AP Calculus Free-Response Questions
AP Calculus Free-Response Questions
BC 2000-present
[pic]
AP Calculus Free-Response Questions Calculator Questions are Highlighted.
2000
420. The Taylor series about x = 5 for a certain function f converges to f(x) for all x in the interval of convergence.
The nth derivative of f at x = 5 is given by f(n)(5) = [pic], and f(5) = [pic].
a. Write the third-degree Taylor polynomial for f about x = 5.
b. Find the radius of convergence of the Taylor series for f about x = 5.
c. Show that the sixth-degree Taylor polynomial for f about x = 5 approximates f(6) with error ≤ .0001.
421. A moving particle has position (x(t), y(t)) at time t. The position of the particle at time t = 1 is (2, 6)
and the velocity vector at any time t > 0 is given by [pic].
a. Find the acceleration vector at time t = 3.
b. Find the position of the particle at time t = 3.
c. For what time t > 0 does the line tangent to the path of the particle at (x(t), y(t)) have a slope of 8?
d. The particle approaches a line as t [pic]. Find the slope of this line. Show the work that leads to
your conclusion.
[pic]
422. Consider the differential equation given by [pic] = x(y – 1)2.
a. Sketch a slope field for the given differential equation at the 11 points: (-2, 1), (-1, -1), (-1, 0)
(-1, 1), (0, -1), (0, 0), (0, 1), (1,-1), (1, 0), (1,1), (2, 1).
b. Use the slope field for the given differential equation to explain why a solution could not have
the graph shown above.
c. Find the particular solution y = f(x) to the given differential equation with initial condition f(0) = -1.
d. Find the range of the solution found in part c.
2001
423. An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t with
[pic]= cos (t3) and [pic]= 3 sin (t2) for 0 ≤ t ≤ 3. At time t = 2, the object is at position (4, 5).
a. Write an equation for the line tangent to the curve at (4, 5).
b. Find the speed of the object at time t = 2.
c. find the total distance traveled by the object over the time interval 0 ≤ t ≤ 1.
d. Find the position of the object at time t = 3.
424. Let f be the function satisfying f ‘(x) = -3xf(x) , for all real numbers x, with f(1) = 4 and [pic].
a. Evaluate [pic] Show the work that leads to your answer.
b. Use Euler’s method, starting at x = 1 with a step size of 0.5, to approximate f(2).
c. Write an expression for y = f(x) by solving the differential equation [pic]= -3xy with the initial
condition f(1) = 4.
425. A function f is defined by f(x) = [pic] for all x in the interval of convergence
of the given power series.
a. Find the interval of convergence for this power series. Show the work that leads to your answer.
b. Find [pic]
c. Write the first three nonzero terms and the general term for an infinite series that represents
[pic]
d. Find the sum of the series determined in part c.
2002
[pic]
426. The figure above shows the path traveled by a roller coaster car over the time interval 0 ≤ t ≤ 18 seconds.
The position of the car at time t seconds can be modeled parametrically by
x(t) = 10t + 4 sin t
y(t) = (20 – t)(1 – cos t),
where x and y are measured in meters. The derivatives of these functions are given by
x’(t) = 10 + 4 cos t
y’(t) = (20 – t)sin t + cos t – 1.
a. Find the slope of the path at time t = 2. Show the computations that lead to your answer.
b. Find the acceleration vector of the car at the time when the car’s horizontal position is x = 140.
c. Find the time t at which the car is at its maximum height, and find the speed, in m/sec, of the car
at this time.
d. For 0 < t < 18, there are two times at which the car is at ground level (y = 0). Find these two times and
write an expression that gives the average speed, in m/sec, of the car between these two times.
Do not evaluate the expression.
427. Consider the differential equation [pic]= 2y – 4x
a. The slope field for the given differential equation is provided below. Sketch the solution curve that
passes through the point (0,1) and sketch the solution curve that passes through the point (0, -1).
[pic]
b. Let f be the function that satisfies the given differential equation with the initial condition f(0) = 1.
