y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions

Size: px
Start display at page:

Download "y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions"

Transcription

1 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E) ( 4) d (4 ) d ( 4) d ( + 4) d (4 ) d y= y y= d t 4 t d dt = (A) t 4 t (B) 4 (C) 4 4 (D) (E) 4

2 . π 4 sin d + π 4 cos d = (A) (B) (C) (D) (E) 4. If f() d = A and 5 f() d = B, then 5 f() d = (A) A + B (B) A B (C) (D) B A (E) 5. The average value of the function f() = ( ) on the interval from = to = 5 is (A) 6 (B) 6 (C) 64 (D) 66 (E) The average value of the function f() = ln on the interval [, 4] is (A).4 (B).4 (C).59 (D).48 (E) d d cos t dt = (A) sin (B) sin (C) cos (D) sin (E) cos

3 8. If the definite integral the Trapezoid Rule with n = 4, the error is (A) (B) 7 (C) ( + ) d is approimated by using (D) 65 6 (E) ln d = ln (A) + C ln (B) + C (C) ln + C (D) ln + C (E) ln + C. d = (A) C (B) C (C) 5 + C (D) + C (E) 5 + C. The average value of f() = e 4 on the interval [ 4, 4 ] is (A).7 (B).545 (C).9 (D).8 (E) 4.6

4 . Find the distance traveled (to three decimal places) in the first four seconds, for a particle whose velocity is given by v(t) = 7e t, where t stands for time. (A).976 (B) 6.4 (C) 6.59 (D).7 (E) 7.. Approimate to three decimal places. sin d using the Trapezoid Rule with n = 4 (A).77 (B).7 (C).555 (D).9 (E).9 4. Use the Trapezoid Rule with n = 4 to approimate the area between the curve y = and the -ais over the interval [, 4]. (A) 5.66 (B) (C) 6.6 (D).56 (E) The graph of the function f shown below consists of a semicircle and a straight line segment. Then y=f() 4 y 4 f() d = (A) π + (B) (π ) (C) π + (D) (π + ) (E) π +

5 6. d 5 cos t dt = d (A) 5 cos 5 cos (B) 5 sin 5 sin (C) cos 5 cos (D) sin 5 sin (E) 5 cos 5 sin 7. d d 5 cos t dt = (A) cos 5 5 cos (B) 5 cos cos 5 (C) cos 5 cos 5 (D) 5 cos 5 cos (E) 5 cos 5 + cos 8. Given that the function f is continuous on the interval [, ), and that f(t) dt =, then f (t) dt = (A) (B) /4 (C) 4 ln (D) 4 (ln ) (E) Can t be determined 9. Given that the function f is continuous on the interval [, ), and that f(t) dt =, then f (t) dt = (A) (B) 6 (C) 4 (D) (E)

6 Part II. Free-Response Questions. On the graph below, shade in the appropriate area indicated by the integral f()d. In the graph to the right, indicate the rectangles that would used in computing a RRAM (right rectangular approimation) to f()d using five rectangles, each of width. (Don t try to compute anything; just draw the relevant picture.) Does it appear the result gives an overestimate or an underestimate of the true area?

7 4 y y=f() The graph of the function f, consisting of three line segments, is given above. Let g() = f(t) dt. (a) Compute g(4) and g( ). g(4) = g( ) = f(t) dt = 4 f(t) dt = 6. f(t) dt = 5 ; (b) Find the instantaneous rate of change of g, with respect to, at =. By the Fundamental Theorem of Calculus, we have g () = f() = 4. (c) Find the minimum value of g on the closed interval [, 4]. Justify your answer. We have g () = f() = when = or when =. Comparing with endpoints we have g( ) = 6 (from part (a)), g() = f(t) dt =, and g(4) = 5 (from part (a)). Therefore, the minimum value of g on the closed interval [, 4] is 6 (which happens at = ).

