Question

For full credit be sure to show all work and units.

1. A 20 foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at 2 ft/sec.

a. How fast is the top of the ladder sliding down the wall when the top of the ladder is 12 feet from the ground?

b. Interpret the meaning of the sign.

2. The current speed record for weight loss is 487 pounds to 130 pounds in 8 months. Show using the Mean Value Theorem that at some instant during this period the rate of weight loss exceeded 44 pounds per month.

3. Given the function ?(?) = (x^3/3)-(3x^2/2)+2x:

a. Find all critical numbers.

b. Identify which, if any, critical numbers are local max or min and explain your answer.

c. Find any inflection points and give the x value.

d. On the interval [0.6, 2.6] identify the absolute max and min, if any. and justify your answer.

e. Give the interval where the curve is concave up and justify your answer.

4. This problem is about some function. All we know about the function is that it exists everywhere and we also know the information given below about the derivative of the function. Answer each of the following questions about this function. Be sure to justify your answers. f ′(−5) = 0 f ′(−2) = 0 f ′(4) = 0 f ′(8) = 0 f ′(x) < 0 on (−5,−2), (−2,4), (8,∞) f ′(x) > 0 on (−∞,−5), (4,8)

a. Identify the critical points of the function.

b. Determine the intervals on which the function increases and decreases.

c. Classify the critical points as relative maximums, relative minimums or neither.

5. Use Newton’s Method to determine a root of x to four decimal places for the following polynomial if your initial guess is -3.5: ?^3 − ?^2 − 15? + 1. Be sure to show all work including intermediate values.

6. We want to construct a rectangular box with a square bottom and no top that will have a volume of 108 cm3. Determine the dimensions of the box that will minimize the amount of material needed to construct the box. The volume of this rectangular box is V=w^2h and the surface area is S = 4wh + w^2 where w is the width of the box and h is the height.

Answer 1

b. This negative sign shows decreasing of the angle of ladder from ground.