1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,"

Transcription

1 1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s) of x where f is not differentiable but is continuous. x = 2, x = 3 c. Find the value(s) of x where f is not continuous, but has a limit. x = 1, x = 4 d. Find the value(s) of x where f does not have a limit. x = 4, x = 3, x = 1, x = 2, x = 5

2 2. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 5, x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3 x = 4, x = 5 b. Find the value(s) of x where f is not differentiable but is continuous. x = 1, x = 2, x = 3, x = 4 c. Find the value(s) of x where f is not continuous, but has a limit. x = 5, x = 3, x = 1 d. Find the value(s) of x where f does not have a limit. x = 4, x = 2, x = 5

3 3. Find f (x), or, as appropriate: dx a. f(x) = 3 f (x) = 0 b. f(x) = 4x + 1 f (x) = 4 c. f(x) = 2x 2 3x + 1 f (x) = 4x 3 d. f(x) = 3x3 4 f (x) = 9 4 x2 e. f(x) = x 1 f (x) = 1 2 x f. f(x) = x x x f (x) = 5 2 x3/ x x g. f(x) = e x f (x) = e x h. f(x) = e f (x) = 0 i. f(x) = 3xe x f (x) = 3e x + 3xe x x + 1 j. y = 3x 4 dx = (3x 4)( 1 2 x ) ( x + 1)(3) (3x 4) 2 = 3x 6 x 4 (3x 4) 2 (2 x) k. y = (4x + 3) 3 dx = 3(4x + 3)2 (4) = 12(4x + 3) 2 l. y = (x 2 3x 2) 5 dx = 5(x2 3x 2) 4 (2x 3) m. f(x) = 4x 3 2x 2 + 4x + 5 f (x) = n. f(x) = e 3x 1 f (x) = 3e 3x 1 o. y = e x2 dx = 2 2xex p. y = ln x dx = 1 x q. y = ln(x 2 + 4x 1) 12x 2 4x x 3 2x 2 + 4x + 5 = 6x 2 2x + 2 4x3 2x 2 + 4x + 5

4 dx = 2x + 4 x 2 + 4x 1 r. y = ln 1 x dx = 1 x s. y = ln(ln x) dx = 1 x ln x t. y = 1 ln x dx = 1 x(ln x) 2 u. y = sin x f (x) = cos x ( ) 3π v. f(x) = sin 4 f (x) = 0 w. f(x) = 3x sin x + sin x cos x f (x) = 3 sin x + 3x cos x + cos 2 x sin 2 x x. f(x) = 3x 2 e x + 4x tan x f (x) = 6xe x + 3x 2 e x + 4x sec 2 x + 4 tan x y. f(x) = sin x cos xe x f (x) = sin x cos xe x sin 2 xe x + cos 2 xe x z. y = 3x 2 sin x cos xe x dx = 6x sin x cos xex + 3x 2 cos 2 xe x 3x 2 sin 2 xe x + 3x 2 sin x cos xe x aa. y = e 4 sin(x2 1) sin(x = 2 1) ( e4 4 cos ( x 2 1 ) 2x ) = 8xe 4 1) ( sin(x2 cos ( x 2 1 )) dx bb. y = 3ex sin x cos x 4e x dx = (cos x 4ex )(3e x cos x) (3e x sin x)( sin x 4e x ) (cos x 4e x ) 2 = (cos x 4ex )(3e x cos x) + (3e x sin x)(sin x + 4e x ) (cos x 4e x ) 2 cc. y = 9x2 e x + 4x cos x 2 sin x + xe x (2 sin x + xe x )[9(2xe x + x 2 e x ) + 4(cos x x sin x)] = dx (2 sin x + (9x2 e x x e + 4x cos x)(2 cos x + 2 x + xe x ) xe x ) 2 (2 sin x + xe x ) 2 dd. y = ex sin x x cos x 3x cos x + 4xe x dx = (3x cos x + 4xex )(e x cos x + e x sin x cos x + x sin x) (3x cos x + 4xe x ) 2 (ex sin x x cos x)[3(cos x x sin x) + 4e x + 4xe x ] (3x cos x + 4xe x ) 2 ee. f(x) = sin(3x 2 + 1) f (x) = 6x cos(3x 2 + 1) ff. f(x) = cos (sin (4x 2)) f (x) = 4 sin(sin(4x 2))(cos(4x 2)) gg. y = sin(e 3x )