8 . Let f be the function depicted in the graph to the right. Arrange the following numbers in increasing order: y y=f() f() d, f() d, f () d. f () d < f() d < f() d. 4. Assume that the average value of a function f over the interval [, ] is.5. Compute f()d. f()d is equal to 4 times its average value over this interval. Therefore, f()d = 4.5 =. 5. Epress the area of a circle of radius r as a definite integral. r r This is clearly r d = 4 r d. r 6. Find an antiderivative for each function below: (a) f() = + f() d = + + C. (b) g() = e g() d = + e + C. (c) v(t) = gt + v (here, v is just a constant) v(t) dt = gt + v t + C

9 (d) h() = /, where < (This is not hard, you just need to be careful!) h() d = ln( ) + C (This says that, in d general, = ln + C.) (e) k() = ( + )/ k() d = ln + + C. (f) θ() = /( + ) θ() d = ln + + C. 7. For each definite integral below, sketch the area represented. (Don t try to compute anything, just draw the relevant picture.) y (a) (b) e π π ln d cos d 4 Equation : y=ln y π π/ π/ π Equation : y=cos y (c) (d) t( t) dt Equation : y=( ) 4 f() d, where { if f() = if >. 4 y Equation : y=² Equation : y= Equation : y=² (( ))/ (( )) Equation 4: y=( ) (( )( ))/ (( )( ))

10 8. Compute the definite integrals (a) (9 )d = (9 ) d = 9 = 7 9 = 8. π (b) sin d = cos π = (c) 4 e d = e 4 = e 4. d (d) = sin = π/ d (e) + = tan = π/4 (f) d + = ln( + ) = ln 9. Solve for a in each of the below: a (a) ( )d = ( ) = a = a a a = a d (b) + = π 4 = a tan = tan a a = (c) a d = = ln a a = ln a a = e.. Suppose that f is an even function and that where a >. Compute a a f()d. Since f is an even function, we have a a f()d = a a f()d = =. f()d =,. Suppose that f is an odd function and that a >. Compute a a f()d. This is =. To prove this, note that a f() d =

11 a a a a f() d u= = f()d = f()d = a a a a f( u) du = f(u) du. That is to say, f() d, and so f()d+ a a a f()d = f() d+ f() d =.. Solve the differential equations: (a) y =, y() =. y = ( ) d = + C. Also, = y() = + C C =. Therefore, y =. (b) y = e, y() =. y = ( e ) d = + e + C. Also, = y() = + e + C C =. Therefore, y = + e. (c) ( + )y d =, y() =. y = = ln + + C. Also, + = y() = ln + C C =. Therefore, y = ln +. (d) y = y, y() =. Write this as dy = y, which separates as d dy y = d ln y = + C. Therefore, y = e +C ; it s better to write this as y = Ke, where K is a (possibly negative) arbitrary constant. From = y() = Ke = K, get K = and so y = e. (e) y = y, y() =. Argue almost eactly as in (d) to arrive at the solution y = e.. Suppose that you know that f () =. What can you say about f()? Write down the most general form that f() could have. Clearly the most general solution must be of the form f() = A + B, where A and B are arbitrary constants. 4. Suppose that the velocity of a particle is given by v(t) = + sin t, t, where v is given in units of cm/sec. How far has this particle traveled after sec?

12 We know that the TOTAL DISTANCE the particle travels is given by the integral s(t) dt, where s(t) is the speed of the particle. Net, since s(t) = v(t), we have that the total distance traveled is v(t) dt = +sin t dt = (+sin t) dt = t cos t cos.84 cm. = 5. Assume that a water pump is pumping water into a large tank at a variable rate: after t hours, water is being pumped at a rate of v(t) = 5t + t gallons/hour. (a) Write down the definite integral that epresses how much water has been pumped into the vessel after 4 hours. (Don t 4 5t dt compute this.) This is clearly + t. (b) Compute how much water is in the tank after 4 hours, assuming that the tank was empty to begin with and that F (t) = 5 (t ln( + t)) is an antiderivative for v(t). The amount pumped into the tank over the first 4 hours is 4 5t dt + t = 5(t ln(+t)) 4 = 5(4 ln 5) 9.6 gallons.

