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  • AP Calculus AB · Unit 1: Limits and Continuity
Kolay

What is the value of limₓ→₀ [sin(3x)/(2x)], where angles are measured in radians?

A)
1
B)
3/2
C)
2/3
D)
0

Açıklama

Correct answer: B. Using limᵤ→₀ (sin u)/u = 1, rewrite sin(3x)/(2x) as (3/2)[sin(3x)/(3x)]. As x approaches 0, the bracketed factor approaches 1, so the limit is 3/2.
b
  • AP Calculus AB · Unit 2: Differentiation: Definition and Fundamental Properties
Kolay

Let f be differentiable, with f(2) = 3 and f′(2) = −1. If g(x) = x²f(x), what is g′(2)?

A)
−4
B)
4
C)
8
D)
12

Açıklama

Correct answer: C. By the product rule, g′(x) = 2xf(x) + x²f′(x). Therefore, g′(2) = 2(2)(3) + 2²(−1) = 12 − 4 = 8.
c
  • AP Calculus BC · Unit 1: Limits and Continuity
Kolay

The function f is defined by f(x) = (x² − 4)/(x − 2) for x ≠ 2 and f(2) = k. For which value of k is f continuous at x = 2?

A)
0
B)
2
C)
4
D)
No value of k

Açıklama

Correct answer: C. For x ≠ 2, factor the numerator: (x² − 4)/(x − 2) = (x − 2)(x + 2)/(x − 2) = x + 2. Therefore, limₓ→₂ f(x) = 4. Continuity at x = 2 requires f(2) to equal this limit, so k = 4.
c
  • AP Calculus BC · Unit 2: Differentiation: Definition and Fundamental Properties
Kolay

Let f(x) = x³ − 2x. Which of the following is the limit definition of f′(2)?

A)
limₕ→₀ {[(2 + h)³ − 2(2 + h)] − 4}/h
B)
limₕ→₀ {[(2 + h)³ − 2(2 + h)] + 4}/h
C)
limₕ→₀ [3(2 + h)² − 2]/h
D)
limₕ→₀ [f(2 + h) − f(h)]/h

Açıklama

Correct answer: A. The derivative at x = 2 is f′(2) = limₕ→₀ [f(2 + h) − f(2)]/h. Since f(2) = 2³ − 2(2) = 4 and f(2 + h) = (2 + h)³ − 2(2 + h), substituting these expressions gives choice A.
a
  • AP Calculus BC · Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
Orta

The curve defined by x² + xy + y² = 7 passes through the point (1, 2). What is dy/dx at (1, 2)?

A)
4/5
B)
−4/5
C)
−5/4
D)
5/4

Açıklama

Correct answer: B. Differentiate implicitly with respect to x: 2x + x(dy/dx) + y + 2y(dy/dx) = 0. Solving for dy/dx gives dy/dx = −(2x + y)/(x + 2y). At (1, 2), dy/dx = −(2 + 2)/(1 + 4) = −4/5.
b
  • AP Physics 1 · Work and Energy
Orta

A 2.0 kg block slides at 3.0 m/s on a horizontal surface. A constant kinetic friction force of magnitude 4.0 N is the only horizontal force acting on the block. How far does the block travel before coming to rest?

A)
0.75 m
B)
1.125 m
C)
2.25 m
D)
4.50 m

Açıklama

Correct answer: C. Apply the work–energy theorem: W_net = K_f − K_i. The normal force and gravity do no work because they are perpendicular to the horizontal displacement. K_i = ½mv² = ½(2.0)(3.0)² = 9.0 J, and K_f = 0. The work done by friction is −f_k d, so −4.0d = 0 − 9.0. Thus d = 9.0/4.0 = 2.25 m. The negative sign of the work indicates that friction removes kinetic energy; the stopping distance is positive.
c
  • AP Physics 1 · Kinematics
Orta

An object moves along a straight line. During a time interval, its velocity-versus-time graph is a straight line with a constant negative slope, and the velocity remains positive. Which statement correctly describes the motion during this interval?

A)
The object moves in the positive direction and slows down at a constant rate.
B)
The object moves in the positive direction and speeds up at a constant rate.
C)
The object moves in the negative direction and slows down at a constant rate.
D)
The object moves in the negative direction and speeds up at a constant rate.

Açıklama

Correct answer: A. The sign of velocity gives the direction of motion, so the object moves in the positive direction. The slope of a velocity-versus-time graph is acceleration: a = Δv/Δt. A constant negative slope therefore indicates constant negative acceleration. Because velocity and acceleration have opposite signs, the speed decreases. A negative slope does not, by itself, mean that the object is moving in the negative direction.
a
  • AP Physics C: Mechanics · Rotational Inertia
Kolay

A uniform solid disk has mass M and radius R. What is its rotational inertia about an axis through its center and perpendicular to the plane of the disk?

