The number of gallons of water in a 500,000 gallon tank t minutes after the tank started to drain is G(t) = 200(50-4t)2.

a.) What is the average rate at which the water flows out during the first 10 minutes.

b.) How fast is the water running out of the tank at the end of 7 minutes?

Answer

a.) -48,000 gal/min

b.) -35,200 gal/min

The number of gallons of water in a 500,000 gallon tank t minutes after the tank started to drain is G(t) = 200(50-4t)2.

How many gallons of water are in the tank when the water is flowing at a rate of -54,400 gallons per minute?

Answer

231,200 gallons

The number of gallons of water in a 500,000 gallon tank t minutes after the tank started to drain is G(t) = 200(50-3t)2.

How long does it take to empty the tank? How fast is the water running out at this time?

Answer

a.) 12.5 mins

b.) 0 gal/min

The cost of producing x widgets is given by the function C(x) = 23x + 234.

The revenue generated by producing x widgets is R(x) = 119x – 6x2.

Maximum profit is achieved when marginal cost is equal to marginal revenue. How many widgets should be produced in order to maximize profit?

Answer

8 widgets

If the cost of producing x items is represented by C(x) = 2x + 5000 dollars, and the revenue produced is represented by R(x) = 10x - .001x2, find the maximum profit using calculus.

Answer

$110,000

Dan shoots a rock straight up in the air out of a cannon with a launch velocity of 100 ft/sec. It reaches a height s(t)=15 + 100t – 10t2 after t seconds.

a.) Find the average velocity of the first 9 seconds.

b.) Find the instantaneous velocity with

t = 3. c.) Find the speed of the rock when

t = 10.

Answer

a.) 10 ft/sec

b.) 40 ft/sec

c.) 100 ft/sec

Dan shoots a rock straight up in the air out of a cannon with a launch velocity of 100 ft/sec. It reaches a height s(t)=15 + 100t – 10t2 after t seconds.

a.) Using calculus, find the maximum height reached by the rock?

b.) How high is the rock when the velocity is 75 ft/sec?

Answer

a.) 265 ft

b.) 124.375 ft

Write the equation of the line that is tangent to f(x) = 3x2

- 4x + 10 and is parallel to the the secant line that connects x = -4 and x = 2.

Answer

y = -10x + 7

a.) When is the particle stopped?b.) When does the particle change direction?c.) When does the particle move the fastest?

Answer

a.) t = 0, t = 6, (8, 10)

b.) t = 8

c.) t = 7

a.) When does the particle slow down?b.) When does the particle have negative acceleration?

Answer

a.) (4, 6) U (7, 8)

b.) (4, 6) U (6, 7)

Find the values of a and b so that the following function is differentiable at x = 2

Answer

A = 1

B = 2

Find the derivative of the following function using the formal definition. Then write the equation of the normal line when x = 5.

Answer

a.) f’(x) = 1/(2sqrt(x+4))

f’(5) = 1/6

b.) y = -6x + 33

What are the four reasons that a function, f(x) would not be differentiable at a point x = a? Draw a picture of each.

Answer

a.) corner

b.) cusp

c.) discontinuity

d.) vertical tangent

(look in notes for pictures and examples)

The length of a rectangle is (2x+3) and the width is (4x – 5). Write a formula to represent the change in the

area with respect to x.

Answer

dA/dx = 16x + 2