Calculus AB Review!

derivatives

100

given y=3sin(x), find dy/dx

dy/dx= 3cos(x)

200

If xy² = -y³ - x³ then find dy/dx in terms of x and y.

dy/dx= (-y²-3x²)/(3y²+2xy)

300

f is a differentiable function with f(3)=-4 and f^'(3)=-2. Let the function g(x)=f(x)/(2x-1). Write the equation of the line tangent to the graph of g at the point where x=3.

y+(4/5)=-(2/25)(x-3)

400

Given the function f(x)= 5/(-9x-6)³, find f^'(x) 

f^'(x)=-15(-9x-6)^-4(-9)

500

if -y=-2x³-y² then find d²y/dx² at the point (-1,2) in simplest form.

4/3

related rates

100

A right triangle has legs of 21 inches and 28 inches whose sides are changing. The short leg is increasing by 8 in/sec and the long leg is growing at 2 in/sec. What is the rate of change of the area?

dA/dt= 133 in²/sec

200

A rectangle has a length of 11 inches and a width of 12 inches whose sides are changing. The length is increasing by 10 in/sec and the width is shrinking at 5 in/sec. What is the rate of change of the area?

dA/dt = 65 in²/sec

300

A right triangle has legs of 30 inches and 40 inches whose sides are changing. The short leg is decreasing by 2 in/sec and the long leg is growing at 5 in/sec. What is the rate of change of the hypotenuse?

dc/dt= 2.8 in/sec

400

A rectangle has a length of 8 inches and a width of 12 inches whose sides are changing. The length is decreasing by 9 in/sec and the width is growing at 4 in/sec. What is the rate of change of the area?

dA/dt = -76 in²/sec

500

The height and radius of a cylinder are both changing. The height of the cylinder is increasing at a constant rate of 1 foot per second. The volume remains a constant 216 cubic feet. At the instant when the height of the cylinder is 8 feet, what is the rate of change in the radius? The volume of a cylinder can be found with the equation V=πr²h Round your answer to three decimal places.

dr/dt = -0.183 ft/sec

particle motion

100

A particle moves along the x-axis so that at time t, is greater than or equal to, t≥0 its velocity is given by v(t)=3t²−4t−20. Determine if the particle is speeding up or slowing down at t = 3

slowing down

200

A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=−t²−8t+44. Determine the acceleration of the particle at t=9.

-2

300

A particle moves along the x-axis so that at time t≥0 its velocity is given by v(t)=2t+8. Determine if the particle is speeding up or slowing down at t=3.

speeding up

400

A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=t³−3t²−9t. Determine all values of t when the particle is at rest.

3

500

A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=−t³+9t²−24t. Determine the speed of the particle at t=1.

9

Indefinite integrals radicals

100

∫(3/√x)dx

6x^1/2+C

200

∫√x⁵ dx

(2/7)x^7/2+C

300

∫(5√x³)/4 dx

(1/2)x^5/2+C

400

∫√x⁵/4 dx

(1/14)x^7/2+C

500

∫(6 ⁴√x)/5 dx

(24/25)x^5/4+C

Product, Quotient, chain

100

Given the function f(x) = x² - xsinx, find f^'(x) in any form

f^'(x)=2x - sinx - xcosx

200

Given the function f(x) = x/(5-3x⁴), find f^'(x) in simplified form.

f^'(x)= (9x⁴+5)/(5-3x⁴)²

300

Given the function f(x) = 2√(cosx), find f^'(x).

f'(x)= -sinx/√cosx

400

given the function y=4cos(4x²-6), find dy/dx

dy/dx=-32xsin(4x²-6)

500

Given the function f(x)=x/(5-2x⁴), find f^'(x) in simplified form.

f^'(x)=(6x⁴+5)/((5-2x⁴)²)