Math 2413 Final Exam Review 1. Evaluate, giving exact values when possible.

Size: px
Start display at page:

Download "Math 2413 Final Exam Review 1. Evaluate, giving exact values when possible."

Transcription

1 Math 4 Final Eam Review. Evaluate, giving eact values when possible. sin cos cos sin y. Evaluate the epression. loglog 5 5ln e. Solve for. 4 6 e 4. Use the given graph of f to answer the following: y f f f d. f e. f f. f g. List all -coordinates where f is not continuous and eplain your answer mathematically. h. List all -coordinates where f is not differentiable and eplain why.

2 5. Find the average rate of change of the function on the given interval. g 8 tan,, 4 4 f cot,, 6 6. Evaluate each it, if it eists. d. e. f. g. h. i. j. k. l. h This is a conceptual problem. h h h e e This is a conceptual problem. h h tan h tan This is a conceptual problem. h h h 4 6 h h 9h h h ln csc lne m. n. o. ln

3 5 p. ln q. r. cos 7. Find constants a and/or b so that f will be continuous for all values of. f a if 5 8 if 5 b if 5 f b, 8 b b, 8. Given the graph of f, sketch the graph of f '. y 9. Determine if each of the given functions is differentiable and continuous everywhere it is defined. 4 if f if if f if if f if d. if f if e. if f tan if 4

4 . Consider the function f. Use the definition of derivative to determine f. Find an equation of the line tangent to the graph of f at.. Differentiate the following. Simplify your answers completely. f 5 f 9. Find f 5 d. f sec4 e. f ln 5 4 f. f e g. f sin sec tan f cos sin f tan sec h. f i. j. k. f f 4 log 6 of the following. 5 f 5 f f sin d. 5 f e e. f cos 4

5 . Suppose f and g are defined so that the following table of values holds true. Evaluate the derivative of h at the indicated -value. f g f g h 5f 4g h f g h g f / 5 / 4 ; ; ; d. h g ; 4. Calculate the instantaneous rate of change of the given function at the specified value., f, f sin cos 5. Find the slope of the line tangent to f tan at Find the velocity and acceleration at time t when 4 s t t 4t 8t. 7. Find the -coordinate of each point where the graph of f horizontal tangent line. 8. Find the equation for the line tangent to y sin y at,. has a 9. Find dy for each of the following. d y y 5 d. ln y e y sec y y e. ysin f. y tan

6 . An airplane is flying at 5 ft/s at an altitude of ft in a direction that will take it directly over an observer at ground level. Find the rate of change of the angle of elevation between the observer and the plane when the plane is directly over a point on the ground that is ft from the observer.. A 5-ft ladder is placed against a large building. The base of the ladder is resting on an oil spill, and it slips away from the building at a rate of ft per minute. Find the rate of change of the height of the top of the ladder above the ground at the instant when the base of the ladder is ft from the base of the building.. A sand storage tank used by the highway department for winter storms is leaking. As the sand leaks out, it forms a conical pile. The radius of the base of the pile increases at the rate of.75 in. per minute. The height of the pile is always twice the radius of the base. Find the rate at which the volume of the pile increasing at the instant the radius of the base is 6 in. Leave your answer in terms of π.. Find all absolute etrema for f on the given interval.,,, f sin cos 5 4 f f,,, 4. Find all interval(s) on which f is increasing or decreasing. f 9 f 4 f 5. Find all points of inflection and state all intervals of concavity for the following functions. 6 4 f 5 f ( ) 6. The function f a b 7 has a relative etrema at and. Find a and b. Use the Second Derivative Test for Relative Etrema to classify each etremum as a relative maimum or a relative minimum. Determine the relative etrem

