CALCULUS AB SECTION I, Pa~t A Time-- 50 minutes Number of questions ~ 25 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems, using the available space for scratchwork. After exam- ining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything w,~itten in the test book. Do not spend too much time on anY one problem. tn this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real ~umbers x for which f(x) is a real number. 2 I. i (4xs - 6x) dx = (A) 2 (B) 4 (C) 6 (D) 36 (E) 42
Transcript
CALCULUS ABSECTION I, Pa~t ATime-- 50 minutes
Number of questions ~ 25
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION.
Directions: Solve each of the following problems, using the available space for scratchwork. After exam-ining the form of the choices, decide which is the best of the choices given and fill in the correspondingoval on the answer sheet. No credit will be given for anything w,~itten in the test book. Do not spend toomuch time on anY one problem.
tn this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real~umbers x for which f(x) is a real number.
14. L~t f tm a differ~ntiable function such that f(3) = 2 and f’(3) = 5. If the tangent line to the graphof f at z = 3 is used to find an approximation to a zaro of f, nat approximation is
(A) 0.4 (B) 0.5 (C7) 2.6. (D) 3.4 (’E) 5.5
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3
2
1
O
Calculus AB ~Part A I
15. The graph of the function f is shown in the figure above. Which of the following statements about fis true?
(A) lim f(x)= lim f(x)
(B) lim f(x) = 2
(D) lira f(x) I
(E) lira f(x) does not exist.
The area of the region enclosed by the graph of y = x-’ + I and the line y = 5 is
14 16 28 32(a) T - (~) 5- (c) -5 (D) 5-
I7. If x-~ + 3’" = 25, what is the value of ~ at the point (4, 3) ?
25 7 7 25
18.
(A) 0 ~) ~ (C) e- 1
Calculus AB ~Part A1
(D) e (E) e + I
19. If f(x) = ln[x-" - l I, then f’(#) =
(A) 2~
2x(B) ~
(D) x2 _ I
[ ~ ]
20. The average valu~ of cos x on the interval
sin 5 - sin 3(A)
sin 5 - sin 3
sin 3 - sin 5(C) 2
t )~D\sin3 + sin52
sin 3 ~- sin 5(g) 8
[-3, 5] is
x21. lira ~ is
(A) o 0~) -~ (c) 1 (D) e 0E) honexistent
22. What are all values of .x for which the function f defined by f(x) = (x~ - 3)e-x is increasing?
(A) There are no such values of x.(B) x<-I and x>3(C) -3 < x < 1(D) -1 < x < 3CE) All values of x
23. If the region enclosed by the y-axis, the line y = 2, and the curve y = ~ is revolved about the, ).-axis, the volume of the solid generated is
8= (E) ~16= 16= (D) -~-32= (B) "-y- (C) ~(A) -7-
24. The e×pres~ion ~I
1
CB)~ ~&!
~ofo
is a Riemann sum approximation for
LPartA ]
END OF PART A OF SEC-YION I
YOU FINISH BEFORE TIME IS CALLED, YOU MAY CH~CK YOUR WORK ON THIS PART ONLY.DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO
CALCULUS ABSECTION I, Part B
Time-- 40 minutesNumber of questions-- 15
A GR.~::~HtNG CALCULATOR IS REQUIRED FOR SOME QUESTIONS ONTHIS PART OF THE EXAMINATION.
Directions: Solve each of the following problems, using the available space for scratchwork. After exam-ining the form of the Choices, decide which is the best of the choices given and fill in the correspondingoval on the answer sheet. No credit will be given for anything written in the test book. Do not spend toomuch time on any one problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANS~TER SHEET TO RECORD YOUR ANS~ERSTO QUESTIONS NUMBERED 76-90.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANS%’ER SHEET.
In this test:
(1) The exact’numerical value of the correct answer does not always appear among the choices given.When this happens, select from among the choices the number that best approximates the exactnumerical val~ae.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers xfor which f(x) is a real number.
A GRAPHING C.~LCULATOR IS REQUINED FOR SOME QUES-i’IONS ONTHIS PART OF THE EXAMINATION,
Directions: Solve each of the following problems, using the available space for scratchwork. After exam-ining the form of the choices, decide which is the best of the choices given and fit! in the correspondingoval on the answer sheet. No credit will be given for anything written in the test book. Do not spend toomuch time on any one problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANNWER SHEET TO RECORD YOUR ANSV~rERSTO QUESTIONS NUMBERED 76-90.
YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWVER SHEET.
tn this test:
(I) The exact numerical value of the correct answer does not always appear among the choices given.When this happens, select from among the choices the number that best approximates tt~e exactnumerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers xfor which f(x) is a real number.
(C) e=eaZ(2z + 1)(D) z=
e:X(2z- 1)(E) 2xa
77. The graph of the function y = x~ + 6x~ + 7x - 2 cos x changes concavity at x =
(A) -1.58 (B) -1.63 (C) -!.67 (D) -1.g9 (E) -2.33
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78. The graph of f is shown in the fi~re above. If fl f(x) dx = 2.3 and F’(x) = f(x), then
F(3) - F(0) =
(A) 0.3 t’B) 1.3 (C) 3.3 (D) 4.3 (E) 53
79. I~t f be a function such that Iimf(2 + h) -f(2) _ 5 Which of the following must be true?
I. f is Continuous at x = 2-II. f is diffemntiable at x = 2.l~I. The derivath~ of f is continuous at x = 2.
(A) I only(B) II only(C) I and II only(D) I ant: lII only(E) II and : only
80. Let f be the function given by f(x) = 2e4~. For what value of x is the slope of the line tangent tothe graph of f at (x, f(x)) eq~ad to 3 ?
