 - Lesson 20 Instructor: Sal Barone School of Mathematics Georgia Tech August 19 - August 6, 2013 (updated October 4, 2013) L9: DIFFERENTIATION RULES Covered sections:
More informationIf y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy. du du. If y = f (u) then y = f (u) u
Section 3 4B The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy du du dx or If y = f (u) then f (u) u The Chain Rule with the Power
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More informationWorkbook 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 informationPlease do not start working until instructed to do so. You have 50 minutes. You must show your work to receive full credit. Calculators are OK.
Loyola University Chicago Math 131, Section 009, Fall 2008 Midterm 2 Name (print): Signature: Please do not start working until instructed to do so. You have 50 minutes. You must show your work to receive
More informationUNIT 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 informationMth 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 informationEdexcel past paper questions. Core Mathematics 4. Parametric Equations
Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran Email: kvkumaran@gmail.com C4 Maths Parametric equations Page 1 Co-ordinate Geometry A parametric equation of
More informationMthSc 107 Test 1 Spring 2013 Version A Student s Printed Name: CUID:
Student s Printed Name: CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More information