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

Save this PDF as:
 WORD  PNG  TXT  JPG

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.

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

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

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

(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

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 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

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 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

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

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

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

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

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

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

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

(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

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

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

1969 AP Calculus BC: Section I

1969 AP Calculus BC: Section I 969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric

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

Limits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4

Limits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4 Limits and Continuity t+ 1. lim t - t + 4. lim x x x x + - 9-18 x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4 6. x

More information

Calculus with the Graphing Calculator

Calculus with the Graphing Calculator Calculus with the Graphing Calculator Using a graphing calculator on the AP Calculus exam Students are expected to know how to use their graphing calculators on the AP Calculus exams proficiently to accomplish

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

Section 4.4 The Fundamental Theorem of Calculus

Section 4.4 The Fundamental Theorem of Calculus Section 4.4 The Fundamental Theorem of Calculus The First Fundamental Theorem of Calculus If f is continuous on the interval [a,b] and F is any function that satisfies F '() = f() throughout this interval

More information

Review Sheet for Second Midterm Mathematics 1300, Calculus 1

Review Sheet for Second Midterm Mathematics 1300, Calculus 1 Review Sheet for Second Midterm Mathematics 300, Calculus. For what values of is the graph of y = 5 5 both increasing and concave up? >. 2. Where does the tangent line to y = 2 through (0, ) intersect

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

with the initial condition y 2 1. Find y 3. the particular solution, and use your particular solution to find y 3.

with the initial condition y 2 1. Find y 3. the particular solution, and use your particular solution to find y 3. FUNDAMENTAL THEOREM OF CALCULUS Given d d 4 Method : Integrate with the initial condition. Find. 4 d, and use the initial condition to find C. Then write the particular solution, and use our particular

More information

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

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

More information

Arkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam

Arkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam 0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker

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

Distance And Velocity

Distance And Velocity Unit #8 - The Integral Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Distance And Velocity. The graph below shows the velocity, v, of an object (in meters/sec). Estimate

More information

Calculus 1: A Large and In Charge Review Solutions

Calculus 1: A Large and In Charge Review Solutions Calculus : A Large and n Charge Review Solutions use the symbol which is shorthand for the phrase there eists.. We use the formula that Average Rate of Change is given by f(b) f(a) b a (a) (b) 5 = 3 =

More information

Math 113 Final Exam Practice Problem Solutions. f(x) = ln x x. lim. lim. x x = lim. = lim 2

Math 113 Final Exam Practice Problem Solutions. f(x) = ln x x. lim. lim. x x = lim. = lim 2 Math 3 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

MTH 252 Lab Supplement

MTH 252 Lab Supplement Fall 7 Pilot MTH 5 Lab Supplement Supplemental Material by Austina Fong Contents Antiderivatives... Trigonometric Substitution... Approimate Integrals Technology Lab (Optional)... 4 Error Bound Formulas...

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions.

More information

AP Calculus AB Winter Break Packet Happy Holidays!

AP Calculus AB Winter Break Packet Happy Holidays! AP Calculus AB Winter Break Packet 04 Happy Holidays! Section I NO CALCULATORS MAY BE USED IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of

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.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove

More information

MAC Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below.

MAC Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. MAC 23. Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. (x, y) y = 3 x + 4 a. x = 6 b. x = 4 c. x = 2 d. x = 5 e. x = 3 2. Consider the area of the

More information

Solutions to Math 41 Final Exam December 10, 2012

Solutions to Math 41 Final Exam December 10, 2012 Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)

More information

1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3

1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3 [STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)

More information

AP Calculus Worksheet: Chapter 2 Review Part I

AP Calculus Worksheet: Chapter 2 Review Part I AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative

More information

I. Horizontal and Vertical Tangent Lines

I. Horizontal and Vertical Tangent Lines How to find them: You need to work with f " x Horizontal tangent lines: set f " x Vertical tangent lines: find values of x where f " x I. Horizontal and Vertical Tangent Lines ( ), the derivative of function

More information

6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2.

6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2. 5 THE ITEGRAL 5. Approimating and Computing Area Preliminar Questions. What are the right and left endpoints if [, 5] is divided into si subintervals? If the interval [, 5] is divided into si subintervals,

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

One of the most common applications of Calculus involves determining maximum or minimum values.

One of the most common applications of Calculus involves determining maximum or minimum values. 8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..