13 6. The rate at which people enter an amusement park on a given day is modeled by the function E defined by E(t) = 56 t 4t + 5. the rate at which people leave the same amusement park on the same day is modeled by the function L defined by L(t) = 989 t 8t + 7. Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 t, the hours during which the park is open. At time t = 9, there are no people in the park. (a) How many people have entered the park by 5: P.M. (t = 7)? Round your answer to the nearest whole number. This is the integral of the rate of people entering the park: dt, 746 people, where the approimation was done via numerical integration on a TI-84 calcula- t 4t + 5 tor. (b) The price of admission to the park is $5 until 5: P.M. (t = 7). After 5: P.M., the price of admission to the park is $. How many dollars are collected from admission to the park on the given day? Round your answer to the nearest whole number. The total in receipts would be the sum $5 7 7 E(t) dt + $ 56 dt = 5 t 4t $ = $5, $, 56 = $7, E(t) dt 56 dt t 4t + 5 E(t) dt =

14 (c) Let H(t) = t 9 (E() L()) d for 9 t. The value of H(7) to the nearest whole number is 75. Find the value of H (7) and eplain the meaning of H(7) and H (7) in the contet of the amusement park. The meaning of H(7) is the total number of people having arrived in the park between 9 a.m. and 5 p.m. less the total number of people having left the park during this same time. In other words, H(7) is the total number of people in the park at 5 p.m. H (7) = E(7) L(7) which is the net rate of people (in people per hour) entering the park at 5 p.m. (d) At what time, t, for 9 t, does the model predict that the number of people in the park is a maimum? As H(t) gives the total number of people in the park at time t, the maimum number of people in the park will happen when H (t) =, i.e., when E() = L(). Using a graphics calculator, one infers that this happens when t 6.7, i.e. at roughly 4: p.m.

15 (, ) Graph of f (, ) 7. The graph of the function f shown above consists of two line segments. Let g be the funtion given by g() = (a) Find g( ), g ( ), and g ( ). g( ) =.5. g ( ) = f( ) = ; g ( ) = f ( ) =. f(t) dt. f(t) dt = (b) Over which interval(s) within (, ) is g increasing? Eplain your reasoning. g is increasing where g () >, i.e., where f() >. This latter condition is satisfied on the interval < <. (c) Over which interval(s) within (, ) is the graph of g concave down? Eplain your reasoning. The graph of g is concave down where g () <, i.e., where f () <. This happens on the interval < <. (d) On the aes provided, sketch the graph of g on the closed interval [, ]. f(t) dt =

16 t (minutes) R(t) (gallons per minute) The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of t. A table of selected values of R(t), for the time interval t 9 minutes, is shown above. (a) Use data from the table to find an approimation for R (45). Show the computations that lead to your answer. Indicate units of measure. R R(5) R(4) (45) = =.5 (gallons/min ). (b) The rate of fuel consumption is increasing fastest at time t = 45. What is the value of R (45)? Eplain your reasoning. For the rate of fuel consumption to be increasing at its fastest, we must have that R (t) is a maimum. This happens (since R is twice-differentiable) when R (t) = ; therefore, we infer that R (45) =. (c) Approimate the value of 9 R(t) dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approimation less than the value of 9 9 R(t) dt? Eplain your reasoning. We have R(t) dt =, 7 gallons. Since R is a strictly-increasing function of t, we conclude that the left Riemann sum must give an underestimate of the total fuel consumption over the first 9 minutes.