5 dx = cos ( e 3x) 3e 3x hh. f(x) = sin(4x) f (x) = 4 cos 4x ii. y = sin 3 x dx = 3 sin2 x cos x jj. y = cos 4 (x 2 + 1) dx = 8x cos3 (x 2 + 1)(sin(x 2 + 1)) kk. y = e cos2 x 1 = 2 sin x cos 2 x 1 xecos dx ll. f(x) = tan 3 ( e 2x 1) f (x) = 6 tan 2 (e 2x 1 )(sec 2 (e 2x 1 ))(e 2x 1 ) mm. f(x) = cos(2x 3 4)e cos x f (x) = e cos x ( sin(2x 3 4)(6x 2 )) + cos(2x 3 4)(e cos x )( sin x) nn. f(x) = x tan(x 2 + 1)e 3x f (x) = tan(x 2 + 1)e 3x + x[sec 2 (x 2 + 1)(2x)e 3x + 3e 3x tan(x 2 + 1)] oo. f(x) = x 2 e sin x2 + e x sin(cos(3x 1)) f (x) = 2xe sin(x2) + x 2 (e sin(x2) )(cos(x 2 ))(2x)+ e x (cos(cos(3x 1)))( sin(3x 1)(3)) + sin(cos(3x 1))e x pp. y = e + 3xex2 2 e 4 sin x cos 2 x dx = (e 4 sin x cos2 x)[3xe x 2 2 (2x) + 3e x2 2 ] (e + 3xe x2 2 )[ 4 cos 3 x + 8 sin 2 x cos x] (e 4 sin x cos 2 x) 2 qq. y = tan 5 ( sec 3 ( x )) dx = 5 ( tan4 sec 3 ( x )) sec 2 ( sec 3 ( x )) 3 sec 2 ( x ) sec ( x ) tan ( x ) (2x) rr. y = arccos x dx = 1 1 x 2 ss. y = sin 5 ( e 3π+5 + 4e 3) x 2 dx = 2 sin5 ( e 3π+5 + 4e 3) x tt. y = arcsin(x 2 5x + 1) dx = 2x 5 1 (x2 5x + 1) 2 uu. y = arctan 2 (ln 3x) vv. y = dx = 2 arctan(ln(3x)) 1 x[1 + (ln(3x)) 2 ] ln(3x) sin(ln x 2 ) dx = sin(ln x2 ) 1 x ln(3x)(cos(ln(x2 )) 2 x ) (sin(ln(x 2 ))) 2 ww. y = 3e4x sin 1 (3x 2 + 1) ln x sin 5x + 3x 2 cos 2 x

6 dx = ( ) (ln x sin(5x) + 3x 2 cos 2 x)[12e 4x (sin 1 (3x 2 + 1) + 3e 4x 6x 1 (3x 2 +1) 2 (ln x sin(5x) + 3x 2 cos 2 x) 2 (3e 4x sin 1 (3x 2 + 1)) [ 1 x sin(5x) + 5 ln x cos(5x) + 3(2x cos2 x 2x 2 cos x sin x) ] (ln x sin(5x) + 3x 2 cos 2 x) 2 4. Find the equation of the line tangent to the given function f at the given point x = a: a. f(x) = 2, a = 3 y = 2 b. f(x) = 5x 1, a = 2 y = 5x 1 c. f(x) = x 2 2x + 4, a = 1 y = 3 d. f(x) = ln x, a = 3 y ln 3 = 1 (x 3) 3 e. f(x) = e 3x, a = 2 y 1 e 6 = 3 (x + 2) e6 f. f(x) = xe x ln 4x, a = 1 y (e ln 4) = (2e 1)(x 1) g. f(x) = sin x, a = π 6 y 1 2 = 3 2 ( x π ) 6 5. a. Define a function that has a left and right handed limit at a point a, but f does not have a limit at a. b. Define a function that has a limit at a point b but is undefined at b. c. Define a function that has a limit at point c, is defined at c, but is discontinuous at c. d. Define a function that is continuous at point d but is not differentiable at d. e. Define a function that is first differentiable at point p but is not second differentiable at point p. f. Draw the graph of a function which has all of the properties in a, b, c, d, e, above. There are more than one correct answer. The function defined below satisfies all conditions a through e: 3 if x 2 2 if 2 < x < 1 f(x) = sin x x if 1 x < 1 x 2 4 x 2 if 1 x < 3, x 2 5 if x = 2 3 x 4 if 3 x 5 3 (x 6) 4 if 5 < x