More information

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

Find all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) = Math 90 Final Review Find all points where the function is discontinuous. ) Find all vertical asymptotes of the given function. x(x - ) 2) f(x) = x3 + 4x Provide an appropriate response. 3) If x 3 f(x)

More information

AP Exam Practice Questions for Chapter 3

AP Exam Practice Questions for Chapter 3 AP Eam Practice Questions for Chapter AP Eam Practice Questions for Chapter f + 6 7 9 f + 7 0 + 6 0 ( + )( ) 0,. The critical numbers of f are and. So, the answer is B.. Evaluate each statement. I: Because

More information

NO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity.

NO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity. AP CALCULUS BC NO CALCULATORS: MIDTERM REVIEW 1. Find lim 7x 6x x 7 x 9. 1 B) 0 C) D). Find the points of discontinuity of the function y of discontinuity. x 9x 0. For each discontinuity identify the type

More information

Solutions to Homework Assignment #2

Solutions to Homework Assignment #2 Solutions to Homework Assignment #. [4 marks] Evaluate each of the following limits. n i a lim n. b lim c lim d lim n i. sin πi n. a i n + b, where a and b are constants. n a There are ways to do this

More information

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt. Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,

More information

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y. 90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y

More information

7.1 Indefinite Integrals Calculus

7.1 Indefinite Integrals Calculus 7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential

More information

AP Calculus AB 2015 Free-Response Questions

AP Calculus AB 2015 Free-Response Questions AP Calculus AB 015 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

Mat 210 Business Calculus Final Exam Review Spring Final on April 28 in COOR HALL 199 at 7:30 AM

Mat 210 Business Calculus Final Exam Review Spring Final on April 28 in COOR HALL 199 at 7:30 AM f ( Mat Business Calculus Final Eam Review Spring Final on April 8 in COOR HALL 99 at 7: AM. A: Find the limit (if it eists) as indicated. Justify your answer. 8 a) lim (Ans: 6) b) lim (Ans: -) c) lim

More information

Summer Review Packet (Limits & Derivatives) 1. Answer the following questions using the graph of ƒ(x) given below.

Summer Review Packet (Limits & Derivatives) 1. Answer the following questions using the graph of ƒ(x) given below. Name AP Calculus BC Summer Review Packet (Limits & Derivatives) Limits 1. Answer the following questions using the graph of ƒ() given below. (a) Find ƒ(0) (b) Find ƒ() (c) Find f( ) 5 (d) Find f( ) 0 (e)

More information

Mathematics 2005 HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics 2005 HIGHER SCHOOL CERTIFICATE EXAMINATION 005 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators may be used A table of standard

More information

Sections 5.1: Areas and Distances

Sections 5.1: Areas and Distances Sections.: Areas and Distances In this section we shall consider problems closely related to the problems we considered at the beginning of the semester (the tangent and velocity problems). Specifically,

More information

(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.

(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N. Calculus Review Packet 1. Consider the function f() = 3 3 2 24 + 30. Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point,

More information

abc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES

abc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS

More information

Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)

Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x

More information

Average rates of change May be used to estimate the derivative at a point

Average rates of change May be used to estimate the derivative at a point Derivatives Big Ideas Rule of Four: Numerically, Graphically, Analytically, and Verbally Average rate of Change: Difference Quotient: y x f( a+ h) f( a) f( a) f( a h) f( a+ h) f( a h) h h h Average rates

More information

Math 1325 Final Exam Review. (Set it up, but do not simplify) lim

Math 1325 Final Exam Review. (Set it up, but do not simplify) lim . Given f( ), find Math 5 Final Eam Review f h f. h0 h a. If f ( ) 5 (Set it up, but do not simplify) If c. If f ( ) 5 f (Simplify) ( ) 7 f (Set it up, but do not simplify) ( ) 7 (Simplify) d. If f. Given

More information

Math 2300 Calculus II University of Colorado

Math 2300 Calculus II University of Colorado Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,

More information

AP Calculus AB 2002 Free-Response Questions

AP Calculus AB 2002 Free-Response Questions AP Calculus AB 00 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be

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

18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then

18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then 8. Final Answers. (a) By the product rule, ( e ) = e + e = e ( + ). (b) If f() = sin(), then f (7) () = 8 cos() since: f () () = cos() f () () = 4 sin() f () () = 8 cos() f (4) () = 6 sin() f (5) () =

More information

( ) for t 0. Rectilinear motion CW. ( ) = t sin t ( Calculator)

( ) for t 0. Rectilinear motion CW. ( ) = t sin t ( Calculator) Rectilinear motion CW 1997 ( Calculator) 1) A particle moves along the x-axis so that its velocity at any time t is given by v(t) = 3t 2 2t 1. The position x(t) is 5 for t = 2. a) Write a polynomial expression

More information

Calculus 1 Exam 1 MAT 250, Spring 2012 D. Ivanšić. Name: Show all your work!