7 7. Let f be a polynomial function that has critical numbers, and. Furthermore, assume f 4 5. For which critical numbers does the Second Derivative Test indicate the function f has a relative minimum? 8. Sketch the graph of a function with the following properties. ', 5, f on f '( ) on 5, f ' 5 f ' f 5 4 f 9. Sketch the graph of a function with the following properties. Continuous and differentiable on, g 4 g d. g on, e. g on,. Consider the points A, B, and C. A company wishes to run a utility cable from point A to point B. Point A lies 9 miles directly east of point C along a straight shoreline. Point B is located on an island 6 miles directly north of point C. It costs $4 per mile to run the cable on land and $5 per mile underwater. Assume that the cable starts at point A and runs along the shoreline, then angles and runs underwater to the island. Find the distance from point A at which the cable line should begin to run underwater in order to yield the minimum total cost.. An open bo is to be made from a square piece of cardboard measuring 5 inches on a side by cutting a square from each corner and folding up the sides. Find the dimensions for which the volume of the resulting bo is maimized.. A man is floating in a rowboat mile from the (straight) shoreline of a large lake. A town is located on the shoreline mile from the point on the shoreline closest to the man. He intends to row in a straight line to some point P on the shoreline and then walk the remaining distance to the town. To what point should he row in order to reach his destination in the least amount of time if he can walk 5 mph and row mph? State your answer as a distance from the town.

8 . Two posts, one meters high and one 5 meters high, stand 6 meters apart. They are tied to a single stake by two wires running from ground level to the top of each post. How far from the -m post should the stake be placed to use the least amount of wire? m 5 m 6 m 4. Evaluate the following. csc cot d d. 8t 5t dt t d d 4 e. f. 6 e d d sec t tan t dt g. sin 7 h. d 49 i. 9 4 d j. k. l. 6 4 sec d d 4 d m. 5 a d

9 5. Evaluate the sums using known summation formulas. 9 k k 5 4 k k 5 k6k 5 k 6. Use the definition of the definite integral to evaluate 4 Riemann sums. 4 d using right 7. Simplify the following epressions showing all steps and using correct mathematical notation. d t dt d t d t cos t dt d d sec t tan tdt d 8. Find the area of the region under each of the given curves for the given interval. y 6,4 on y cos on y, 6, 4 4 on 9. Assume at represents the acceleration of an object at time t, vt represents the velocity of the object at time t, and st represents the position of the object at time t. If at sin t, v 4 m s, and s 5 m, find st.

10 Topics Limits and Continuity Be able to find its by simplifying the epressions. Consider reducing rational epressions, multiplying by conjugates, or by simplifying an absolute value epression. Be able to find its at infinity by dividing by the highest power of in the denominator. Know when this process will work better than using L'Hôpital's Rule. Know L'Hôpital's Rule and be able to use it to find its of indeterminate forms. Know when L'Hôpital's Rule applies and when it doesn't. Be able to recognize all of the indeterminate forms. Be able to use the natural log function to work with eponential functions. Be able to use the definition of the derivative to evaluate a it. Know the definition of continuity. Be able to decide if a function is continuous and to be able to find constants so that a function is continuous. Derivatives Know the definition of a derivative. Be able to find the derivative using only the definition. Be able to find a it by recognizing it as a derivative. Know all of the derivative short-cut rules. Be able to use implicit differentiation. Remember that the chain, product, and quotient rules still apply. Be able to find a derivative by using the Second Fundamental Theorem of Calculus. Applications of the Derivative Be able to find the slope of a tangent or a normal line. Know the relationship between a position (distance) function, the velocity and the acceleration. Be able to solve related rates problems. Look for the question "how fast" or "what rate". Be able to find absolute and local etrem Be able to decide if your answer should be a y-value or if it should be a point. Be able to use both the first and second derivative tests. Know what information derivatives give about the shape of a function. Be able to find intervals of increase/decrease, intervals of concavity, vertical tangents and cusps. Be able to solve applied optimization problems. Integrals Know all of the antiderivative rules. Be able to evaluate definite and indefinite integrals. Given an acceleration or velocity function and some initial values, be able to find the position function. Be able to use integration by substitution to evaluate definite and indefinite integrals. Know the Fundamental Theorem of Calculus -- both parts. Be able to evaluate a definite integral by using the First Fundamental Theorem of Calculus. Be able to find the derivative of a function defined as a definite integral by using the Second Fundamental Theorem of Calculus.

Math 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.

Math 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n. . Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.