7 lim f(x) = 3 x 2 lim f(x) = 2 x 2 + lim f(x) Does not exist. x 2 lim x 0 f(x) = 1, f(0) is undefined. 10 lim f(x) = 4, f(2) = 5, f is discontinuous at 5. x 2 9 f is continuous at x = 4 but not differentiable at 4. f is first differentiable at x = 6 but 8 not second differentiable at 6. The graph of f is drawn below y Think About: -6 a. Given the graph of the original function f(x), how can you draw the graph of its derivative, f (x)? b. Given the graph of the derivative -7 of a function, f (x), how can you draw the graph of the original function, f(x)? Find dx a. y sin x = x cos y by implicite differentiation: -9 cos y y cos x = dx sin x + x sin y b. x 2 y + y 2 x = 5 2xy y2 = dx x 2 + 2xy c. 2y 4x sin x = e y 4 sin x + 4x cos x = dx 2 e y

8 d. ln(xy) + x y = 1 dx = y2 xy xy x 2 e. e xy + ln(x + y) = 2xy dx = 2y(x + y) yexy (x + y) 1 2x(x + y) + xe xy (x + y) Find f (x) and f (x): a. f(x) = 4 f (x) = 0 f (x) = 0 f (x) = 0 b. f(x) = 3x 2 f (x) = 6x f (x) = 6 f (x) = 0 c. f(x) = e x f (x) = e x f (x) = e x f (x) = e x d. f(x) = ln x f (x) = 1 x f (x) = 1 x 2 f (x) = 2 x 3 e. f(x) = x2 + 1 x 1 f (x) = x2 2x 1 (x 1) 2 f 4 (x) = (x 1) 3 f (x) = 12 (x 1) 4 f. f(x) = x f (x) = 1 2 x f (x) = 1 4 x 3 f (x) = 3 8 x 5 g. f(x) = x 5/4 f (x) = 5 4 x1/4 f (x) = 5 16 x 3/4 f (x) = x 7/4 h. f(x) = arccos x f 1 (x) = f x (x) = 1 x 2 (1 x 2 ) f (x) = (1 x 2 ) 3/2 3x 2 (1 x 2 ) 5/2 3/2 i. f(x) = sin x f (x) = cos x f (x) = sin x f (x) = cos x j. f(x) = e sin x f (x) = cos xe sin x f (x) = e sin x (cos 2 x sin x) f (x) = e sin x (cos 3 x 3 sin x cos x cos x) 9. Find the linearization of the given function at the given point a: a. f(x) = 2x 3, a = 2 L(x) = 2x 3, the linearization of a line is itself. b. f(x) = 2x 3, a = 4 L(x) = 2x 3, the linearization of a line is itself. c. f(x) = 2x 3, a = 10 L(x) = 2x 3, the linearization of a line is itself. d. f(x) = 4x 2 x + 3, a = 1 L(x) = 9x 1

9 e. f(x) = sin x, a = 0 L(x) = x f. f(x) = cos x, a = π ( L(x) = 2 x π ) 2 4 g. f(x) = e x, a = 0 L(x) = x + 1 h. f(x) = e x, a = 1 L(x) = e + e(x 1) i. f(x) = ln x, a = 1 L(x) = x 1 j. f(x) = ln x, a = 2 L(x) = (ln 2) + 1 (x 2) 2 k. f(x) = arctan x, a = 0 L(x) = x

Math 250 Skills Assessment Test

Math 250 Skills Assessment Test Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).