Calculus 1 Exam 1 MAT 250, Spring 2012 D. Ivanšić. Name: Show all your work! Calculus 1 Exam 1 MAT 250, Spring 2012 D. Ivanšić Name: Show all your work! 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 1 f(x)

More information

y = (x2 +1) cos(x) 2x sin(x) d) y = ln(sin(x 2 )) y = 2x cos(x2 ) by the chain rule applied twice. Once to ln(u) and once to

y = (x2 +1) cos(x) 2x sin(x) d) y = ln(sin(x 2 )) y = 2x cos(x2 ) by the chain rule applied twice. Once to ln(u) and once to M408N Final Eam Solutions, December 13, 2011 1) (32 points, 2 pages) Compute dy/d in each of these situations. You do not need to simplify: a) y = 3 + 2 2 14 + 32 y = 3 2 + 4 14, by the n n 1 formula.

More information

AP Calculus Free-Response Questions 1969-present AB

AP Calculus Free-Response Questions 1969-present AB AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions

More information

Write the make and model of your calculator on the front cover of your answer booklets e.g. Casio fx-9750g, Sharp EL-9400, Texas Instruments TI-85.

Write the make and model of your calculator on the front cover of your answer booklets e.g. Casio fx-9750g, Sharp EL-9400, Texas Instruments TI-85. INTERNATIONAL BACCALAUREATE BACCALAURÉAT INTERNATIONAL BACHILLERATO INTERNACIONAL M01/520/S(2) MATHEMATICAL METHODS STANDARD LEVEL PAPER 2 Tuesday 8 May 2001 (morning) 2 hours INSTRUCTIONS TO CANDIDATES

More information

CHAPTER 72 AREAS UNDER AND BETWEEN CURVES

CHAPTER 72 AREAS UNDER AND BETWEEN CURVES CHAPTER 7 AREAS UNDER AND BETWEEN CURVES EXERCISE 8 Page 77. Show by integration that the area of the triangle formed by the line y, the ordinates and and the -ais is 6 square units. A sketch of y is shown

More information

Math 117. Study Guide for Exam #1

Math 117. Study Guide for Exam #1 Math 117 Study Guide for Eam #1 Structure of the Eam 0 to 1 problem of finding the derivative of a function by definition (most likely a polynomial) 3 to problems of finding the derivative of functions

More information

AP Calculus BC Summer Review

AP Calculus BC Summer Review AP Calculus BC 07-08 Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it

More information

4.1 Analysis of functions I: Increase, decrease and concavity

4.1 Analysis of functions I: Increase, decrease and concavity 4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval

More information

University of Waterloo Final Examination MATH 116 Calculus 1 for Engineering

University of Waterloo Final Examination MATH 116 Calculus 1 for Engineering Last name (Print): First name (Print): UW Student ID Number: University of Waterloo Final Eamination MATH 116 Calculus 1 for Engineering Instructor: Matthew Douglas Johnston Section: 001 Date: Monday,

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

Bonus Homework and Exam Review - Math 141, Frank Thorne Due Friday, December 9 at the start of the final exam.

Bonus Homework and Exam Review - Math 141, Frank Thorne Due Friday, December 9 at the start of the final exam. Bonus Homework and Exam Review - Math 141, Frank Thorne (thornef@mailbox.sc.edu) Due Friday, December 9 at the start of the final exam. It is strongly recommended that you do as many of these problems

More information

Calculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić. Name: Show all your work!

Calculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić. Name: Show all your work! Calculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić Name: Show all your work! 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 2 f(x)

More information

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Solve the 4 multiple-choice problems below. A graphing calculator is required for some questions

More information

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14. Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +

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

Review Problems for the Final

Review Problems for the Final Review Problems for the Final Math 6-3/6 3 7 These problems are intended to help you study for the final However, you shouldn t assume that each problem on this handout corresponds to a problem on the

More information

Particle Motion Problems

Particle Motion Problems Particle Motion Problems Particle motion problems deal with particles that are moving along the x or y axis. Thus, we are speaking of horizontal or vertical movement. The position, velocity, or acceleration