More information

Technical Calculus I Homework. Instructions

Technical Calculus I Homework. Instructions Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the

More information

Solutions Exam 4 (Applications of Differentiation) 1. a. Applying the Quotient Rule we compute the derivative function of f as follows:

Solutions Exam 4 (Applications of Differentiation) 1. a. Applying the Quotient Rule we compute the derivative function of f as follows: MAT 4 Solutions Eam 4 (Applications of Differentiation) a Applying the Quotient Rule we compute the derivative function of f as follows: f () = 43 e 4 e (e ) = 43 4 e = 3 (4 ) e Hence f '( ) 0 for = 0

More information

Final Exam Review / AP Calculus AB

Final Exam Review / AP Calculus AB Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f 5 4 1 y - - -1 - - -4-5 ) lim f(), - lim f(), and + lim f 8 6 4 y -4 - - -1-1 4 5-4

More information

Mathematics 1161: Midterm Exam 2 Study Guide

Mathematics 1161: Midterm Exam 2 Study Guide Mathematics 1161: Midterm Eam 2 Study Guide 1. Midterm Eam 2 is on October 18 at 6:00-6:55pm in Journalism Building (JR) 300. It will cover Sections 3.8, 3.9, 3.10, 3.11, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6,

More information

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2 Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice

More information

( ) 7 ( 5x 5 + 3) 9 b) y = x x

( ) 7 ( 5x 5 + 3) 9 b) y = x x New York City College of Technology, CUNY Mathematics Department Fall 0 MAT 75 Final Eam Review Problems Revised by Professor Kostadinov, Fall 0, Fall 0, Fall 00. Evaluate the following its, if they eist:

More information

AP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015

AP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015 AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use

More information

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity. Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive

More information

3.1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY

3.1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY MATH00 (Calculus).1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY Name Group No. KEYWORD: increasing, decreasing, constant, concave up, concave down, and inflection point Eample 1. Match the

More information

( ) 9 b) y = x x c) y = (sin x) 7 x d) y = ( x ) cos x

( ) 9 b) y = x x c) y = (sin x) 7 x d) y = ( x ) cos x NYC College of Technology, CUNY Mathematics Department Spring 05 MAT 75 Final Eam Review Problems Revised by Professor Africk Spring 05, Prof. Kostadinov, Fall 0, Fall 0, Fall 0, Fall 0, Fall 00 # Evaluate

More information

Find the following limits. For each one, if it does not exist, tell why not. Show all necessary work.

Find the following limits. For each one, if it does not exist, tell why not. Show all necessary work. Calculus I Eam File Spring 008 Test #1 Find the following its. For each one, if it does not eist, tell why not. Show all necessary work. 1.) 4.) + 4 0 1.) 0 tan 5.) 1 1 1 1 cos 0 sin 3.) 4 16 3 1 6.) For

More information

Math 231 Final Exam Review

Math 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 information

lim 2 x lim lim sin 3 (9) l)

lim 2 x lim lim sin 3 (9) l) MAC FINAL EXAM REVIEW. Find each of the following its if it eists, a) ( 5). (7) b). c). ( 5 ) d). () (/) e) (/) f) (-) sin g) () h) 5 5 5. DNE i) (/) j) (-/) 7 8 k) m) ( ) (9) l) n) sin sin( ) 7 o) DNE

More information

Solutions to Test 2 Spring = y+x dy dx +0 = ex+y x+y dy. e x = dy dx (ex+y x) = y e x+y. dx = y ex+y e x+y x

Solutions to Test 2 Spring = y+x dy dx +0 = ex+y x+y dy. e x = dy dx (ex+y x) = y e x+y. dx = y ex+y e x+y x 12pt 1 Consider the equation e +y = y +10 Solutions to Test 2 Spring 2018 (a) Use implicit differentiation to find dy d d d (e+y ) = d ( (y+10) e+y 1+ dy ) d d = y+ dy d +0 = e+y +y dy +e d = y+ dy d +y

More information

The Fundamental Theorem of Calculus Part 3

The Fundamental Theorem of Calculus Part 3 The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative

More information

3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13

3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13 Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................