Use Euler’s method, starting at x = 0 with a step size of 0.1, to approximate f(0.2). Show the work
that leads to your answer.
c. Find the value of b for which y = 2x + b is a solution to the given differential equation.
Justify your answer.
d. Let g be the function that satisfies the given differential equation with the initial condition g(0) = 0.
Does the graph of g have a local extremum at the point (0, 0)? If so, is the point a local maximum or
a local minimum? Justify your answer.
428. The Maclaurin series for the function f is given by
f(x) = [pic]
on its interval of convergence.
a. Find the interval of convergence of the Maclaurin series for f. Justify your answer.
b. Find the first four terms and the general term for the Maclaurin series for f ‘ (x).
c. Use the Maclaurin series you found in part (b) to find the value of f ‘[pic].
2003
429. [pic]
A particle starts at point A on the positive x-axis at time t = 0 and travels along the curve from
A to B to C to D, as shown above. The coordinates of the particle’s position (x(t), y(t)) are
differentiable functions of t, where x′(t) = [pic] = -9cos[pic] sin[pic] and y′(t) = [pic] is not
explicitly given. At time t = 9, particle reaches its final position at point D on the positive x-axis.
a. At point C, is [pic]positive? At point C, is [pic] positive? Give a reason for each answer.
b. The slope of the curve is undefined at point B. At what time t is the particle at point B?
c. The line tangent to the curve at the point (x(8), y(8)) has equation y = [pic]x − 2. Find the
velocity vector and the speed of the particle at this point.
d. How far apart are points A and D, the initial and final positions, respectively, of the particle?
[pic]
430. The figure above shows the graphs of the line x = [pic] and the curve C given by x = [pic] Let S
be the shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at
point P.
a. Find the coordinates of point P and the value of [pic] for curve C at point P.
b. Set up and evaluate an integral expression with respect to y that gives the area of S.
c. Curve C is a part of the curve x2 − y2 = 1. Show that x2 − y2 = 1 can be written as the polar equation
r2 = [pic]
d. Use the polar equation given in part (c) to set up an integral expression with respect to the polar
angle θ that represents the area of S.
431. The function f is defined by the power series
f(x) = [pic]
for all real numbers x.
a. Find f′(0) and f′′(0). Determine whether f has a local maximum, a local minimum, or neither at x = 0.
Give a reason for your answer.
b. Show that 1 - [pic] approximates f(1) with error less than [pic].
c. Show that y = f(x) is a solution to the differential equation xy′ + y = cos x.
2004
432. An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t ≥ 0 with
[pic] The derivative [pic]is not explicitly given. At time t = 2, the object is at position (1, 8).
a. Find the x-coordinate of the position of the object at time t = 4.
b. At time t = 2, the value of [pic] is -7. Write the equation for the line tangent to the curve at the point
(x(2), y(2)).
c. Find the speed of the object at time t = 2.
d. For t ≥ 3, the line tangent to the curve at (x(t), y(t)) has a slope of 2t + 1. Find the acceleration
vector of the object at time t = 4.
433. A population is modeled by a function P that satisfies the logistic differential equation
[pic]
a. If P(0)= 3, what is [pic]
If P(0) = 20, what is [pic]
b. If P(0) = 3, for what value of P is the population growing the fastest?
c. A different population is modeled by a function Y that satisfies the separable differential equation
[pic]
Find Y(t) if Y(0) = 3.
d. For the function Y found in part c, what is [pic]
434. Let f be the function given by f(x) = sin[pic], and let P(x) be the third-degree Taylor polynomial
for f about x = 0.
a. Find P(x).
b. Find the coefficient of x22 in the Taylor series for f about x = 0.
c. Use the Lagrange error bound to show that [pic].
d. Let G be the function given by G(x) = [pic]. Write the third-degree Taylor polynomial for G
about x = 0.
