Final Exam Review

Differentiation

100

Find the derivative of 

f(x)=6x^3−9x+4

a.f′(x)=18x^2

b. f′(x)=18x^2−9

c. f′(x)=9x−9

d. f′(x)= 0

b. f′(x)=18x^2−9

200

Find the derivative of y=√x+8* 3√x−2 *4√x

a.dy/dx = 1/2x^−1/2+8/3x^-2/3−1/2x^−3/4

b. dy/dx= DNE

c. dy/dx = 1/2x^−1/2+8/3x^-1/3−1/2x^−2/4

d. dy/dx= 0

a. dy/dx = 1/2x^−1/2+8/3x^-2/3−1/2x^−3/4

300

Find the derivative of f(y)= y⁵−5y³+2y

                                        ------------

                                              y³

a. f'(y)= y−4y^-3

b. f'(y)= 0

c. f'(y) = 2y−4y^-3

d. f'(y)= 4y^-3

c. f'(y) = 2y−4y^-3

400

The position of an object at any time t is given by s(t)=3t⁴−40t³+126t²−9. Determine the velocity of the object at any time t.

a.s′(t)=12t²

b.s′(t)=DNE

c. s′(t)=12t(t−1)

d. s′(t)=12t(t−3)(t−7)

d. s′(t)=12t(t−3)(t−7)

Integrals

100

Evaluate each of the following indefinite integral

∫6x⁵−18x²+7dx

x⁶−6x³+7x+c

200

Evaluate each of the following indefinite integral

∫6x⁵dx−18x²+7

x⁶+c−18x²+7

300

Evaluate each of the following indefinite integral.

∫40x³+12x²−9x+14dx

10x⁴+4x³−9/2x²+14x+c

400

Evaluate each of the following indefinite integral.

∫40x³+12x²−9xdx+14

10x⁴+4x³−9/2x²+c+14

Monotonicty

100

Determine where the function h(z)=6+40z³−5z⁴−4z⁵ is increasing and decreasing.

Increasing :−3<z<0,  0<z<2

Decreasing :−∞<z<−3,  2<z<∞

200

Determine where the function R(x)=(x+1)(x−2)² is increasing and decreasing.

Increasing :−∞<x<0,  2<x<∞

Decreasing :0<x<2

300

Determine where, if anywhere, the function y=2z⁴−z³−3z² is not changing.

z = 0, 1.07359, -0.69859

400

Determine where, if anywhere, the function f(x)=x³+9x²−48x+2 is not changing.

x= -8, 2

Random

100

In the following assume that x and y are both functions of t. Given x=−2, y=11 and x′=−4 determine y′ for the following equation.

6y²+x²=2−x³e^4−4y

y′=8/11

200

2. In the following assume that x, y and z are all functions of t. Given x=4, y=−2, z=1, x′=9 and y′=−3 determine z′ for the following equation.

x(1−y)+5z³=y²z²+x²−3

z'= 45/11

300

For a certain rectangle the length of one side is always three times the length of the other side.

If the shorter side is decreasing at a rate of 2 inches/minute at what rate is the longer side decreasing?

y′=−6

400

A tank of water in the shape of a cone is being filled with water at a rate of 12 m³/sec. The base radius of the tank is 26 meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing when the radius of the top of the water is 10 meters?

h′=3/25π