Final Exam Review

Integrals

100

∫3x(x²+6x+30)dx

3/4x⁴+6x³+45x²+C

200

∫30 sqrt(x³) dx

12x^5/2+C

300

₀∫⁴ 5x+2sqrt(x) dx

152/3

Limits

100

lim x->4 (x²+6)/(x²+2x-3)

22/21

200

lim->∞ (9x²+2x+3)/(9x+6)

∞

300

lim x->2 (x²-4)/(2x²+4x-16)

1/3

Optimization

100

Tell me the constraint and objective equations of the following problem:

You have 400ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?

Constraint: 2x+2y=400

Objective: xy = A

200

You have 800ft of fencing to make a pen for hogs. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?

x = 200ft, y=400ft

300

The sum of the length, width, and height of a box does not exceed 96 inches. What is the greatest volume of the box under these conditions?

32768in³

Trig

100

Find the derivative of the function:

y = cot(12x²+4sqrt(x))

-csc²(12x²+4sqrt(x))(24x+2x^-1/2)

200

Find the derivative of the function:

c(x) = sinx/1+cosx

(1+cosx)(cosx)-sin(x)(-sinx)/(1+cosx)² OR 1/1+cosx

300

Find the integral:

_-𝛑/2∫⁰sinx + 4x² dx

4.167 or -1 + 𝛑³/6

Random

100

Find the critical points of the function:

f(x) = 6x²+2x-2

-1/6

200

Find the derivative:

f(x) = (2x²+2x+4)(cosx)

(4x+2)cosx-(2x²+2x+4)sinx

300

Find the derivative of the function:

f(x) = xe^-2x

x(-2e^-2x)+e^-2x