More information

LINEAR APPROXIMATION, LIMITS, AND L'HOPITAL'S RULE v.05

LINEAR APPROXIMATION, LIMITS, AND L'HOPITAL'S RULE v.05 LINEAR APPROXIMATION, LIMITS, AND L'HOPITAL'S RULE v.05 Linear Approimation Nearby a point at which a function is differentiable, the function and its tangent line are approimately the same. The tangent

More information

Math 147 Exam II Practice Problems

Math 147 Exam II Practice Problems Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab

More information

Department of Mathematical x 1 x 2 1

Department of Mathematical x 1 x 2 1 Contents Limits. Basic Factoring Eample....................................... One-Sided Limit........................................... 3.3 Squeeze Theorem.......................................... 4.4

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

Week In Review #8 Covers sections: 5.1, 5.2, 5.3 and 5.4. Things you must know

Week In Review #8 Covers sections: 5.1, 5.2, 5.3 and 5.4. Things you must know Week In Review #8 Covers sections: 5.1, 5.2, 5.3 and 5. Things you must know Know how to get an accumulated change by finding an upper or a lower estimate value Know how to approximate a definite integral

More information

Worksheets for MA 113

Worksheets for MA 113 Worksheet # : Review Worksheets for MA 3 http://www.math.uky.edu/~ma3/s.3/packet.pdf Worksheet # 2: Functions, Logarithms, and Intro to Limits Worksheet # 3: Limits: A Numerical and Graphical Approach

More information

1. Taylor Polynomials of Degree 1: Linear Approximation. Reread Example 1.

1. Taylor Polynomials of Degree 1: Linear Approximation. Reread Example 1. Math 114, Taylor Polynomials (Section 10.1) Name: Section: Read Section 10.1, focusing on pages 58-59. Take notes in your notebook, making sure to include words and phrases in italics and formulas in blue

More information

Amherst College, DEPARTMENT OF MATHEMATICS Math 11, Final Examination, May 14, Answer Key. x 1 x 1 = 8. x 7 = lim. 5(x + 4) x x(x + 4) = lim

Amherst College, DEPARTMENT OF MATHEMATICS Math 11, Final Examination, May 14, Answer Key. x 1 x 1 = 8. x 7 = lim. 5(x + 4) x x(x + 4) = lim Amherst College, DEPARTMENT OF MATHEMATICS Math, Final Eamination, May 4, Answer Key. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value,

More information

Analyzing Functions. Implicit Functions and Implicit Differentiation

Analyzing Functions. Implicit Functions and Implicit Differentiation Analyzing Functions Implicit Functions and Implicit Differentiation In mathematics, an implicit function is a generalization of the concept of a function in which the dependent variable, say, has not been

More information

PreCalculus Final Exam Review Revised Spring 2014

PreCalculus Final Exam Review Revised Spring 2014 PreCalculus Final Eam Review Revised Spring 0. f() is a function that generates the ordered pairs (0,0), (,) and (,-). a. If f () is an odd function, what are the coordinates of two other points found

More information

AP Calculus Chapter 3 Testbank (Mr. Surowski)

AP Calculus Chapter 3 Testbank (Mr. Surowski) AP Calculus Chapter 3 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.). If f(x) = 0x 4 3 + x, then f (8) = (A) (B) 4 3 (C) 83 3 (D) 2 3 (E) 2

More information

AP CALCULUS AB - Name: Summer Work requirement due on the first day of class

AP CALCULUS AB - Name: Summer Work requirement due on the first day of class AP CALCULUS AB - Name: Summer Work For students to successfully complete the objectives of the AP Calculus curriculum, the student must demonstrate a high level of independence, capability, dedication,

More information

8. Set up the integral to determine the force on the side of a fish tank that has a length of 4 ft and a heght of 2 ft if the tank is full.

8. Set up the integral to determine the force on the side of a fish tank that has a length of 4 ft and a heght of 2 ft if the tank is full. . Determine the volume of the solid formed by rotating the region bounded by y = 2 and y = 2 for 2 about the -ais. 2. Determine the volume of the solid formed by rotating the region bounded by the -ais

More information

AP Calculus Testbank (Chapter 6) (Mr. Surowski)

AP Calculus Testbank (Chapter 6) (Mr. Surowski) AP Calculus Testbank (Chapter 6) (Mr. Surowski) Part I. Multiple-Choice Questions 1. Suppose that f is an odd differentiable function. Then (A) f(1); (B) f (1) (C) f(1) f( 1) (D) 0 (E). 1 1 xf (x) =. The

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus Objectives Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for Integrals. Find the average value of

More information

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

More information