A)
½MR²
B)
MR²
C)
⅔MR²
D)
¼MR²

Açıklama

Correct answer: A. Rotational inertia is I = ∫ r² dm, where r is the perpendicular distance from the rotation axis. For a uniform disk, the surface mass density is M/(πR²). A thin ring of radius r and width dr has mass dm = (2M/R²)r dr. Therefore, I = (2M/R²)∫₀ᴿ r³ dr = ½MR². The axis matters: MR² applies to a thin hoop about its central perpendicular axis, while ¼MR² applies to a thin uniform disk about a diameter in its plane.
a
  • AP Physics C: Mechanics · Work and Energy
Orta

A particle moves along the x-axis under a force with x-component Fₓ(x) = (3 N/m²)x² − (2 N/m)x. What is the work done by this force as the particle moves from x = 1 m to x = 3 m?

A)
8 J
B)
16 J
C)
18 J
D)
42 J

Açıklama

Correct answer: C. The work done by a position-dependent force is W = ∫ Fₓ(x) dx over the displacement. With x expressed in meters and Fₓ in newtons, W = ∫₁³ (3x² − 2x) dx = [x³ − x²]₁³ = (27 − 9) − (1 − 1) = 18 J. The force is not constant, so its value at one endpoint cannot be multiplied by the displacement. Work is the signed area under the force-versus-position graph.
c
  • AP Calculus BC · Unit 4: Contextual Applications of Differentiation
Orta

The radius of a circle is increasing at a constant rate of 3 centimeters per second. At the instant when the radius is 4 centimeters, at what rate is the area of the circle increasing?

A)
12π cm²/s
B)
16π cm²/s
C)
24π cm²/s
D)
48π cm²/s

Açıklama

Correct answer: C. The area is A = πr², so differentiating with respect to time gives dA/dt = 2πr(dr/dt). At the given instant, r = 4 cm and dr/dt = 3 cm/s. Thus dA/dt = 2π(4)(3) = 24π cm²/s.
c
  • AP Calculus BC · Unit 5: Analytical Applications of Differentiation
Orta

Suppose f is continuous on [1, 5] and differentiable on (1, 5), with f(1) = 3 and f(5) = 11. Which of the following must be true?

A)
There is a number c in (1, 5) such that f′(c) = 2.
B)
There is a number c in (1, 5) such that f′(c) = 8.
C)
There is a number c in (1, 5) such that f(c) = 2.
D)
For every x in (1, 5), f′(x) = 2.

Açıklama

Correct answer: A. By the Mean Value Theorem, there is at least one c in (1, 5) for which f′(c) equals the average rate of change on [1, 5]. That rate is [f(5) − f(1)]/(5 − 1) = (11 − 3)/4 = 2.
a
  • AP Calculus BC · Unit 6: Integration and Accumulation of Change
Orta

What is the value of the definite integral ∫₀¹ x eˣ dx?

A)
e − 1
B)
1
C)
e
D)
e + 1

Açıklama

Correct answer: B. Use integration by parts, ∫u dv = uv − ∫v du. Choose u = x and dv = eˣ dx, so du = dx and v = eˣ. Then ∫x eˣ dx = x eˣ − eˣ + C = eˣ(x − 1) + C. Evaluating at both limits gives ∫₀¹ x eˣ dx = [eˣ(x − 1)]₀¹ = e(1 − 1) − 1(0 − 1) = 1.
b
  • AP Calculus BC · Unit 7: Differential Equations
Orta

A population P satisfies dP/dt = 0.2P(1 − P/500), where t is measured in years. For which value of P is the population increasing at the greatest rate?

A)
100
B)
200
C)
250
D)
500

Açıklama

Correct answer: C. The growth rate is the quadratic function g(P) = 0.2P(1 − P/500). It is zero at P = 0 and P = 500 and reaches its maximum halfway between these values. Therefore, the population increases at the greatest rate when P = 250.
c
  • AP Calculus BC · Unit 8: Applications of Integration
Orta

The region bounded by y = √x, y = 0, and x = 4 is revolved about the x-axis. What is the volume of the resulting solid?

A)
4π
B)
8π
C)
16π/3
D)
32π/3

Açıklama

Correct answer: B. Using the disk method, V = π∫₀⁴(√x)² dx = π∫₀⁴x dx. Evaluating gives V = π[x²/2]₀⁴ = 8π.
b
  • AP Calculus BC · Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
Orta

A curve is given by x(t) = t² + 1 and y(t) = t³ − 3t. What is dy/dx at t = 2?

