Calculus practice quiz with answers

What is d/dx (x⁵)?

1. 5x⁴
2. x⁴
3. 5x⁵
4. 4x⁵

Answer: A. Power rule: bring the exponent down and subtract 1: 5x⁴.

What is d/dx sin(x)?

1. cos(x)
2. −sin(x)
3. tan(x)
4. −cos(x)

Answer: A. The derivative of sin(x) is cos(x). The derivative of cos(x) is −sin(x).

What is d/dx e³ˣ?

1. 3xe³ˣ⁻¹
2. e³
3. e³ˣ
4. 3e³ˣ

Answer: D. Chain rule: the derivative of eᵘ is eᵘ·u′. Here u = 3x, so u′ = 3 and the answer is 3e³ˣ.

What is d/dx ln(x)?

1. 1/ln(x)
2. ln(x)/x
3. 1/x
4. eˣ

Answer: C. d/dx ln(x) = 1/x for x > 0.

Using the product rule, what is d/dx [x² cos(x)]?

1. 2x cos(x) − x² sin(x)
2. −2x sin(x)
3. 2x cos(x) + x² sin(x)
4. 2x − sin(x)

Answer: A. u = x², v = cos(x). u′v + uv′ = 2x·cos(x) + x²·(−sin(x)) = 2x cos(x) − x² sin(x).

Using the chain rule, what is d/dx (3x + 1)⁴?

1. 12(3x + 1)³
2. 12x³
3. (3x + 1)³
4. 4(3x + 1)³

Answer: A. Outer: 4(3x + 1)³. Inner derivative: 3. Multiply: 12(3x + 1)³.

What is d/dx tan(x)?

1. sec(x)tan(x)
2. cot(x)
3. sec²(x)
4. −csc²(x)

Answer: C. d/dx tan(x) = sec²(x).

What is the limit of (x² − 4)/(x − 2) as x → 2?

1. 2
2. 0
3. 4
4. Undefined

Answer: C. Factor: (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2. As x → 2, this approaches 4.

What is the limit of sin(x)/x as x → 0?

1. 0
2. 1
3. ∞
4. It does not exist

Answer: B. This is a standard limit: sin(x)/x → 1 as x → 0.

What is the limit of (3x² + 1)/(x² − 5) as x → ∞?

1. 3
2. ∞
3. 1
4. 0

Answer: A. The degrees are equal, so the limit is the ratio of the leading coefficients: 3/1 = 3.

What is ∫ x³ dx?

1. x⁴ + C
2. x³/3 + C
3. 3x² + C
4. x⁴/4 + C

Answer: D. Reverse power rule: add 1 to the exponent and divide by the new exponent: x⁴/4 + C.

What is ∫ cos(x) dx?

1. −cos(x) + C
2. −sin(x) + C
3. sin(x) + C
4. cos(x) + C

Answer: C. Because d/dx sin(x) = cos(x), the antiderivative of cos(x) is sin(x) + C.

What is the value of ∫ from 0 to 2 of 3x² dx?

1. 12
2. 8
3. 6
4. 4

Answer: B. An antiderivative is x³. Evaluate: 2³ − 0³ = 8.

What is the slope of the tangent line to f(x) = x² at x = 3?

1. 6
2. 3
3. 2
4. 9

Answer: A. f′(x) = 2x, so f′(3) = 6.

If f′(c) = 0 and f″(c) > 0, what does f have at x = c?

1. An inflection point
2. A vertical asymptote
3. A local maximum
4. A local minimum

Answer: D. Second derivative test: a positive second derivative means the graph is concave up, so the critical point is a local minimum.