17 b (d) For < b 9 minutes, eplain the meaning of in terms of fuel consumption for the plane. meaning of b b R(t) dt Eplain the R(t) dt in terms of fuel consumption for this plane. Indicate units of measure in both answers. b R(t) dt is the total fuel consumption (in gallons) over the first b minutes, where < b 9. The integral R(t) dt is the average fuel consumption (in gallons per minute) over the first b b minutes. b

18 y y=8- y=f() R S 9. Let f be the function given by f() = 4, and let L be the line y = 8, where L is tangent to the graph of f. Let R be the region bounded by the graph of f and the -ais, and let S be the region bounded by the graph of f, the line l, and the -ais, as shown above. (a) Show that L is tangent to the graph of y = f() at the point =. We have that f () = 8 = = 4 7 =, and so the line has the correct slope. Also, f() = 4 = 9 = 8, and so the line and the graph of y = f() both intersect where =. These two facts imply that the given straight line is tangent to the graph of y = f() at the indicated point. (b) Find the area of S. Area (S) = 6 = 7 = 7 4 (8 ) d (4 ) d ( 4 ) ( ) ( ) 4 = 95.

19 4. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by F (t) = sin ( ) t for t, where F (t) is measured in cars per minute and t is measured in minutes. (a) To the nearest whole number, how many cars pass through the intersection over the -minute period? The total number of cars passing through the given intersection over the -minute period is the integral: F (t) dt = ( sin ( )) t dt, 474 cars. (b) Is the traffic flow increasing or decreasing at t = 7? Give a reason for your answer. The traffic flow is increasing or decreasing at t = 7 according as to whether F (7) > or F (7) <. However, ( ) 7 F (7) = cos.87 <, and so the traffic flow is decreasing at t = 7. (c) What is the average value of the traffic flow over the time interval t 5? Indicate units of measure. The average traffic flow over the time interval t 5 is 5 5 ( sin ( t )) dt 8.9 cars/min (d) What is the average rate of change of the traffic flow over the time interval t 5? Indicate units of measure. The average rate of change of the traffic flow over the time interval t 5 is the difference quotient F (5) F () 5 = 5 ( 4 sin ( ) 5 4 sin ( )).5 cars/min

20 4 y y=f() Let f be the function defined on the closed interval [, 7]. The graph of f, consisting of four line segments, is shown above. Let g be the function given by g() = (a) Find g(), g (), and g (). g() = f() =, g () = f () =. f(t) dt. f(t) dt =, g () = (b) Find the average rate of change of g on the interval g() g(). This is = ( 4) = 7. (c) For how many values c, where < c <, is g (c) equal to the average rate found in part (b)? Eplain your reasoning. As g (c) = f(c), we are seeking the number of solutions of f(c) = 7 on the interval < c <. An inspection of the graph reveals that there are two such solutions. (d) Find the -coordinate of each point of inflection of the graph of g on the interval < < 7. Justify your answer. There are no solutions of g () = f () = on the interval < < 7. However, there are changes in the concavity of the graph of y = g() at = and at = 5 since the sign of f () changes sign at these two values of.

21 y y=f() R - y=g() 4. Let f and g be the functions given by f() = ( ) and g() = ( ) for. The graphs of f and g are shown in the figure above. Find the area of the region R enclosed by the graphs of f and g. This area is given by the integral (f() g()) d = = (( ) ( ) ) d ( 6 ) 5 5/ + / = = 7 5.

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)

More information

AP Calculus AB Free-Response Scoring Guidelines

AP Calculus AB Free-Response Scoring Guidelines Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per

More information

CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums

CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums INTEGRAL AND AREA BY HAND (APPEAL TO GEOMETRY) I. Below are graphs that each represent a different f()

More information

CALCULUS AB SECTION II, Part A

CALCULUS AB SECTION II, Part A CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank

More information

AP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66

AP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66 AP Calculus BC Chapter 4 REVIEW 4.1 4.4 Name Date Period NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW. 1. 4 d dt (3t 2 + 2t 1) dt = 2 (A) 12 (B) 4 (C) 46 (D) 55 (E) 66 2. The velocity of a particle

More information

ANOTHER FIVE QUESTIONS:

ANOTHER FIVE QUESTIONS: No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e ln ln @ @ Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE

More information

cos 5x dx e dt dx 20. CALCULUS AB WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator.

cos 5x dx e dt dx 20. CALCULUS AB WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator. WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator. Find the derivative. Do not leave negative eponents or comple fractions in our answers. 4. 8 4 f

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus 6.. Worksheet Estimating with Finite Sums All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Suppose an oil pump is producing 8 gallons per hour

More information

Name: Period: For full credit, show all step by step work required to support your answers on your own paper.