More information

2008 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

2008 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION 8 CALCULUS AB 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 of the following problems. After eamining the form

More information

CALCULUS I. Practice Problems. Paul Dawkins

CALCULUS I. Practice Problems. Paul Dawkins CALCULUS I Practice Problems Paul Dawkins Table of Contents Preface... iii Outline... iii Review... Introduction... Review : Functions... Review : Inverse Functions... 6 Review : Trig Functions... 6 Review

More information

AP CALCULUS BC - FIRST SEMESTER EXAM REVIEW: Complete this review for five extra percentage points on the semester exam.

AP CALCULUS BC - FIRST SEMESTER EXAM REVIEW: Complete this review for five extra percentage points on the semester exam. AP CALCULUS BC - FIRST SEMESTER EXAM REVIEW: Complete this review for five etra percentage points on the semester eam. *Even though the eam will have a calculator active portion with 0 of the 8 questions,

More information

Solutions to Math 41 Exam 2 November 10, 2011

Solutions to Math 41 Exam 2 November 10, 2011 Solutions to Math 41 Eam November 10, 011 1. (1 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it is or.

More information

+ 2 on the interval [-1,3]

+ 2 on the interval [-1,3] Section.1 Etrema on an Interval 1. Understand the definition of etrema of a function on an interval.. Understand the definition of relative etrema of a function on an open interval.. Find etrema on a closed

More information

Math 2250 Exam #3 Practice Problem Solutions 1. Determine the absolute maximum and minimum values of the function f(x) = lim.

Math 2250 Exam #3 Practice Problem Solutions 1. Determine the absolute maximum and minimum values of the function f(x) = lim. Math 50 Eam #3 Practice Problem Solutions. Determine the absolute maimum and minimum values of the function f() = +. f is defined for all. Also, so f doesn t go off to infinity. Now, to find the critical

More information

AP Calculus BC Summer Assignment 2018

AP Calculus BC Summer Assignment 2018 AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different

More information

Calculus BC AP/Dual Fall Semester Review Sheet REVISED 1 Name Date. 3) Explain why f(x) = x 2 7x 8 is a guarantee zero in between [ 3, 0] g) lim x

Calculus BC AP/Dual Fall Semester Review Sheet REVISED 1 Name Date. 3) Explain why f(x) = x 2 7x 8 is a guarantee zero in between [ 3, 0] g) lim x Calculus BC AP/Dual Fall Semester Review Sheet REVISED Name Date Eam Date and Time: Read and answer all questions accordingly. All work and problems must be done on your own paper and work must be shown.

More information

Math 180, Final Exam, Spring 2008 Problem 1 Solution. 1. For each of the following limits, determine whether the limit exists and, if so, evaluate it.

Math 180, Final Exam, Spring 2008 Problem 1 Solution. 1. For each of the following limits, determine whether the limit exists and, if so, evaluate it. Math 80, Final Eam, Spring 008 Problem Solution. For each of the following limits, determine whether the limit eists and, if so, evaluate it. + (a) lim 0 (b) lim ( ) 3 (c) lim Solution: (a) Upon substituting

More information

In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work.

In #1-5, find the indicated limits. For each one, if it does not exist, tell why not. Show all necessary work. Calculus I Eam File Fall 7 Test # In #-5, find the indicated limits. For each one, if it does not eist, tell why not. Show all necessary work. lim sin.) lim.) 3.) lim 3 3-5 4 cos 4.) lim 5.) lim sin 6.)

More information

1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2.

