ap07_calculus_bc_form_b_frq

AP? Calculus BC

2007 Free-Response Questions

Form B

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CALCULUS BC

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 R be the region bounded by the graph of 2

2-=x x y e and the horizontal line 2,=y and let S be the region bounded by the graph of 22-=x x y e and the horizontal lines 1=y and 2,=y as shown above.

(a) Find the area of R .

(b) Find the area of S .

(c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line 1.=y

WRITE ALL WORK IN THE EXAM BOOKLET.

2. An object moving along a curve in the xy -plane is at position ()()(),x t y t at time t with

()arctan 1=+dx t dt t and ()

2ln 1=+dy t dt for 0.≥t At time 0,=t the object is at position ()3,4.-- (Note: 1tan arctan -=x x )

(a) Find the speed of the object at time 4.=t

(b) Find the total distance traveled by the object over the time interval 0 4.￡￡t

(c) Find ()4.x

(d) For 0,>t there is a point on the curve where the line tangent to the curve has slope 2. At what time t is the object at this point? Find the acceleration vector at this point.

WRITE ALL WORK IN THE EXAM BOOKLET.

3. The wind chill is the temperature, in degrees Fahrenheit ()F ,∞ a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v , in miles per hour ()mph . If the air temperature is 32F,∞ then the wind chill is given by ()0.1655.622.1=-W v v and is valid for 560.￡￡v

(a) Find ()20.￠W Using correct units, explain the meaning of ()20￠W in terms of the wind chill.

(b) Find the average rate of change of W over the interval 560.￡￡v Find the value of v at which the

instantaneous rate of change of W is equal to the average rate of change of W over the interval 560.￡￡v (c) Over the time interval 04￡￡t hours, the air temperature is a constant 32F.∞ At time 0,=t the wind velocity is 20=v mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at 3=t hours? Indicate units of measure.

WRITE ALL WORK IN THE EXAM BOOKLET.

END OF PART A OF SECTION II

CALCULUS BC

SECTION II, Part B

Time—45 minutes

Number of problems—3

No calculator is allowed for these problems.

4. Let f be a function defined on the closed interval 55

f= The graph of ,f￠ the derivative

-￡￡ with ()1 3.

x

of f, consists of two semicircles and two line segments, as shown above.

(a) For 55,

-<< find all values x at which f has a relative maximum. Justify your answer.

x

(b) For 55,

-<< find all values x at which the graph of f has a point of inflection. Justify your answer.

x

(c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning.

(d) Find the absolute minimum value of ()

-￡￡ Explain your reasoning.

x

f x over the closed interval 5 5.

WRITE ALL WORK IN THE EXAM BOOKLET.

5. Consider the differential equation 32 1.dy x y dx

=++ (a) Find 22d y dx

in terms of x and y . (b) Find the values of the constants m , b , and r for which rx y mx b e =++ is a solution to the differential equation.

(c) Let ()y f x = be a particular solution to the differential equation with the initial condition ()0 2.f =- Use

Euler’s method, starting at 0x = with a step size of 1,2

to approximate ()1.f Show the work that leads to your answer.

(d) Let ()y g x = be another solution to the differential equation with the initial condition ()0,g k = where k is a constant. Euler’s method, starting at 0x = with a step size of 1, gives the approximation ()10.g a Find the value of k .

6. Let f be the function given by ()36x f x e -= for all x .

(a) Find the first four nonzero terms and the general term for the Taylor series for f about 0.x = (b) Let g be the function given by ()()0.x

g x f t dt =

ú Find the first four nonzero terms and the general term for the Taylor series for g about 0.x =

(c) The function h satisfies ()()h x k f ax =￠ for all x , where a and k are constants. The Taylor series for h

about 0x = is given by

()231.2!3!!n x x x h x x n =++++++""

Find the values of a and k .

WRITE ALL WORK IN THE EXAM BOOKLET.

END OF EXAM