A)
3/4
B)
4/9
C)
9/4
D)
3

Açıklama

Correct answer: C. For a parametrically defined curve, dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. Here, dx/dt = 2t and dy/dt = 3t² − 3. At t = 2, dx/dt = 4 and dy/dt = 9, so dy/dx = 9/4.
c
  • AP Calculus BC · Unit 10: Infinite Sequences and Series
Orta

Let f(x) = ln(1 + x). Which of the following is the third-degree Taylor polynomial for f about x = 0?

A)
x − x²/2 + x³/3
B)
1 + x − x²/2 + x³/3
C)
x − x² + 2x³
D)
x − x²/2 − x³/3

Açıklama

Correct answer: A. The Maclaurin series for ln(1 + x) is x − x²/2 + x³/3 − ⋯ for −1 < x ≤ 1. Therefore, the third-degree Taylor polynomial about x = 0 is x − x²/2 + x³/3.
a
  • AP Calculus AB · Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
Orta

The differentiable function f is one-to-one, with f(2) = 5 and f′(2) = 3. If g = f⁻¹, what is g′(5)?

A)
1/5
B)
1/3
C)
2/3
D)
3

Açıklama

Correct answer: B. For an inverse function, g′(a) = 1/f′(g(a)). Since f(2) = 5, g(5) = 2. Thus g′(5) = 1/f′(2) = 1/3.
b
  • AP Calculus AB · Unit 4: Contextual Applications of Differentiation
Orta

A 10-foot ladder rests against a vertical wall. The bottom of the ladder moves away from the wall at 2 feet per second. When the bottom is 6 feet from the wall, at what rate is the top of the ladder moving?

A)
−3/2 ft/s
B)
−4/3 ft/s
C)
3/2 ft/s
D)
8/3 ft/s

Açıklama

Correct answer: A. Let x be the distance of the ladder's bottom from the wall and y the height of its top. Since x² + y² = 100, differentiating gives x(dx/dt) + y(dy/dt) = 0. When x = 6, y = 8. Therefore, dy/dt = −[6(2)]/8 = −3/2 ft/s. The negative sign indicates that the top is moving downward.
a
  • AP Calculus AB · Unit 5: Analytical Applications of Differentiation
Orta

Let f(x) = x³ − 6x² + 9x + 2. Which statement correctly identifies and classifies all critical points of f?

A)
A relative maximum at x = 1 and a relative minimum at x = 3
B)
A relative minimum at x = 1 and a relative maximum at x = 3
C)
A relative maximum at x = 0 and a relative minimum at x = 3
D)
The only critical point is x = 2, where f has a relative minimum

Açıklama

Correct answer: A. Since f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3), the critical numbers are x = 1 and x = 3. The derivative changes from positive to negative at x = 1, so f has a relative maximum there. It changes from negative to positive at x = 3, so f has a relative minimum there.
a
  • AP Calculus AB · Unit 6: Integration and Accumulation of Change
Orta

Let g(x) = ∫₁ˣ² (t³ + 1) dt. What is g′(1)?

A)
2
B)
4
C)
6
D)
8

Açıklama

Correct answer: B. By the Fundamental Theorem of Calculus and the chain rule, g′(x) = [(x²)³ + 1](2x). At x = 1, g′(1) = (1 + 1)(2) = 4.
b
  • AP Calculus AB · Unit 7: Differential Equations
Orta

Which of the following is the solution to the differential equation dy/dx = 2xy with the initial condition y(0) = 3?

A)
y = 3eˣ²
B)
y = 3e²ˣ
C)
y = eˣ² + 2
D)
y = 3x² + 3

Açıklama

Correct answer: A. Separate variables: (1/y)dy = 2x dx. Integrating gives ln|y| = x² + C, so y = Ceˣ². Using y(0) = 3 gives C = 3. Therefore, y = 3eˣ².
a
  • AP Calculus AB · Unit 8: Applications of Integration
Zor

The region bounded by y = 4 − x² and the x-axis is revolved about the x-axis. What is the volume of the resulting solid?

A)
256π/15
B)
512π/15
C)
64π/3
D)
128π/5

Açıklama

Correct answer: B. The curve intersects the x-axis at x = −2 and x = 2. Using the disk method, V = π∫₋₂²(4 − x²)² dx. Expanding and integrating gives π∫₋₂²(16 − 8x² + x⁴)dx = 512π/15.
b