2005
[pic]
435. The curve above is drawn in the xy-plane and is described by the equation in polar coordinates r = θ + sin(2θ)
for 0 ( θ ( π, where r is measured in meters and θ is measured in radians. The derivative of r with respect to
θ is given by [pic]= 1 + 2cos(2θ).
a. Find the area bounded by the curve and the x-axis.
b. Find the angle θ that corresponds to the point on the curve with x-coordinate −2.
c. For [pic] is negative. What does this fact say about r? What does this fact say about the
curve?
d. Find the value of θ in the interval 0 ( θ ( [pic] that corresponds to the point on the curve in the first
quadrant with the greatest distance from the origin. Justify your answer.
436. Consider the differential equation [pic]= 2x − y.
a. On the axes below, sketch a slope field for the given differential equation at the twelve points indicated,
and sketch the solution curve that passes through the point (0, 1).
[pic]
b. The solution curve that passes through the point (0, 1) has a local minimum at x = ln[pic]. What is the
y-coordinate of this local minimum?
c. Let y = f(x) be the particular solution to the given differential equation with the initial condition f(0) = 1.
Use Euler’s method, starting at x = 0 with two steps of equal size, to approximate f(-0.4).
Show the work that leads to your answer.
d. Find [pic] in terms of x and y. Determine whether the approximation found in part (c) is less than or
greater than f(-0.4). Explain your reasoning.
437. Let f be a function with derivatives of all orders and for which f(2) = 7. When n is odd, the nth derivative
of f at x = 2 is 0. When n is even and n ( 2, the nth derivative of f at x = 2 is given by f(n)(2) = [pic].
a. Write the sixth-degree Taylor polynomial for f about x = 2.
b. In the Taylor series for f about x = 2, what is the coefficient of (x-2)2n for n ( 1?
c. Find the interval of convergence of the Taylor series for f about x = 2. Show the work that leads to your
answer.
2006
438. An object moving along a curve in the xy-plane is at position (x(t), y(t)) at time t, where
[pic]
for t ( 0. At time t = 2, the object is at the point (6, -3). (Note: sin-1x = arcsin x)
a. Find the acceleration vector and the speed of the object at time t = 2.
b. The curve has a vertical tangent line at one point. At what time t is the object at this point?
c. Let m(t) denote the slope of the line tangent to the curve at the point (x(t), y(t)). Write an expression
for m(t) in terms of t and use it to evaluate [pic]
d. The graph of the curve has a horizontal asymptote y = c. Write, but do not evaluate, an expression
involving an improper integral that represents this value c.
439. Consider the differential equation [pic]= 5x2 ( [pic] for y ( 2. Let y = f(x) be the particular solution to this
differential equation with the initial condition f(-1) = -4.
a. Evaluate [pic] at (-1, -4).
b. Is it possible for the x-axis to be tangent to the graph of f at some point? Explain why or why not.
c. Find the second-degree Taylor polynomial for f about x = -1.
d. Use Euler’s method, starting at x = -1 with two steps of equal size to approximate f(0). Show the work
that leads to your answer.
440. The function f is defined by the power series
f(x) = ([pic]
for all real numbers x for which the series converges. The function g is defined by the power series
g(x) = 1 ( [pic]
for all real numbers x for which the series converges.
a. Find the interval of convergence of the power series for f. Justify your answer.
b. The graph of y = f(x) ( g(x) passes through the point (0,-1). Find y((0) and y(((0). Determine whether
y has a relative minimum, a relative maximum, or neither at x = 0. Give a reason for your answer.
2007
441. The graphs of the polar curves r = 2 and r = 3 +2cos ( are shown on the board. The curves intersect when
( = [pic] and ( = [pic].
a. Let R be the region that is inside the graph of r = 2 and also inside the graph of r =2 and also inside the
graph of r = 3 + 2cos ( as shown. Find the area of R.
b. A particle moving with nonzero velocity along the polar curve given by r = 3 + 2cos ( has position
(x(t), y(t)) at time t, with ( = 0 when t = 0. This particle moves along the curve so that [pic]. Find
the value of [pic] at ( = [pic] and interpret you answer in terms of the motion of the particle.
c. For the particle described in part (b), [pic]. Find the value of [pic] at ( = [pic] and interpret your answer
in terms of the motion of the particle.