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

SET 1. (1) Solve for x: (a) e 2x = 5 3x

SET 1. (1) Solve for x: (a) e 2x = 5 3x () Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x

More information

Math 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx

Math 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first

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

Inverse Trigonometric Functions. September 5, 2018

Inverse Trigonometric Functions. September 5, 2018 Inverse Trigonometric Functions September 5, 08 / 7 Restricted Sine Function. The trigonometric function sin x is not a one-to-one functions..0 0.5 Π 6, 5Π 6, Π Π Π Π 0.5 We still want an inverse, so what

More information

A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15

A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15 A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15 The scoring for this section is determined by the formula [C (0.25 I)] 1.8 where C is the

More information

Week #6 - Taylor Series, Derivatives and Graphs Section 10.1

Week #6 - Taylor Series, Derivatives and Graphs Section 10.1 Week #6 - Taylor Series, Derivatives and Graphs Section 10.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used by

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

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

Math 222 Spring 2013 Exam 3 Review Problem Answers

Math 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 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

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

Inverse Trig Functions

Inverse Trig Functions 6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)

More information

1 Solution to Homework 4

1 Solution to Homework 4 Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value

More information

dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)

dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy) Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =

More information

Worksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra

Worksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation

More information

Homework Problem Answers

Homework Problem Answers Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln

More information

Final Exam 2011 Winter Term 2 Solutions

Final Exam 2011 Winter Term 2 Solutions . (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L

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

(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 information

MATH 104 Practice Problems for Exam 2

MATH 104 Practice Problems for Exam 2 . Find the area between: MATH 4 Practice Problems for Exam (a) x =, y = / + x, y = x/ Answer: ln( + ) 4 (b) y = e x, y = xe x, x = Answer: e6 4 7 4 (c) y = x and the x axis, for x 4. x Answer: ln 5. Calculate

More information

Solutions to Exam 2, Math 10560

Solutions to Exam 2, Math 10560 Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If

More information

3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:

3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y: 3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable

More information

Solution: APPM 1350 Final Exam Spring 2014

Solution: APPM 1350 Final Exam Spring 2014 APPM 135 Final Exam Spring 214 1. (a) (5 pts. each) Find the following derivatives, f (x), for the f given: (a) f(x) = x 2 sin 1 (x 2 ) (b) f(x) = 1 1 + x 2 (c) f(x) = x ln x (d) f(x) = x x d (b) (15 pts)

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

MATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS

MATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS Name (print): Signature: MATH 5, FALL SEMESTER 0 COMMON EXAMINATION - VERSION B - SOLUTIONS Instructor s name: Section No: Part Multiple Choice ( questions, points each, No Calculators) Write your name,

More information

Mathematics 1052, Calculus II Exam 1, April 3rd, 2010

Mathematics 1052, Calculus II Exam 1, April 3rd, 2010 Mathematics 5, Calculus II Exam, April 3rd,. (8 points) If an unknown function y satisfies the equation y = x 3 x + 4 with the condition that y()=, then what is y? Solution: We must integrate y against

More information

MATH 151, SPRING 2018

MATH 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 information

c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0

c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0 Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to

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

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

Math 147 Exam II Practice Problems

Math 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 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

Unit #3 - Differentiability, Computing Derivatives, Trig Review

Unit #3 - Differentiability, Computing Derivatives, Trig Review Unit #3 - Differentiability, Computing Derivatives, Trig Review Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Derivative Interpretation and Existence 1. The cost, C (in

More information

Solution to Review Problems for Midterm II

Solution to Review Problems for Midterm II Solution to Review Problems for Midterm II Midterm II: Monday, October 18 in class Topics: 31-3 (except 34) 1 Use te definition of derivative f f(x+) f(x) (x) lim 0 to find te derivative of te functions