Name: Period: For full credit, show all step by step work required to support your answers on your own paper. Name: Period: For full credit, show all step by step work required to support your answers on your own paper. 1. The temperature outside a house during a 4-hour period is given by t F t 8 1cos 1, t 4 F

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

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

Ex. Find the derivative. Do not leave negative exponents or complex fractions in your answers.

Ex. Find the derivative. Do not leave negative exponents or complex fractions in your answers. CALCULUS AB THE SECOND FUNDAMENTAL THEOREM OF CALCULUS AND REVIEW E. Find the derivative. Do not leave negative eponents or comple fractions in your answers. 4 (a) y 4 e 5 f sin (b) sec (c) g 5 (d) y 4

More information

AP Calculus AB 2nd Semester Homework List

AP Calculus AB 2nd Semester Homework List AP Calculus AB 2nd Semester Homework List Date Assigned: 1/4 DUE Date: 1/6 Title: Typsetting Basic L A TEX and Sigma Notation Write the homework out on paper. Then type the homework on L A TEX. Use this

More information

Final Value = Starting Value + Accumulated Change. Final Position = Initial Position + Displacement

Final Value = Starting Value + Accumulated Change. Final Position = Initial Position + Displacement Accumulation, Particle Motion Big Ideas Fundamental Theorem of Calculus and Accumulation AP Calculus Course Description Goals page 6 Students should understand the meaning of the definite integral both

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

Chapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry

Chapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry AP Calculus Chapter Review Name: Block:. [No Calculator] Evaluate using the FTOC (the evaluation part) a) 7 8 4 7 d b) 9 4 7 d. [No Calculator] Evaluate using geometry a) d c) 6 8 d. [No Calculator] Evaluate

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus 5. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Suppose an oil pump is producing 8 gallons per hour for the first 5 hours of

More information

A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the

A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the attached packet of problems, and turn it in on Monday, August

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

(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

Chapter 6: The Definite Integral

Chapter 6: The Definite Integral Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives

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

(A) 47 ft/sec (B) 52 ft/sec (C) 120 ft/sec (D) 125 ft/sec (E) 141 ft/sec

(A) 47 ft/sec (B) 52 ft/sec (C) 120 ft/sec (D) 125 ft/sec (E) 141 ft/sec Name Date Period Worksheet 6.1 Integral as Net Change Show all work. Calculator Permitted, but show all integral set ups. Multiple Choice 1. The graph at right shows the rate at which water is pumped from

More information

Motion with Integrals Worksheet 4: What you need to know about Motion along the x-axis (Part 2)

Motion with Integrals Worksheet 4: What you need to know about Motion along the x-axis (Part 2) Motion with Integrals Worksheet 4: What you need to know about Motion along the x-axis (Part 2) 1. Speed is the absolute value of. 2. If the velocity and acceleration have the sign (either both positive

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

Part 1: Integration problems from exams

Part 1: Integration problems from exams . Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating

More information

2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part

2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 1. Let R be the region in the first and second quadrants bounded

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

a t of a car from 0 to 15 seconds are given in the table below. If the

a t of a car from 0 to 15 seconds are given in the table below. If the Name Date Period Worksheet 8.1 Integral as Net Change Show all work. Calculator Permitted, but show all integral set ups. Multiple Choice 1. The graph at right shows the rate at which water is pumped from