442. Let f be the function defined for x > 0, with f(e) = 2 and f′, the first derivative of f, given by f′(x) = x2 ln x.
a. Write an equation for the line tangent to the graph of f at the point (e, 2).
b. Is the graph of f concave up or concave down on the interval 1 < x < 3? Give a reason for your answer.
c. Use antidifferentiation to find f(x).
443. Let f be the function given by f(x) = [pic].
a. Write the first four nonzero terms and the general term of the Taylor series for f about x = 0.
b. Use your answer to part (a) to find [pic]
c. Write the first four nonzero terms of the Taylor series for [pic] about x = 0. Use the first two
terms of your answer to estimate [pic]
d. Explain why the estimate found in part (c) differs from the actual value of [pic] by less than [pic]
2008 Shaded problem is Calculator active
|x |h(x) |h′(x) |h’’(x) |h′′′(x) |h(4)x |
|1 |11 |30 |42 |99 |18 |
|2 |80 |128 |[pic] |[pic] |[pic] |
|3 |317 |[pic] |[pic] |[pic][pic] |[pic][pic] |
444.
Let h be a function having derivatives of all orders for x > 0. Selected values of h and its first four
derivatives are indicated in the table above. The function h and these four derivatives are increasing on the
interval 1 ≤ x ≤ 3.
a. Write the first-degree Taylor polynomial for h about x = 2 and use it to approximate h(1.9). Is this
approximation greater than or less than h(1.9)? Explain your reasoning.
b. Write the third-degree Taylor polynomial for h about x = 2 and use it to approximate h(1.9).
c. Use the Lagrange error bound to show that the third-degree Taylor polynomial for h about x = 2
approximates h(1.9) with error less than 3 X 10-4.
445. The derivative of a function f is given by f′(x) = (x − 3)ex for x > 0, and f(1) = 7.
(a) The function f has a critical point at x = 3. At this point, does f have a relative minimum, a relative
maximum, or neither? Justify your answer.
(b) On what intervals, if any, is the graph of f both decreasing and concave up? Explain your reasoning.
(c) Find the value of f(3).
446. Consider the logistic differential equation [pic]= [pic](6 − y). Let y = f(t) be the particular solution to the
differential equation with f(0) = 8.
a. A slope field for this differential equation is given below. Sketch possible solution curves through
points (3, 2) and (0, 8).
[pic]
b. Use Euler’s method, starting at t = 0 with two steps of equal size, to approximate f(1).
c. Write the second-degree Taylor polynomial for f about t = 0, and use it to approximate f(1).
d. What is the range of f for t [pic] 0?
447. [pic]
A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position
of the diver and her position at a later time. At time t seconds after she leaps, the horizontal distance from
the front edge of the platform to the diver’s shoulders is given by x(t), and the vertical distance from the
water surface to her shoulders is given by y(t), where x(t) and y(t) are measured in meters. Suppose the
diver’s shoulders are 11.4 meters above the water when she makes her leap and that
[pic]
for 0 ≤t ≤ A, where A is the time that the diver’s shoulders enter the water.
(a) Find the maximum vertical distance from the water surface to the diver’s shoulders.
(b) Find A, the time that the diver’s shoulders enter the water.
(c) Find the total distance traveled by the diver’s shoulders from the time she leaps from the platform until
the time her shoulders enter the water.
(d) Find the angle θ, 0 < θ < [pic], between the path of the diver and the water at the instant the diver’s
shoulders enter the water.
448. Consider the differential equation [pic]= 6x2 − x2y. Let y = f(x) be a particular solution to this differential
equation with the initial condition f(-1) = 2.
(a) use Euler’s method with two steps of equal size, starting at x = -1, to approximate f(0). Show the work
that leads to your answer.
(b) At the point (-1, 2) the value of [pic] is -12. Find the second-degree Taylor polynomial for f about x = -1.
(c) Find the particular solution y = f(x) to the given differential equation with the initial condition f(-1) = 2.
449. The Maclaurin series for ex is ex = [pic]. The continuous function f is defined by
f(x) = [pic] for x[pic]1 and f(1) = 1. The function f has derivatives of all orders at x = 1.