1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2. . Find A and B so that f Ae B has a local minimum of 6 when.. The graph below is the graph of f, the derivative of f; The domain of the derivative is 5 6. Note there is a cusp when =, a horizontal tangent

More information

AP Calculus (BC) Summer Assignment (169 points)

AP Calculus (BC) Summer Assignment (169 points) AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion

More information

Chapter 8: Radical Functions

Chapter 8: Radical Functions Chapter 8: Radical Functions Chapter 8 Overview: Types and Traits of Radical Functions Vocabulary:. Radical (Irrational) Function an epression whose general equation contains a root of a variable and possibly

More information

It s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4]

It s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4] It s Your Turn Problems I. Functions, Graphs, and Limits. Here s the graph of the function f on the interval [ 4,4] f ( ) =.. It has a vertical asymptote at =, a) What are the critical numbers of f? b)

More information

( ) as a fraction. If both numerator and denominator are

( ) as a fraction. If both numerator and denominator are A. Limits and Horizontal Asymptotes What you are finding: You can be asked to find lim f x x a (H.A.) problem is asking you find lim f x x ( ) and lim f x x ( ). ( ) or lim f x x ± ( ). Typically, a horizontal

More information

AP Calculus (BC) Summer Assignment (104 points)

AP Calculus (BC) Summer Assignment (104 points) AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion

More information

AP Calc AB First Semester Review

AP Calc AB First Semester Review AP Calc AB First Semester Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. 1) lim (7-7) 7 A) -4 B) -56 C) 4 D) 56 1) Determine

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Chapter Practice Dicsclaimer: The actual eam is different. On the actual eam ou must show the correct reasoning to receive credit for the question. SHORT ANSWER. Write the word or phrase that best completes

More information

AP CALCULUS BC SUMMER ASSIGNMENT

AP CALCULUS BC SUMMER ASSIGNMENT AP CALCULUS BC SUMMER ASSIGNMENT Work these problems on notebook paper. All work must be shown. Use your graphing calculator only on problems -55, 80-8, and 7. Find the - and y-intercepts and the domain

More information

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26. Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

More information

1998 AP Calculus AB: Section I, Part A

1998 AP Calculus AB: Section I, Part A 55 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. What is the -coordinate of the point

More information

Key- Math 231 Final Exam Review

Key- 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 information

x f(x)

x f(x) CALCULATOR SECTION. For y + y = 8 find d point (, ) on the curve. A. B. C. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) = e, A(t) is measured in tons of

More information

AP Calculus 2 Summer Review Packet

AP Calculus 2 Summer Review Packet AP Calculus Summer Review Packet This review packet is to be completed by all students enrolled in AP Calculus. This packet must be submitted on the Monday of the first full week of class. It will be used

More information

x f(x)

x f(x) CALCULATOR SECTION. For y y 8 find d point (, ) on the curve. A. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) e, A(t) is measured in tons of silver and

More information

Chapter 2 THE DERIVATIVE

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

More information

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental

More information

1985 AP Calculus AB: Section I

1985 AP Calculus AB: Section I 985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of

More information

BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points)

BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points) BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: United and Continuous! ( points) For #- below, find the its, if they eist.(#- are pt each) ) 7 ) 9 9 ) 5 ) 8 For #5-7, eplain why

More information

1. sin 2. Honors Pre-Calculus Final Exam Review 2 nd semester June TRIGONOMETRY Solve for 0 2. without using a calculator: 2. csc 2 3.

1. sin 2. Honors Pre-Calculus Final Exam Review 2 nd semester June TRIGONOMETRY Solve for 0 2. without using a calculator: 2. csc 2 3. Honors Pre-Calculus Name: Final Eam Review nd semester June 05 TRIGONOMETRY Solve for 0 without using a calculator:. sin. csc. tan. cos ) ) ) ) Solve for in degrees giving all solutions. 5. sin 6. cos

More information

1. sin 2. csc 2 3. tan 1 2. Cos 8) Sin 10. sec. Honors Pre-Calculus Final Exam Review 2 nd semester. TRIGONOMETRY Solve for 0 2

1. sin 2. csc 2 3. tan 1 2. Cos 8) Sin 10. sec. Honors Pre-Calculus Final Exam Review 2 nd semester. TRIGONOMETRY Solve for 0 2 Honors Pre-Calculus Final Eam Review nd semester Name: TRIGONOMETRY Solve for 0 without using a calculator: 1 1. sin. csc 3. tan 1 4. cos 1) ) 3) 4) Solve for in degrees giving all solutions. 5. sin 1

More information

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

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine from the graph whether the function has an absolute etreme values on the interval

More information

Solutions to Math 41 Final Exam December 9, 2013

Solutions to Math 41 Final Exam December 9, 2013 Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain

More information

Helpful Concepts for MTH 261 Final. What are the general strategies for determining the domain of a function?

