12th Grade Calculus Spiral Review: Derivatives, Integrals, and Limits

Question 1

  1. Find the derivative of f(x) = 3x^3 - 5x^2 + 2x - 7.

Answer: f'(x) = 9x^2 - 10x + 2.

Question 2

  1. Evaluate the limit: ( \lim_{x \to 2} (x^2 - 4) / (x - 2) ).

Answer: The limit is 4.

Question 3

  1. Calculate the integral: ( \int (2x^3 - 3x + 1) dx ).

Answer: ( \frac{1}{2}x^4 - \frac{3}{2}x^2 + x + C ) where C is the constant of integration.

Question 4

  1. Determine the second derivative of f(x) = x^4 - 8x^3 + 18x^2.

Answer: f''(x) = 12x^2 - 48x + 36.

Question 5

  1. Find the derivative of f(x) = e^(2x) sin(x).

Answer: f'(x) = e^(2x)(2sin(x) + cos(x)).

Question 6

  1. Evaluate the limit: ( \lim_{x \to 0} (\sin(5x)/x) ).

Answer: The limit is 5.

Question 7

  1. Find the area under the curve y = x^2 from x = 1 to x = 4.

Answer: Area = 21.

Question 8

  1. If f(x) = x^2 + 3x + 2, find f'(x).

Answer: f'(x) = 2x + 3.

Question 9

  1. Find the limit: ( \lim_{x \to \infty} (3x^2 - x)/(5x^2 + 2) ).

Answer: The limit is ( \frac{3}{5} ).

Question 10

  1. Integrate: ( \int (4x^2 - 2) dx ).

Answer: ( \frac{4}{3}x^3 - 2x + C ).

Question 11

  1. Find the critical points of f(x) = x^3 - 6x^2 + 9x.

Answer: Critical points are x = 0, 3.

Question 12

  1. Evaluate ( \int (5sin(x) + 3cos(x)) dx ).

Answer: ( -5cos(x) + 3sin(x) + C ).

Question 13

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

Answer: h'(x) = ( \frac{2x}{x^2 + 1} ).

Question 14

  1. Find the limit: ( \lim_{x \to 2} (2x^2 - 8)/(x - 2) ).

Answer: The limit is 8.

Question 15

  1. Evaluate ( \int_0^1 x^3 dx ).

Answer: ( \frac{1}{4} ).

Question 16

  1. Determine the first derivative of f(x) = x^5 + 2x^4 - x + 7.

Answer: f'(x) = 5x^4 + 8x^3 - 1.

Question 17

  1. Evaluate the limit: ( \lim_{x \to 0} (1 - cos(x))/x^2 ).

Answer: The limit is ( \frac{1}{2} ).

Question 18

  1. Compute the derivative of f(x) = 3x^2 ln(x).

Answer: f'(x) = 6xln(x) + 3x.

Question 19

  1. Find ( \int_1^2 (3x^2 + 4) dx ).

Answer: ( \frac{19}{3} ).

Question 20

  1. What is the limit of the sequence defined by a_n = 1/n as n approaches infinity?

Answer: The limit is 0.

Question 21

  1. Calculate the second derivative of g(x) = x \sin(x).

Answer: g''(x) = x \sin(x).

Question 22

  1. Use the Fundamental Theorem of Calculus to evaluate: ( \int_1^3 (2x) dx ).

Answer: The value is 4.

Question 23

  1. Find the derivative of y = cos(x^2).

Answer: y' = -2xsin(x^2).

Question 24

  1. Evaluate ( \lim_{x \to 2} (x^2 - 4)/(x - 2) ) using L'Hospital's Rule.

Answer: The limit is 4.

Question 25

  1. Compute the integral ( \int (6x + 2) dx ).

Answer: ( 3x^2 + 2x + C ).

Question 26

  1. Find the maximum point of f(x) = -2x^2 + 4x + 1.

Answer: Maximum at x = 1.

Question 27

  1. Find the limit: ( \lim_{x \to 0} (tan(x)/x) ).

Answer: The limit is 1.

Question 28

  1. Evaluate ( \int (x^2 - 4x + 3) dx ).

Answer: ( \frac{1}{3}x^3 - 2x^2 + 3x + C ).

Question 29

  1. What is the derivative of f(x) = 1/(x + 1)?

Answer: f'(x) = -1/(x + 1)^2.

Question 30

  1. Find the limit: ( \lim_{x \to 0} (x^2 \cdot sin(1/x)) ).

Answer: The limit is 0.

Notes

This spiral review includes a variety of problems covering key concepts in derivatives, integrals, and limits in AP Calculus AB. The problems are designed to reinforce understanding and build fluency in calculus topics.