More information

LSU AP Calculus Practice Test Day

LSU AP Calculus Practice Test Day LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part A AP CALCULUS AB: PRACTICE EXAM SECTION I: PART A NO CALCULATORS ALLOWED. YOU HAVE 60 MINUTES. 1. If y = ( 1 + x 5) 3

More information

Unit #6 Basic Integration and Applications Homework Packet

Unit #6 Basic Integration and Applications Homework Packet Unit #6 Basic Integration and Applications Homework Packet For problems, find the indefinite integrals below.. x 3 3. x 3x 3. x x 3x 4. 3 / x x 5. x 6. 3x x3 x 3 x w w 7. y 3 y dy 8. dw Daily Lessons and

More information

AP Calculus Prep Session Handout. Table Problems

AP Calculus Prep Session Handout. Table Problems AP Calculus Prep Session Handout The AP Calculus Exams include multiple choice and free response questions in which the stem of the question includes a table of numerical information from which the students

More information

Chapter 4 Overview: Definite Integrals

Chapter 4 Overview: Definite Integrals Chapter Overview: Definite Integrals In this chapter, we will study the Fundamental Theorem of Calculus, which establishes the link between the algebra and the geometry, with an emphasis on the mechanics

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

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

AP Calculus BC Multiple-Choice Answer Key!

AP Calculus BC Multiple-Choice Answer Key! Multiple-Choice Answer Key!!!!! "#$$%&'! "#$$%&'!!,#-! ()*+%$,#-! ()*+%$!!!!!! "!!!!! "!! 5!! 6! 7!! 8! 7! 9!!! 5:!!!!! 5! (!!!! 5! "! 5!!! 5!! 8! (!! 56! "! :!!! 59!!!!! 5! 7!!!! 5!!!!! 55! "! 6! "!!

More information

Sample Final Exam 4 MATH 1110 CALCULUS I FOR ENGINEERS

Sample Final Exam 4 MATH 1110 CALCULUS I FOR ENGINEERS Dept. of Math. Sciences, UAEU Sample Final Eam Fall 006 Sample Final Eam MATH 0 CALCULUS I FOR ENGINEERS Section I: Multiple Choice Problems [0% of Total Final Mark, distributed equally] No partial credit

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

Chapter 4 Overview: Definite Integrals

Chapter 4 Overview: Definite Integrals Chapter Overview: Definite Integrals In the Introduction to this book, we pointed out that there are four tools or operations in Calculus. This chapter presents the fourth the Definite Integral. Where

More information

e x Improper Integral , dx

e x Improper Integral , dx Improper Integral ff() dddd aa bb, ff() dddd, ff() dddd e, d An improper integral is a definite integral that has. an infinite interval of integration.. They have a discontinuity on the interior of the

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

Chapter 4 Overview: Definite Integrals

Chapter 4 Overview: Definite Integrals Chapter Overview: Definite Integrals In the Introduction to this book, we pointed out that there are four tools or operations in Calculus. This chapter presents the fourth the Definite Integral. Where

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

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2) . f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula

More information

AP Calculus BC Fall Final Part IIa

AP Calculus BC Fall Final Part IIa AP Calculus BC 18-19 Fall Final Part IIa Calculator Required Name: 1. At time t = 0, there are 120 gallons of oil in a tank. During the time interval 0 t 10 hours, oil flows into the tank at a rate of

More information

AP Calculus Exam Format and Calculator Tips:

AP Calculus Exam Format and Calculator Tips: AP Calculus Exam Format and Calculator Tips: Exam Format: The exam is 3 hours and 15 minutes long and has two sections multiple choice and free response. A graphing calculator is required for parts of

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

Examples of the Accumulation Function (ANSWERS) dy dx. This new function now passes through (0,2). Make a sketch of your new shifted graph.