Helpful Concepts for MTH 261 Final. What are the general strategies for determining the domain of a function? Helpful Concepts for MTH 261 Final What are the general strategies for determining the domain of a function? How do we use the graph of a function to determine its range? How many graphs of basic functions

More information

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2 Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice

More information

CHAPTER 3: DERIVATIVES

CHAPTER 3: DERIVATIVES (Exercises for Section 3.1: Derivatives, Tangent Lines, and Rates of Change) E.3.1 CHAPTER 3: DERIVATIVES SECTION 3.1: DERIVATIVES, TANGENT LINES, and RATES OF CHANGE In these Exercises, use a version

More information

K. Function Analysis. ). This is commonly called the first derivative test. f ( x) is concave down for values of k such that f " ( k) < 0.

K. Function Analysis. ). This is commonly called the first derivative test. f ( x) is concave down for values of k such that f  ( k) < 0. K. Function Analysis What you are finding: You have a function f ( ). You want to find intervals where f ( ) is increasing and decreasing, concave up and concave down. You also want to find values of where

More information

MATH 1325 Business Calculus Guided Notes

MATH 1325 Business Calculus Guided Notes MATH 135 Business Calculus Guided Notes LSC North Harris By Isabella Fisher Section.1 Functions and Theirs Graphs A is a rule that assigns to each element in one and only one element in. Set A Set B Set

More information

3. Go over old quizzes (there are blank copies on my website try timing yourself!)

3. Go over old quizzes (there are blank copies on my website try timing yourself!) final exam review General Information The time and location of the final exam are as follows: Date: Tuesday, June 12th Time: 10:15am-12:15pm Location: Straub 254 The exam will be cumulative; that is, it

More information

Review of elements of Calculus (functions in one variable)

Review of elements of Calculus (functions in one variable) Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints

More information

Calculus AB Topics Limits Continuity, Asymptotes

Calculus AB Topics Limits Continuity, Asymptotes Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3

More information

1998 AP Calculus AB: Section I, Part A

1998 AP Calculus AB: Section I, Part A 998 AP Calculus AB: 55 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. What is the -coordinate

More information

WW Prob Lib1 Math course-section, semester year

WW Prob Lib1 Math course-section, semester year Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment due /4/03 at :00 PM..( pt) Give the rational number whose decimal form is: 0 7333333 Answer:.( pt) Solve the following inequality:

More information

Solutions to Math 41 First Exam October 12, 2010

Solutions to Math 41 First Exam October 12, 2010 Solutions to Math 41 First Eam October 12, 2010 1. 13 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it

More information

AP Calculus AB/BC ilearnmath.net

AP Calculus AB/BC ilearnmath.net CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or

More information

UBC-SFU-UVic-UNBC Calculus Exam Solutions 7 June 2007

UBC-SFU-UVic-UNBC Calculus Exam Solutions 7 June 2007 This eamination has 15 pages including this cover. UBC-SFU-UVic-UNBC Calculus Eam Solutions 7 June 007 Name: School: Signature: Candidate Number: Rules and Instructions 1. Show all your work! Full marks

More information

Calculus 1st Semester Final Review

Calculus 1st Semester Final Review Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ), c /, > 0 Find the limit: lim 6+

More information

Math 261 Final Exam - Practice Problem Solutions. 1. A function f is graphed below.