More information

Final Exam. Math 3 December 7, 2010

Final 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 information

MathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited

MathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited Everything You Need to Know A Level Edexcel C4 March 4 Copyright 4 Elite Learning Limited Page of 4 Further Binomial Expansion: Make sure it starts with a e.g. for ( x) ( x ) then use ( + x) n + nx + n(n

More information

Terms of Use. Copyright Embark on the Journey

Terms of Use. Copyright Embark on the Journey Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means - electronic, mechanical, photo-copies, recording, or otherwise

More information

(1) Find derivatives of the following functions: (a) y = x5 + 2x + 1. Use the quotient and product rules: ( 3 x cos(x)) 2

(1) Find derivatives of the following functions: (a) y = x5 + 2x + 1. Use the quotient and product rules: ( 3 x cos(x)) 2 Calc 1: Practice Exam Solutions Name: (1) Find derivatives of the following functions: (a) y = x5 + x + 1 x cos(x) Answer: Use the quotient and product rules: y = xcos(x)(5x 4 + ) (x 5 + x + 1)( 1 x /

More information

Math 131 Final Exam Spring 2016

Math 131 Final Exam Spring 2016 Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing

More information

Multiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question

Multiple 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 information

Math 251, Spring 2005: Exam #2 Preview Problems

Math 251, Spring 2005: Exam #2 Preview Problems Math 5, Spring 005: Exam # Preview Problems. Using the definition of derivative find the derivative of the following functions: a) fx) = e x e h. Use the following lim =, e x+h = e x e h.) h b) fx) = x

More information

11.5. The Chain Rule. Introduction. Prerequisites. Learning Outcomes

11.5. The Chain Rule. Introduction. Prerequisites. Learning Outcomes The Chain Rule 11.5 Introduction In this Section we will see how to obtain the derivative of a composite function (often referred to as a function of a function ). To do this we use the chain rule. This

More information

MA1021 Calculus I B Term, Sign:

MA1021 Calculus I B Term, Sign: MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your

More information

Brief answers to assigned even numbered problems that were not to be turned in

Brief answers to assigned even numbered problems that were not to be turned in Brief answers to assigned even numbered problems that were not to be turned in Section 2.2 2. At point (x 0, x 2 0) on the curve the slope is 2x 0. The point-slope equation of the tangent line to the curve

More information

16 Inverse Trigonometric Functions

16 Inverse Trigonometric Functions 6 Inverse Trigonometric Functions Concepts: Restricting the Domain of the Trigonometric Functions The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Using the Inverse Trigonometric

More information

1d C4 Integration cot4x 1 4 1e C4 Integration trig reverse chain 1

1d C4 Integration cot4x 1 4 1e C4 Integration trig reverse chain 1 A Assignment Nu Cover Sheet Name: Drill Current work Question Done BP Ready Topic Comment Aa C4 Integration Repeated linear factors 3 (x ) 3 (x ) + c Ab C4 Integration cos^ conversion x + sinx + c Ac C4

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

MATH 151 Engineering Mathematics I

MATH 151 Engineering Mathematics I MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section

More information

Mrs. Charnley, J303 Mrs. Zayaitz Ruhf, J305

Mrs. Charnley, J303 Mrs. Zayaitz Ruhf, J305 Mrs. Charnley, J303 charnleyc@eastonsd.org Mrs. Zayaitz Ruhf, J305 zayaitzruhfm@eastonsd.org Part I. Online Quiz/Review, DUE June 30, 016 This is a quick review of some material from Pre-Calculus and other

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

Dr. Sophie Marques. MAM1020S Tutorial 8 August Divide. 1. 6x 2 + x 15 by 3x + 5. Solution: Do a long division show your work.

