Limits and Derivatives

Q. Evaluate the limit: lim (x -> 0) (sin(5x)/x).

Solution

Using the limit property, lim (x -> 0) (sin(kx)/x) = k, thus lim (x -> 0) (sin(5x)/x) = 5.

Correct Answer: C — 5

Q. Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3).

Solution

Factoring gives (x - 3)(x + 3)/(x - 3), canceling gives x + 3, substituting x = 3 gives 6.

Correct Answer: C — 6

Q. Find the derivative of f(x) = 4x^3 - 3x^2 + 2x - 1.

Solution

Using the power rule, f'(x) = 12x^2 - 6x + 2.

Correct Answer: A — 12x^2 - 6x + 2

Q. Find the limit: lim (x -> 0) (1 - cos(x))/x^2.

Solution

Using L'Hôpital's rule, the limit evaluates to 1/2.

Correct Answer: B — 1/2

Q. Find the limit: lim (x -> 1) (x^3 - 1)/(x - 1).

Solution

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1), canceling gives x^2 + x + 1, substituting x = 1 gives 3.

Correct Answer: C — 2

Q. What is the derivative of f(x) = 5x^4 - 4x^3 + 3x - 2?

Solution

Using the power rule, f'(x) = 20x^3 - 12x^2 + 3.

Correct Answer: A — 20x^3 - 12x^2 + 3

Q. What is the derivative of f(x) = e^(2x)?

Solution

Using the chain rule, f'(x) = 2e^(2x).

Correct Answer: A — 2e^(2x)

Q. What is the derivative of f(x) = ln(x^2 + 1)?

Solution

Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).

Correct Answer: A — 2x/(x^2 + 1)

Q. What is the limit of (3x^2 - 2x + 1) as x approaches 2?

Solution

Substituting x = 2 into the expression gives 3(2^2) - 2(2) + 1 = 12 - 4 + 1 = 9.

Correct Answer: B — 7

Showing 1 to 9 of 9 (1 Pages)