Mixed Deck — All AP CALC AB Topics Flashcards

What is the value of ∫ from 0 to 2 of (3x² − 2x) dx?

Answer: 4

An antiderivative is x³ − x². Evaluating from 0 to 2 gives (8 − 4) − 0 = 4.

Which function is the solution to dy/dx = −x/y with initial condition y(0) = 2?

Answer: y = √(4 − x²)

Separating gives y dy = −x dx, so y²/2 = −x²/2 + C and y² + x² = 4. Since y(0) = 2 is positive, y = √(4 − x²).

If f(x) = x² sin x, what is f′(π)?

Answer: −π²

By the product rule, f′(x) = 2x sin x + x² cos x. At x = π this is 2π(0) + π²(−1) = −π².

What is the absolute maximum value of f(x) = x³ − 6x² + 9x + 1 on the closed interval [0, 5]?

Answer: 21

f'(x) = 3(x − 1)(x − 3), so the critical points are x = 1 and x = 3. Evaluating gives f(0) = 1, f(1) = 5, f(3) = 1, f(5) = 21, so the maximum is 21.

Which function is the solution to dy/dx = y − 4 with y(0) = 1?

Answer: y = 4 − 3e^x

Separating gives dy/(y − 4) = dx, so y − 4 = Ce^x. Using y(0) = 1 gives C = −3, so y = 4 − 3e^x.

The base of a solid is the region in the first quadrant bounded by y = 1 − x² and the coordinate axes. Cross-sections perpendicular to the x-axis are squares. What is the volume of the solid?

Answer: 8/15

Each square has side 1 − x², so V = ∫₀¹ (1 − x²)² dx = ∫₀¹ (1 − 2x² + x⁴) dx = 1 − 2/3 + 1/5 = 8/15.

Let f(x) = x³ + x + 1, which is one-to-one. If g is the inverse function of f, what is g′(3)?

Answer: 1/4

Since f(1) = 3, g(3) = 1. Then g′(3) = 1 / f′(1) = 1/(3·1² + 1) = 1/4.

Let f(x) = x³. What is the value of lim_{h→0} [(2 + h)³ − 8] / h?

Answer: 12

The limit is the definition of f′(2) for f(x) = x³. Since f′(x) = 3x², f′(2) = 12.

A particle moves along a line with position s(t) = t³ − 6t² + 9t for t ≥ 0. At which time is the particle speeding up?

Answer: t = 4

v(t) = 3(t − 1)(t − 3) and a(t) = 6t − 12. At t = 4, v = 9 and a = 12 have the same sign, so the speed is increasing. At t = 0.5 and t = 2.5 the signs differ, and at t = 2 the acceleration is 0.

What is the value of ∫ from 0 to 1 of x·e^(x²) dx?

Answer: (e − 1)/2

Let u = x², so du = 2x dx and x dx = du/2. The integral becomes (1/2)∫ from 0 to 1 of e^u du = (e − 1)/2.

A 13-foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant 2 ft/s. How fast is the top of the ladder sliding down the wall at the instant the bottom is 5 feet from the wall?

Answer: 5/6 ft/s

From x² + y² = 169, x(dx/dt) + y(dy/dt) = 0. When x = 5, y = 12, so 5(2) + 12(dy/dt) = 0 and dy/dt = -5/6 ft/s, so the top slides down at 5/6 ft/s.

What is lim(x→2) (x² − 4)/(x − 2)?

Answer: 4

Factor the numerator as (x − 2)(x + 2) and cancel the common factor. That leaves lim(x→2) (x + 2) = 4.

Which of the following is equal to ∫ 2x·cos(x²) dx?

Answer: sin(x²) + C

Let u = x², so du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C.

Let f(x) = x³ − 3x on the interval [0, 2]. At which value of c in (0, 2) does the Mean Value Theorem guarantee that f'(c) equals the average rate of change of f on [0, 2]?

Answer: 2/√3

The average rate of change is (f(2) − f(0))/2 = (2 − 0)/2 = 1. Setting f'(c) = 3c² − 3 = 1 gives c² = 4/3, so c = 2/√3, which lies in (0, 2).

A function y = f(t) satisfies dy/dt = −3y with f(0) = 6. What is f(ln 2)?

Answer: 3/4

The solution is y = 6e^(−3t). At t = ln 2, e^(−3 ln 2) = 2^(−3) = 1/8, so f(ln 2) = 6/8 = 3/4.

What is lim(x→∞) (3x² + 5x)/(2x² − 1)?

Answer: 3/2

The numerator and denominator have the same degree, so the limit is the ratio of the leading coefficients, 3/2.

What is the area of the region enclosed by the graphs of y = x and y = x²?

Answer: 1/6

The graphs intersect at x = 0 and x = 1, and y = x lies above y = x² between them. The area is ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.

Which of the following functions is continuous at x = 0 but not differentiable at x = 0?

Answer: f(x) = x^(2/3)

For f(x) = x^(2/3), f′(x) = (2/3)x^(−1/3), which is unbounded near 0, so there is a cusp at x = 0. The other three functions are differentiable at 0.

What is lim(x→0) (√(x + 9) − 3)/x?

Answer: 1/6

Multiply the numerator and denominator by the conjugate √(x + 9) + 3. The expression becomes 1/(√(x + 9) + 3), which approaches 1/6.

If f(x) = x² / (x + 1), what is f′(1)?

Answer: 3/4

By the quotient rule, f′(x) = [2x(x + 1) − x²] / (x + 1)² = (x² + 2x) / (x + 1)². At x = 1 this is 3/4.