Dr. Sophie Marques. MAM1020S Tutorial 8 August Divide. 1. 6x 2 + x 15 by 3x + 5. Solution: Do a long division show your work. Dr. Sophie Marques MAM100S Tutorial 8 August 017 1. Divide 1. 6x + x 15 by 3x + 5. 6x + x 15 = (x 3)(3x + 5) + 0. 1a 4 17a 3 + 9a + 7a 6 by 3a 1a 4 17a 3 + 9a + 7a 6 = (4a 3 3a + a + 3)(3a ) + 0 3. 1a

More information

6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives

6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives Objectives 1. Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function. 2. Find an Approximate Value of an Inverse Sine Function. 3. Use Properties of Inverse Functions to Find Exact Values

More information

Partial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt

Partial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt Partial Derivatives for Math 229 Our puropose here is to explain how one computes partial derivatives. We will not attempt to explain how they arise or why one would use them; that is left to other courses

More information

Math 106: Review for Exam II - SOLUTIONS

Math 106: Review for Exam II - SOLUTIONS Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present

More information

Math 171 Spring 2017 Final Exam. Problem Worth

Math 171 Spring 2017 Final Exam. Problem Worth Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:

More information

Chapter 5 Notes. 5.1 Using Fundamental Identities

Chapter 5 Notes. 5.1 Using Fundamental Identities Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx

More information

2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where

2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where AP Review Chapter Name: Date: Per: 1. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the

More information

Chapter 7: Techniques of Integration

Chapter 7: Techniques of Integration Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration

More information

MCV4U - Practice Mastery Test #9 & #10

MCV4U - Practice Mastery Test #9 & #10 Name: Class: Date: ID: A MCV4U - Practice Mastery Test #9 & #10 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. x + 2 is a factor of i) x 3 3x 2 + 6x 8

More information

Final Examination 201-NYA-05 May 18, 2018

Final 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 information

Math 106: Review for Exam II - SOLUTIONS

Math 106: Review for Exam II - SOLUTIONS Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present

More information

Homework Solutions: , plus Substitutions

Homework Solutions: , plus Substitutions Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions

More information

Review Problems for Test 1

Review Problems for Test 1 Review Problems for Test Math 6-03/06 9 9/0 007 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And

More information

Grade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12

Grade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12 First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm E: Page of Indefinite Integrals. 9 marks Each part is worth 3 marks. Please

More information

Integration by Parts

Integration by Parts Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u

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

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

5 t + t2 4. (ii) f(x) = ln(x 2 1). (iii) f(x) = e 2x 2e x + 3 4

5 t + t2 4. (ii) f(x) = ln(x 2 1). (iii) f(x) = e 2x 2e x + 3 4 Study Guide for Final Exam 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its expression to be well-defined. Some examples of the conditions are: What

More information

and lim lim 6. The Squeeze Theorem

and lim lim 6. The Squeeze Theorem Limits (day 3) Things we ll go over today 1. Limits of the form 0 0 (continued) 2. Limits of piecewise functions 3. Limits involving absolute values 4. Limits of compositions of functions 5. Limits similar

More information

INVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1.

INVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1. INVERSE FUNCTIONS DERIVATIVES Recall the steps for computing y implicitly: (1) Take of both sies, treating y like a function. (2) Expan, a, subtract to get the y terms on one sie an everything else on

More information

Chapter 22 : Electric potential

Chapter 22 : Electric potential Chapter 22 : Electric potential What is electric potential? How does it relate to potential energy? How does it relate to electric field? Some simple applications What does it mean when it says 1.5 Volts

More information

WeBWorK, Problems 2 and 3

WeBWorK, Problems 2 and 3 WeBWorK, Problems 2 and 3 7 dx 2. Evaluate x ln(6x) This can be done using integration by parts or substitution. (Most can not). However, it is much more easily done using substitution. This can be written

More information

Math 131 Exam 2 Spring 2016

Math 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 information

Differential Equations: Homework 2

Differential Equations: Homework 2 Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y

