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ap06_frq_calculus_ab_b

AP? Calculus AB

2006 Free-Response Questions

Form B

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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.

1. Let f be the function given by ()323cos .432

x x x

f x x =--+ Let R be the shaded region in the second

quadrant bounded by the graph of f , and let S be the shaded region bounded by the graph of f a nd line A , the

line tangent to the graph of f a t 0,x = as shown above. (a) Find the area of R .

(b) Find the volume of the solid generated when R is rotated about the horizontal line 2.y =- (c) Write, but do not evaluate, an integral expression that can be used to find the area of S .

WRITE ALL WORK IN THE EXAM BOOKLET.

2. Let f be the function defined for 0x ≥ with ()05f = and ,f ￠ the first derivative of f , given

by ()()()

x f x e x 42sin .-=￠ The graph of ()y f x =￠ is shown above.

(a) Use the graph of f ￠ to determine whether the graph of f is concave up, concave down, or neither on the

interval 1.7 1.9.x << Explain your reasoning. (b) On the interval 03,x ￡￡ find the value of x a t which f has an absolute maximum. Justify your answer. (c) Write an equation for the line tangent to the graph of f a t 2.x =

WRITE ALL WORK IN THE EXAM BOOKLET.

3. The figure above is the graph of a function of x , which models the height of a skateboard ramp. The function

meets the following requirements.

(i) At 0,x = the value of the function is 0, and the slope of the graph of the function is 0. (ii) At 4,x = the value of the function is 1, and the slope of the graph of the function is 1. (iii) Between 0x = and 4,x = the function is increasing.

(a ) Let ()2,f x ax = where a is a nonzero constant. Show that it is not possible to find a value for a so tha t f meets requirement (ii) above.

(b) Let ()2

,16

x g x cx =- where c is a nonzero constant. Find the value of c so tha t g meets requirement (ii) above. Show the work that leads to your answer.

(d) Let (),n

x h x k

= where k is a nonzero constant and n is a positive integer. Find the values of k a nd n so

that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.

WRITE ALL WORK IN THE EXAM BOOKLET.

END OF PART A OF SECTION II

CALCULUS AB SECTION II, Part B

Time—45 minutes Number of problems—3

No calculator is allowed for these problems.

4. The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the

function f . In the figure above, ()3213

142

f t t t =-++ for 04t ￡￡ and f is piecewise linear for 424.t ￡￡

(a ) Find ()22.f ￠ Indicate units of measure.

(b) For the time interval 024,t ￡￡ at what time t is f increasing at its greatest rate? Show the reasoning that

supports your answer. (c) Find the total number of calories burned over the time interval 618t ￡￡ minutes.

(d) The setting on the machine is now changed so that the person burns ()f t c + calories per minute. For this

setting, find c so that an average of 15 calories per minute is burned during the time interval 618.t ￡￡

WRITE ALL WORK IN THE EXAM BOOKLET.

5. Consider the differential equation

()()2

1cos .dy y x dx

p =- (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

(Note: Use the axes provided in the exam booklet.)

(b) There is a horizontal line with equation y c = that satisfies this differential equation. Find the value of c . (c) Find the particular solution ()y f x = to the differential equation with the initial condition ()10.f =

6. A car travels on a straight track. During the time interval 060t ￡￡ seconds, the car’s velocity v , measured in

feet per second, and acceleration a , measured in feet per second per second, are continuous functions. The table above shows selected values of these functions.

(a) Using appropriate units, explain the meaning of

()60

30v t dt ú in terms of the car’s motion. Approximate

()60

v t dt ú using a trapezoidal approximation with the three subintervals determined by the table.

(b) Using appropriate units, explain the meaning of

()30

a t dt ú in terms of the car’s motion. Find the exact

value of

()30

.a t dt ú

(c) For 060,t << must there be a time t when ()5?v t =- Justify your answer. (d) For 060,t << must there be a time t when ()0?a t = Justify your answer.

WRITE ALL WORK IN THE EXAM BOOKLET.

END OF EXAM

ap06_frq_calculus_ab_b

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