Examples of the Accumulation Function (ANSWERS) dy dx. This new function now passes through (0,2). Make a sketch of your new shifted graph. Eamples of the Accumulation Function (ANSWERS) Eample. Find a function y=f() whose derivative is that f()=. dy d tan that satisfies the condition We can use the Fundamental Theorem to write a function

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

A.P. Calculus BC First Semester Exam Calculators Allowed Two Hours Number of Questions 10

A.P. Calculus BC First Semester Exam Calculators Allowed Two Hours Number of Questions 10 A.P. Calculus BC First Semester Exam Calculators Allowed Two Hours Number of Questions 10 Each of the ten questions is worth 10 points. The problem whose solution you write counted again, so that the maximum

More information

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. AP Calculus AB Exam SECTION I: Multiple Choice 016 DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 71 INTEGRAL AS NET CHANGE Distance versus Displacement We have already seen how the position of an object can be found by finding the integral of the velocity function The change in position is a displacement

More information

AP Calculus AB Riemann Sums

AP Calculus AB Riemann Sums AP Calculus AB Riemann Sums Name Intro Activity: The Gorilla Problem A gorilla (wearing a parachute) jumped off of the top of a building. We were able to record the velocity of the gorilla with respect

More information

+ 1 for x > 2 (B) (E) (B) 2. (C) 1 (D) 2 (E) Nonexistent

+ 1 for x > 2 (B) (E) (B) 2. (C) 1 (D) 2 (E) Nonexistent dx = (A) 3 sin(3x ) + C 1. cos ( 3x) 1 (B) sin(3x ) + C 3 1 (C) sin(3x ) + C 3 (D) sin( 3x ) + C (E) 3 sin(3x ) + C 6 3 2x + 6x 2. lim 5 3 x 0 4x + 3x (A) 0 1 (B) 2 (C) 1 (D) 2 (E) Nonexistent is 2 x 3x

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

Answer Key for AP Calculus AB Practice Exam, Section I

Answer Key for AP Calculus AB Practice Exam, Section I Answer Key for AP Calculus AB Practice Exam, Section I Multiple-Choice Questions Question # Key B B 3 A 4 E C 6 D 7 E 8 C 9 E A A C 3 D 4 A A 6 B 7 A 8 B 9 C D E B 3 A 4 A E 6 A 7 A 8 A 76 E 77 A 78 D

More information

Review Sheet for Exam 1 SOLUTIONS

Review Sheet for Exam 1 SOLUTIONS Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma

More information

A MATH 1225 Practice Test 4 NAME: SOLUTIONS CRN:

A MATH 1225 Practice Test 4 NAME: SOLUTIONS CRN: A MATH 5 Practice Test 4 NAME: SOLUTIONS CRN: Multiple Choice No partial credit will be given. Clearly circle one answer. No calculator!. Which of the following must be true (you may select more than one

More information

Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value

Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value AP Calculus Unit 6 Basic Integration & Applications Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value b (1) v( t) dt p( b) p( a), where v(t) represents the velocity and

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

Calculus AB 2014 Scoring Guidelines

Calculus AB 2014 Scoring Guidelines P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official

More information

1998 Calculus AB Scoring Guidelines

1998 Calculus AB Scoring Guidelines 41 Velocity (feet per second) v(t) 9 8 7 6 5 4 1 O 1998 Calculus AB Scoring Guidelines 5 1 15 5 5 4 45 5 Time (seconds) t t v(t) (seconds) (feet per second) 5 1 1 15 55 5 7 78 5 81 4 75 45 6 5 7. The graph

More information

MA 114 Worksheet #01: Integration by parts

MA 114 Worksheet #01: Integration by parts Fall 8 MA 4 Worksheet Thursday, 3 August 8 MA 4 Worksheet #: Integration by parts. For each of the following integrals, determine if it is best evaluated by integration by parts or by substitution. If

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

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) + 5 + + 5 ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin

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

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

7.2 Trapezoidal Approximation

7.2 Trapezoidal Approximation 7. Trapezoidal Approximation NOTES Write your questions here! Riemann Sum f(x) = x + 1 [1,3] f(x) = x + 1 The Definite Integral b f(x)dx a Trapezoidal Approximation for interval [1,3] with n subintervals