Math 261 Final Exam - Practice Problem Solutions. 1. A function f is graphed below. Math Final Eam - Practice Problem Solutions. A function f is graphed below. f() 8 7 7 8 (a) Find f(), f( ), f(), and f() f() = ;f( ).;f() is undefined; f() = (b) Find the domain and range of f Domain:

More information

Math 180, Exam 2, Spring 2013 Problem 1 Solution

Math 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

Find 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

Find 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

CLEP Calculus. Time 60 Minutes 45 Questions. For each question below, choose the best answer from the choices given. 2. If f(x) = 3x, then f (x) =

CLEP Calculus. Time 60 Minutes 45 Questions. For each question below, choose the best answer from the choices given. 2. If f(x) = 3x, then f (x) = CLEP Calculus Time 60 Minutes 5 Questions For each question below, choose the best answer from the choices given. 7. lim 5 + 5 is (A) 7 0 (C) 7 0 (D) 7 (E) Noneistent. If f(), then f () (A) (C) (D) (E)

More information

4.1 & 4.2 Student Notes Using the First and Second Derivatives. for all x in D, where D is the domain of f. The number f()

4.1 & 4.2 Student Notes Using the First and Second Derivatives. for all x in D, where D is the domain of f. The number f() 4.1 & 4. Student Notes Using the First and Second Derivatives Deinition A unction has an absolute maimum (or global maimum) at c i ( c) ( ) or all in D, where D is the domain o. The number () c is called

More information

Unit #3 Rules of Differentiation Homework Packet

Unit #3 Rules of Differentiation Homework Packet Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified

More information

Review sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston

Review sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston Review sheet Final Eam Math Calculus I Fall 5 UMass Boston The eam is closed tetbook NO CALCULATORS OR ELECTRONIC DEVICES ARE ALLOWED DURING THE EXAM The final eam will contain problems of types similar

More information

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012 The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,

More information

Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016

Calculus 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 information

Math 125 Practice Problems for Test #3

Math 125 Practice Problems for Test #3 Math Practice Problems for Test # Also stud the assigned homework problems from the book. Donʹt forget to look over Test # and Test #! Find the derivative of the function. ) Know the derivatives of all

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Math 1325 Test 3 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the location and value of each relative etremum for the function. 1)

More information

MATH140 Exam 2 - Sample Test 1 Detailed Solutions

MATH140 Exam 2 - Sample Test 1 Detailed Solutions www.liontutors.com 1. D. reate a first derivative number line MATH140 Eam - Sample Test 1 Detailed Solutions cos -1 0 cos -1 cos 1 cos 1/ p + æp ö p æp ö ç è 4 ø ç è ø.. reate a second derivative number

More information

1. The cost (in dollars) of producing x units of a certain commodity is C(x) = x x 2.

1. The cost (in dollars) of producing x units of a certain commodity is C(x) = x x 2. APPM 1350 Review #2 Summer 2014 1. The cost (in dollars) of producing units of a certain commodity is C() 5000 + 10 + 0.05 2. (a) Find the average rate of change of C with respect to when the production

More information

Date Period For each problem, find all points of absolute minima and maxima on the given interval.

Date Period For each problem, find all points of absolute minima and maxima on the given interval. Calculus C o_0b1k5k gkbult_ai nsoo\fwtvwhairkew ULNLuCC._ ` naylflu [rhisg^h^tlsi traesgevrpvfe_dl. Final Eam Review Day 1 Name ID: 1 Date Period For each problem, find all points of absolute minima and

More information

Math 2413 General Review for Calculus Last Updated 02/23/2016

Math 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 information

Purdue University Study Guide for MA Credit Exam

Purdue 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 information

AP Calculus Prep Session Handout. Integral Defined Functions

AP Calculus Prep Session Handout. Integral Defined Functions AP Calculus Prep Session Handout A continuous, differentiable function can be epressed as a definite integral if it is difficult or impossible to determine the antiderivative of a function using known

More information

Curriculum Framework Alignment and Rationales for Answers

Curriculum Framework Alignment and Rationales for Answers The multiple-choice section on each eam is designed for broad coverage of the course content. Multiple-choice questions are discrete, as opposed to appearing in question sets, and the questions do not

More information

AP 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. 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 information

Mat 270 Final Exam Review Sheet Fall 2012 (Final on December 13th, 7:10 PM - 9:00 PM in PSH 153)

Mat 270 Final Exam Review Sheet Fall 2012 (Final on December 13th, 7:10 PM - 9:00 PM in PSH 153) Mat 70 Final Eam Review Sheet Fall 0 (Final on December th, 7:0 PM - 9:00 PM in PSH 5). Find the slope of the secant line to the graph of y f ( ) between the points f ( b) f ( a) ( a, f ( a)), and ( b,