More information

MATH 162. Midterm Exam 1 - Solutions February 22, 2007

MATH 162. Midterm Exam 1 - Solutions February 22, 2007 MATH 62 Midterm Exam - Solutions February 22, 27. (8 points) Evaluate the following integrals: (a) x sin(x 4 + 7) dx Solution: Let u = x 4 + 7, then du = 4x dx and x sin(x 4 + 7) dx = 4 sin(u) du = 4 [

More information

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite

More information

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite

More information

Math 152 Take Home Test 1

Math 152 Take Home Test 1 Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I

More information

MATH 408N PRACTICE FINAL

MATH 408N PRACTICE FINAL 05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results

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

M GENERAL MATHEMATICS -2- Dr. Tariq A. AlFadhel 1 Solution of the First Mid-Term Exam First semester H

M GENERAL MATHEMATICS -2- Dr. Tariq A. AlFadhel 1 Solution of the First Mid-Term Exam First semester H M - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam First semester 38-39 H 3 Q. Let A =, B = and C = 3 Compute (if possible) : A+B and BC A+B is impossible. ( ) BC = 3

More information

Math 180, Final Exam, Fall 2012 Problem 1 Solution

Math 180, Final Exam, Fall 2012 Problem 1 Solution Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.

More information

A A A A A A A A A A A A. a a a a a a a a a a a a a a a. Apples taste amazingly good.

A A A A A A A A A A A A. a a a a a a a a a a a a a a a. Apples taste amazingly good. Victorian Handwriting Sheet Aa A A A A A A A A A A A A Aa Aa Aa Aa Aa Aa Aa a a a a a a a a a a a a a a a Apples taste amazingly good. Apples taste amazingly good. Now make up a sentence of your own using

More information

MA Practice Exam #2 Solutions

MA Practice Exam #2 Solutions MA 123 - Practice Exam #2 Solutions Name: Instructions: For some of the questions, you must show all your work as indicated. No calculators, books or notes of any form are allowed. Note that the questions

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions -8-006 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse

More information

Find all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) =

Find 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 information

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that

More information

" = Y(#,$) % R(r) = 1 4& % " = Y(#,$) % R(r) = Recitation Problems: Week 4. a. 5 B, b. 6. , Ne Mg + 15 P 2+ c. 23 V,

 = Y(#,$) % R(r) = 1 4& %  = Y(#,$) % R(r) = Recitation Problems: Week 4. a. 5 B, b. 6. , Ne Mg + 15 P 2+ c. 23 V, Recitation Problems: Week 4 1. Which of the following combinations of quantum numbers are allowed for an electron in a one-electron atom: n l m l m s 2 2 1! 3 1 0 -! 5 1 2! 4-1 0! 3 2 1 0 2 0 0 -! 7 2-2!

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

A special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule.

A special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule. The Chain Rule A special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule. In order to master the techniques explained here it is vital that

More information

Math 1431 Final Exam Review

Math 1431 Final Exam Review Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back

More information

Chapter 3: Transcendental Functions

Chapter 3: Transcendental Functions Chapter 3: Transcendental Functions Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 32 Except for the power functions, the other basic elementary functions are also called the transcendental

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

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

Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =

Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) = Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists

More information

AP Calculus AB Summer Math Packet

AP Calculus AB Summer Math Packet Name Date Section AP Calculus AB Summer Math Packet This assignment is to be done at you leisure during the summer. It is meant to help you practice mathematical skills necessary to be successful in Calculus

More information

PART A: Solve the following equations/inequalities. Give all solutions. x 3 > x + 3 x

PART A: Solve the following equations/inequalities. Give all solutions. x 3 > x + 3 x CFHS Honors Precalculus Calculus BC Review PART A: Solve the following equations/inequalities. Give all solutions. 1. 2x 3 + 3x 2 8x = 3 2. 3 x 1 + 4 = 8 3. 1 x + 1 2 x 4 = 5 x 2 3x 4 1 4. log 2 2 + log

More information

MATH 1241 Common Final Exam Fall 2010

MATH 1241 Common Final Exam Fall 2010 MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the

More information

Friday 09/15/2017 Midterm I 50 minutes

Friday 09/15/2017 Midterm I 50 minutes Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.

More information