Calculus 12 Final Exam Review (Part 1)

Limits

100

What is the limit as x approaches 5 of the function y = 3x²?

75

200

What is the function as x approaches infinity of the function y = 1/x?

0

300

What is the limit as x approaches infinity of the function f(x) = (2x²+3) / (x²-5x-1)?

2

400

What is the limit as x approaches 3 of the function f(x) = (2x²-5x-3) / (x-3)?

7

500

What is the limit as x approaches 5 of the function f(x) = (x-5) / (sqrt(x+4)-3)?

6

Derivatives - Product & Quotient Rule

100

What is the derivative of the function y = x²(2x⁴-5)?

12x⁵-10x

200

What is the derivative of the function y = (1 + x^3/2)(x^-3 - 2x^1/3)?

-3x^-4 - (3/2)x^-5/2 - (2/3)x^-2/3 - (11/3)x^5/6

300

What is the derivative of the function f(x) = (2x²) / (3x+1)?

(6x²+4x) / (3x+1)²

400

What is the derivative of the function f(x) = (sqrt(x) + 2x) / (7x - 4x²)?

(x/2^-1/2 + 2)(7x - 4x²) - (x^1/2 + 2x)(7 - 8x) / (7x-4x²)

500

What is the derivative of the function f(x) = (5x⁵ - x³ - 4) / (2x² - 5)?

(30x⁶ - 127x⁴ + 15x² + 16x) / (4x⁴ - 20x² + 25)

Derivatives - Chain Rule

100

What is the derivative of the function y = (x³ + 3)⁵?

15x²(x³ + 3)⁴

200

What is the derivative of the function y = (5x² + 3)⁴?

40x(5x² + 3)³

300

What is the derivative of the function f(x) = (3x - 1)(-3x² - 4)^-3?

3(15x² - 6x - 4) / (-3x² - 4)⁴

400

What is the derivative of the function (x² - 3)^1/5 / (-x - 5)?

(3x² - 10x - 15) / 5(-x - 5)²(x² - 3)^4/5

500

What is the derivative of the function f(x) = -sin(-9x² + 3x + 5)?

-cos(-9x² + 3x + 5)(-18x + 3)

Relative Extrema

100

y = x³ + 4x²

What is the x-coordinate of the relative maximum on the interval (-5,0)?

-8/3

200

y = x⁴ - 2x³

For what value of x does f have a relative minimum?

3/2

300

y = x³ - 3x² + 6

For what value(s) of x does f have a relative maximum?

0

400

f(x) = x⁴ - 2x³

Find the relative extrema of the function. 

Relative minimum at (3/2, -27/16)

500

f(x) = 6x⁵ - 50x³ - 120

For what value(s) of x does f have a relative maximum?

-sqrt(5)

Related Rates + Optimization

100

The radius of a circular puddle is growing at a rate of 10 cm/sec. How fast is its area growing when the radius is 14 cm?

280(pi) cm²/sec

200

A sphere-shaped balloon is being inflated so that its diameter is increasing at a rate of 2.9 cm/min. How quickly is the volume of the balloon increasing when the diameter is 17 cm? 

(1445/4)pi cm³/min

300

What are two positive numbers with a product of 100 and the smallest sum possible?

10, 10

400

A farmer is building a fence around a rectangular field that has a river on one of its sides. If he has 2400 ft of fencing and wishes to enclose the largest area possible, what dimensions should he use? 

x = 600, y = 1200

500

A box is made by cutting square pieces of the same size from the corners of a sheet of cardboard and then folding up the sides. If a square sheet of 20 cm x 20 cm cardboard is used, what are the dimensions of a box with the largest possible volume?

(40/3) cm x (40/3) cm x (10/3) cm