High School Calculus practice questions and exam guide

Evaluate lim_{x→0} x² sin(1/x).

1. Does not exist
2. 1
3. 0
4. ∞

Answer: C — 0

C) Correct — Since −x² ≤ x² sin(1/x) ≤ x² and both bounds go to 0, the squeeze theorem gives 0. A) sin(1/x) oscillates, but the x² factor squeezes the product to 0. B) Confuses this with lim (sin x)/x = 1. D) sin(1/x) stays between −1 and 1, so nothing blows up.

Evaluate lim_{x→∞} (5x + 1)/(x² + 4).

1. 5
2. ∞
3. 1/4
4. 0

Answer: D — 0

D) Correct — The denominator has the higher degree, so the fraction shrinks to 0. A) Divides the leading coefficients even though the degrees are different. B) Reverses the degree rule; ∞ happens when the numerator's degree is higher. C) Uses the constant terms, which become negligible as x → ∞.

Which limit defines the derivative f'(x)?

1. lim_{h→0} [f(x + h) − f(x)]/h
2. lim_{h→0} [f(x + h) − f(x)]
3. [f(b) − f(a)]/(b − a)
4. lim_{h→0} f(x + h)/h

Answer: A — lim_{h→0} [f(x + h) − f(x)]/h

A) Correct — The derivative is the limit of the difference quotient. B) Leaves out the division by h; this limit is 0 for continuous f. C) Is the average rate of change, with no limit taken. D) Leaves out the subtraction of f(x).

Find d/dx[x eˣ].

1. eˣ
2. x eˣ
3. (x + 1)eˣ
4. (x − 1)eˣ

Answer: C — (x + 1)eˣ

C) Correct — Product rule: 1·eˣ + x·eˣ = (x + 1)eˣ. A) Multiplies the derivatives: 1·eˣ. B) Differentiates only the eˣ factor. D) Subtracts the two product-rule terms instead of adding.

Find d/dx[2ˣ].

1. x·2^(x−1)
2. 2ˣ ln 2
3. 2ˣ
4. 2ˣ/ln 2

Answer: B — 2ˣ ln 2

B) Correct — For a constant base b, d/dx[bˣ] = bˣ ln b. A) Applies the power rule to an exponential. C) Treats 2ˣ like eˣ, missing the factor ln 2. D) Finds the antiderivative instead.

Implicitly differentiate x² + y² = 25. What is dy/dx?

1. x/y
2. −x/y
3. −y/x
4. 2x + 2y

Answer: B — −x/y

B) Correct — 2x + 2y·dy/dx = 0, so dy/dx = −x/y. A) Drops the negative sign when solving. C) Inverts the fraction. D) Differentiates but forgets the dy/dx on the y-term and never solves for it.

Position is s(t) = t³ − 6t² + 9t. What is the velocity at t = 2?

1. 2
2. −3
3. 3
4. 0

Answer: B — −3

B) Correct — v(t) = 3t² − 12t + 9; v(2) = 12 − 24 + 9 = −3. A) Is the position s(2), not the velocity. C) Drops the negative sign. D) Is the acceleration a(2) = 6(2) − 12.

A circle's radius grows at 0.5 m/s. How fast is its area growing when r = 10 m?

1. 10π m²/s
2. 100π m²/s
3. 5π m²/s
4. 20π m²/s

Answer: A — 10π m²/s

A) Correct — dA/dt = 2πr·dr/dt = 2π(10)(0.5) = 10π. B) Is the area itself, not its rate. C) Forgets the factor 2 from the derivative of r². D) Forgets to multiply by dr/dt = 0.5.

f'(c) = 0 and f''(c) = 0. What does the second derivative test say?

1. c is an inflection point
2. c is a local maximum
3. Nothing; the test is inconclusive
4. c is a local minimum

Answer: C — Nothing; the test is inconclusive

C) Correct — For example, x⁴, −x⁴, and x³ all have f' = f'' = 0 at 0 but behave differently. A) Not always: x⁴ has f''(0) = 0 but a minimum at 0. B) Not always: x⁴ has a minimum at 0. D) Not always: −x⁴ has a maximum at 0.

Find ∫ (3x² − 4x + 1) dx.

1. 6x − 4 + C
2. x³ − 4x² + x + C
3. x³ − 2x² + x + C
4. x³ − 2x² + C

Answer: C — x³ − 2x² + x + C

C) Correct — Integrate term by term. A) Differentiates instead of integrating. B) Forgets to divide 4x by the new exponent 2. D) Drops the antiderivative of 1, which is x.

What is the general solution of dy/dx = 3y?

1. y = 3eˣ + C
2. y = Ceˣ + 3
3. y = Ce^(3x)
4. y = 3x + C

Answer: C — y = Ce^(3x)

C) Correct — dy/y = 3 dx, so ln|y| = 3x + c. A) Moves the 3 outside and adds the constant. B) Adds 3 instead of using it as the growth rate. D) Treats y as a constant on the right side.

Find the area between y = x and y = x² on [0, 1].

1. 1/2
2. 1/3
3. 5/6
4. 1/6

Answer: D — 1/6

D) Correct — ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6. A) Integrates only the top curve, x. B) Integrates only the bottom curve, x². C) Adds the integrals instead of subtracting.

What is the sum/difference rule?

d/dx[f ± g] = f' ± g'.

Evaluate lim_{x→∞} 1/x.

0.

What is a differential equation?

An equation involving a function and its derivatives.

Find f''(x) for f(x) = x⁴.

f'(x) = 4x³; f''(x) = 12x².

What is Rolle's Theorem?

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) with f'(c) = 0.

What is the chain rule?

d/dx[f(g(x))] = f'(g(x))·g'(x).