81. A railroad track and a mad cross m right angles. An obserwr stands on the road 70 meters south of thecrossing and watches an eastbound train trawling at 60 meters per second. At how many meters p~rsecond is the train moving away from the observer 4 seconds after it passes through the intersection?
(A) 57.60 (B) 57.88 (C) 59.20 (D) 60.00 (E) 67,40
82. If )’ = 2x - 8, what is the minimum value of the product xy ?
(A) .7-16 03) -~ (C) _z~(D) 0 (E) 2
83. What is the area of the re,on in the f’g~t quadrant enclosed by the graphs of y = cos :c, y = x, andthe ),-axis?
(A) 0.127 (t3)-0385 (C) 0.400 (D) 0.600 (E) 0.947
84.The base of a solid S is the region enclosed by the -graph of y = l"~’q"~n x, the line x = e, and thex-axis. If the cross sections of S perpenttieutar to the x-axis are squares, then the volume of S is
2 1
(A) -0.46 (B) 0!0 (C) 0.9l (’D) 0.95 (E) 3.73
86. Let f(x) = "~x. If the rate of change of f at zc = es twime its rate of change at ac = 1. then
l ](A) ~ t~) 1 (C) ’~ t~) ~ (E)
Calcu|us AB ~
87, At time t >- 0, the a~-w~leration of a particle moving on the .~axis ~s =(r) = t + sin t. At t = 0, thewlo~iry of the particle is -2. For what value of r wilt the wlocity of the pro-title be z¢ro?
iA) 1.02 (B) 1.48 (C) 1.15 (D) 2.gl (E) 3.14
0 i~ ~-0
~) 3 3 13
z F(=) =--5-
~_ F(=) = ~
IZI. F(=) = ~
(A) ~ only
(c)!n ~(D) I ~d
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88.Let f(x) = fagraph of f ?
(A) v
h(t) dt, where h has the graph shown above. Which of the following could be the
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(c) Y Y
(E) Y
a b c
CALCULUS ABSECTION tI
Time-- 1 hour and 30 minutesNumber of problems-- 6
Pement of total grade-- 50
A GRA~PHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS ORPARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION.
REMEMBER TO SHOW YOUR SETUPS AS DESCRIBED IN THE GENERAL INSTRUCTIONS.
General instructions for this section are printed on the back cover of this booklet.
I. A particle moves along the x-axis so that its velocity at any time t 2 0 is given byv(t) = 3t’- - 2t - 1. The position x(t) is 5 for t = 2.
(a) Write a polynomial expression for the position of the particle at any time. t > 0.
...... (b) For what values of t, 0 _< t _< 3, is the particle’s instantaneous velocity the same as its averagevelocity on the closed interval [0, 3]?
Calculus AB ~
(c) Find the total distance traveled by the particle from time t = 0 until time t = 3.
y
p (o, 3)
~, = f(x)
Q (~, o)X
0
2. Let f be the function given by f(x) = 3 cos x. As shown above, the graph o’f f crosses the
y-axis at point P and the x-axis at point Q.
(a) Write an equation for the line passing through points P and Q.
(b) Write an equation for .the line tangent to the graph of f at point Q. Show the analysis that leadsto your equation.
(c) Find the x-coordinate of the point on the graph of f, between points P and Q, at which theline tangent to the graph of f is parallel to line PQ.
(d) Let R be the region m the first quadrant bounded by the graph of f and line segment PQ.Write an integral expression for the volume of the solid generated by reyolving the region Rabout the x-axis. Do not, evaluate.
Let f be the function given by f(x) = "~ - 3.
(a) On the axes provided below, sketch the graph of f and shade the region R enclosed by thegraph of f, the x-axis, and the vertical line x = 6.
(b) Find the area of the region R described in part
[461
Calculus A~ ~
Rather than using the line x = 6 as in part (a), consider the line x = w, where w can beany number ~eater than3. Let A (~t,) be the area of the region enclosed by the graph of f,the x-axis, and the vertical line x = w. Write an integal expression for ,4
(d) Let A(w) be as described in t:~art (c). Find the rate of change of A with respect to w when
4. Let f be the function given by f(x) = x~ - 6x"~ + p, where p is an arbitrary constant.
(a) Write an expression for f’(x) and use it to find the relative maximum and minimum valuesof f in terms of p. Show the analysis that leads to your conclusion.
(b) For what values of t.he constant, p does f have 3 distinct rea! roots?
(c) Find the value of p such that,the average value of f over the closed interval [-1, 2] is l.
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-5-4-3 -2-1
5. The graph of a function f consists of a semicircle and two line segments as shown above. Let g beX
the Nnction given by g(x) = ~o f(t) dr.
(a) Find g(3).
(b) Find all values of x on the open interval (-2, 5) at which g has a relative maximum. Justifyyour answer.
Calculus AB
(c) Write an equation for the line tangent to the graph of g at .r = 3 .
(d) Find the x-coordinate of each point of inflection of the graph of g on the open interval (-2, 5).Justify your answer.
51
6. Let v’(t) be the velocity, in feet per second, of a skydiver at time t seconds, t 2 0. After her
d~, -2v - 32, with initial conditionparachute opens, her velocity satisfies the differential equation "~ =
v(0) = -50.
(a) Use separation of variables to find an expression for ~’ in terms of t, where t is measuredin seconds.
(b) Terminal :velocity is defined as tim v(t). Find the terminal velocity of the skydiver to the nearestfoot per second, t-~
Calculus AB ~
(c) I~: is safe to land when her speed is 20 feet per second. At what time t does she reach this speed?