More information

MONTGOMERY HIGH SCHOOL CP Pre-Calculus Final Exam Review

MONTGOMERY HIGH SCHOOL CP Pre-Calculus Final Exam Review MONTGOMERY HIGH SCHOOL 01-015 CP Pre-Calculus Final Eam Review The eam will cover the following chapters and concepts: Chapter 1 Chapter 1.1 Functions.1 Power and Radical Functions 1. Analyzing Graphs

More information

Integration Techniques for the AB exam

Integration Techniques for the AB exam For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior

More information

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

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

More information

West Essex Regional School District. AP Calculus AB. Summer Packet

West Essex Regional School District. AP Calculus AB. Summer Packet West Esse Regional School District AP Calculus AB Summer Packet 05-06 Calculus AB Calculus AB covers the equivalent of a one semester college calculus course. Our focus will be on differential and integral

More information

MATH Calculus I - Prerequisite Review

MATH Calculus I - Prerequisite Review MATH 241 - Calculus I - Prerequisite Review Give eact answers unless a problem specifies otherwise. + 5 1. Rationalize the numerator and simplify: 10 2. Simplify and give your answer in simplified radical

More information

sin x (B) sin x 1 (C) sin x + 1

sin x (B) sin x 1 (C) sin x + 1 ANSWER KEY Packet # AP Calculus AB Eam Multiple Choice Questions Answers are on the last page. NO CALCULATOR MAY BE USED IN THIS PART OF THE EXAMINATION. On the AP Eam, you will have minutes to answer

More information

(a) During what time intervals on [0, 4] is the particle traveling to the left?

(a) During what time intervals on [0, 4] is the particle traveling to the left? Chapter 5. (AB/BC, calculator) A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t /

More information

UNIT 2 SIMPLE APPLICATION OF DIFFERENTIAL CALCULUS

UNIT 2 SIMPLE APPLICATION OF DIFFERENTIAL CALCULUS Calculus UNIT 2 SIMPLE APPLICATION OF DIFFERENTIAL CALCULUS Structure 2.0 Introduction 2.1 Objectives 2.2 Rate of Change of Quantities 2.3 Increasing and Decreasing Function 2.4 Maima and Minima of Functions

More information

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least Problems to practice for FINAL. Below is the graph of a function () At which of the marked values ( and ) is: (a) () greatest = (b) () least = (c) () the greatest = (d) () the least = (e) () = = (f) ()

More information

Part Two. Diagnostic Test

Part Two. Diagnostic Test Part Two Diagnostic Test AP Calculus AB and BC Diagnostic Tests Take a moment to gauge your readiness for the AP Calculus eam by taking either the AB diagnostic test or the BC diagnostic test, depending

More information

Solutions to review problems MAT 125, Fall 2004

Solutions to review problems MAT 125, Fall 2004 Solutions to review problems MAT 125, Fall 200 1. For each of the following functions, find the absolute maimum and minimum values for f() in the given intervals. Also state the value where they occur.

More information

First Midterm Examination

First Midterm Examination Çankaya University Department of Mathematics 016-017 Fall Semester MATH 155 - Calculus for Engineering I First Midterm Eamination 1) Find the domain and range of the following functions. Eplain your solution.

More information

lim x c) lim 7. Using the guidelines discussed in class (domain, intercepts, symmetry, asymptotes, and sign analysis to

lim x c) lim 7. Using the guidelines discussed in class (domain, intercepts, symmetry, asymptotes, and sign analysis to Math 7 REVIEW Part I: Problems Using the precise definition of the it, show that [Find the that works for any arbitrarily chosen positive and show that it works] Determine the that will most likely work

More information

AB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2

AB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2 AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6

More information

5.5 Worksheet - Linearization

5.5 Worksheet - Linearization AP Calculus 4.5 Worksheet 5.5 Worksheet - Linearization All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. Consider the function = sin. a) Find the equation

More information

Review: 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. 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 information