More information

Level 1 Calculus Final Exam Day 1 50 minutes

Level 1 Calculus Final Exam Day 1 50 minutes Level 1 Calculus Final Exam 2013 Day 1 50 minutes Name: Block: Circle Teacher Name LeBlanc Normile Instructions Write answers in the space provided and show all work. Calculators okay but observe instructions

More information

Understanding Part 2 of The Fundamental Theorem of Calculus

Understanding Part 2 of The Fundamental Theorem of Calculus Understanding Part of The Fundamental Theorem of Calculus Worksheet 8: The Graph of F () What is an Anti-Derivative? Give an eample that is algebraic: and an eample that is graphical: eample : Below is

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

Practice problems from old exams for math 132 William H. Meeks III

Practice 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 information

AP Calculus BC. Practice Exam. Advanced Placement Program

AP Calculus BC. Practice Exam. Advanced Placement Program Advanced Placement Program AP Calculus BC Practice Eam The questions contained in this AP Calculus BC Practice Eam are written to the content specifications of AP Eams for this subject. Taking this practice

More information

AP Calculus BC 2015 Free-Response Questions

AP Calculus BC 2015 Free-Response Questions AP Calculus BC 05 Free-Response Questions 05 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central

More information

1 Exam 1 Spring 2007.

1 Exam 1 Spring 2007. Exam Spring 2007.. An object is moving along a line. At each time t, its velocity v(t is given by v(t = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = 5. 2. Use the

More information

Calculus 1: Sample Questions, Final Exam

Calculus 1: Sample Questions, Final Exam Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)

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

L. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c)

L. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c) L. Function Analysis What you are finding: You have a function f ( x). You want to find intervals where f ( x) is increasing and decreasing, concave up and concave down. You also want to find values of

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

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

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time 55 minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems,

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

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

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks) Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.

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

(a) Find the area of RR. (b) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(a) Find the area of RR. (b) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus AB Final Review Name: Revised 07 EXAM Date: Tuesday, May 9 Reminders:. Put new batteries in your calculator. Make sure your calculator is in RADIAN mode.. Get a good night s sleep. Eat breakfast

More information

1993 AP Calculus AB: Section I

1993 AP Calculus AB: Section I 99 AP Calculus AB: Section I 9 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among

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

. CALCULUS AB. Name: Class: Date:

. CALCULUS AB. Name: Class: Date: Class: _ Date: _. 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, using

More information

AP Calculus AB. Free-Response Questions

AP Calculus AB. Free-Response Questions 2018 AP Calculus AB Free-Response Questions College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online

More information

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 997 AP Calculus BC: Section I, Part A 5 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..

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

Problem Set Four Integration AP Calculus AB

Problem Set Four Integration AP Calculus AB Multiple Choice: No Calculator unless otherwise indicated. ) sec 2 x 2dx tan x + C tan x 2x +C tan x x + C d) 2tanxsec 2 x x + C x 2 2) If F(x)= t + dt, what is F (x) x 2 + 2 x 2 + 2x x 2 + d) 2(x2 +)

More information

Fundamental Theorem of Calculus

Fundamental Theorem of Calculus NCTM Annual Meeting and Eposition Denver, CO April 8, Presented by Mike Koehler Blue Valley North High School Overland Park, KS I. Approimations with Rectangles (Finding the Area Under Curves by Approimating

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

24. AB Calculus Step-by-Step Name. a. For what values of x does f on [-4,4] have a relative minimum and relative maximum? Justify your answers.

24. AB Calculus Step-by-Step Name. a. For what values of x does f on [-4,4] have a relative minimum and relative maximum? Justify your answers. 24. AB Calculus Step-by-Step Name The figure to the right shows the graph of f!, the derivative of the odd function f. This graph has horizontal tangents at x = 1 and x = 3. The domain of f is!4 " x "

More information

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

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

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

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