(a) Write the first four nonzero terms and the general term of the Taylor series for [pic]about x = 1.
(b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of
the Taylor series fro f about x = 1.
(c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b).
(d) Use the Taylor series for f about x=1 to determine whether the graph of f has any points of inflection.
450. A particle moving along a curve so that its position at time t is (x(t), y(t)), where x(t) = t2 − 4t + 8 and
y(t) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known
that [pic].
(a) Find the speed of the particle at time t = 3 seconds.
(b) Find the total distance traveled by the particle for 0 ≤t ≤ 4 seconds.
(c) Find the time t, 0 ≤t ≤ 4, when the line tangent to the path of the particle is horizontal. Is the direction
of motion of the particle toward the left or toward the right at that time? Give a reason for your ans.
(d) there is a point with x-coordinate 5 through which the particle passes twice. Find each of the following.
(i) The two values of t when that occurs
(ii) The slopes of the lines tangent to the particle’s path at that point.
(iii) The y-coordinate of that point, given y(2) = 3 + [pic]
451. Consider the differential equation [pic]. Let y = f(x) be the particular solution to this differential
equation with the initial condition f(1) = 0. For this particular solution, f(x) < 1 for all values of x.
(a) Use Euler’s method, starting at x =1 with two steps of equal size, to approximate f(0). Show the
work that leads to your answer.
(b) find [pic]. Show the work that leads to your answer.
(c) Find the particular solution y = f(x) to the differential equation [pic]= 1 − y with the initial condition
f(1) = 0.
452. f(x) = [pic]
The function f, defined above, has derivatives of all orders. Let g be the function defined by
g(x) = 1 + [pic]
(a) Write the first three nonzero terms and the general term of the Taylor series for cos x about x = 0.
Use this series to write the first three nonzero terms and the general term of the Taylor series for
f about x = 0.
(b) Use the Taylor series for f about x = 0 found in part (a) to determine whether f has a relative maximum,
relative minimum, or neither at x = 0. Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for g about x = 0.
(d) The Taylor series for g about x = 0, evaluated at x = 1, is an alternating series with individual terms that
decrease in absolute value to 0. Use the third-degree Taylor polynomial for g about x = 0 to estimate
the value of g(1). Explain why this estimate differs from the actual value of g(1) by less than [pic]
453. (Graphing Cal. Required) At time t, a particle moving in the xy-plane is at position (x(t), y(t)), where x(t) and
y(t) are not explicitly given. For t ≥0, [pic]= 4t + 1 and [pic] = sin (t2). At time t = 0, x(0) = 0 & y(0) = -4.
a. Find the speed of the particle at time t = 3, and find the acceleration vector of the particle at t = 3.
b. Find the slope of the line tangent to the path of the particle at time t = 3.
c. Find the position of the particle at time t = 3.
d. Find the total distance traveled by the particle over the time interval 0 ≤t ≤ 3.
[pic]
454. Let f(x) = e2x. Let R be the region in the first quadrant bounded by the graph of f, the coordinate axes and
the vertical line x = k, where k > 0. the region R is shown in the figure above.
a. Write, but do not evaluate, an expression involving an integral the gives the perimeter of R in terms of k.
b. The region R is rotated about the x-axis to form a solid. Find the volume, V, of the solid in terms of k.
c. The volume V, found in part b, changes as k changes. If [pic], determine [pic] when k = [pic].
455.
[pic]
Let f(x) = sin(x2) + cos x. the graph of y = |f(5)(x)| is shown above.
a. Write the first four nonzero terms of the Taylor series for sin x about x = 0, and write the first four
nonzero terms of the Taylor series for sin(x2) about x = 0.
b. Write the first four nonzero terms of the Taylor series for cos x about x = 0. Use this series and the
series for sin(x2), found in part (a), to write the first four nonzero terms of the Taylor series for f
about x = 0.
c. find the value of f(6)(0).
d. Let P4(x) be the fourth-degree Taylor polynomial for f about x = 0. Using the information from the
graph of y = |f(5)(x)| shown above, show that [pic]
456.
[pic]
457.
[pic]
458.
[pic]
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