Assignment 5 (8%)

Assignment 5 is graded with a total of 100 marks and it contributes 8 percent towards your course grade.

1. For f(x) =  x 6  − 3x2

a. Give in interval notation the intervals wheref is increasing and wheref is decreasing.

b. Give in interval notation the intervals wheref is concave up and where f is concave down.

c. Give the coordinates of any points of inflection.

d. Sketch the curve.

2. Repeat question 1 above for  f(x) = e −x2

3. A wire 10 m long is to be cut in two pieces. One piece is bent into a square and the other into a circle. How should the wire  be cut so that the total area enclosed is

a. a maximum?

b. a minimum?

Give exact answers and then to one decimal place.

4. A boat leaves a dock at 2:00 pm and travels south at a speed of 20 km/h. Another boat has been heading east at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two   boats closest together?

5. Consider the cost function, C(X) = 25000 + 100X + 0.1X2        where C is given in dollars. The average cost is given by c(X) =

C(X)

X

.

a. Find the cost, average cost, and marginal cost at a production level of 100 units.

b. Find the production level that will minimize the average cost.

c. Find the minimum average cost.

6. Use Newton’s method to approximate the root of X4 + X − 4 = 0 in the interval [1,2] correct to six decimal places.

7. Use Newton’s method to find the positive root of 3 sin X = X correct to six decimal places.

8. Find f if f" (t) = e2t − 4 cos t  if f(0) =   and  f (2(冗) ) = 0

.

9. Find a function f such that f′ (X) = X5 + 20 and the line 12X + y = −20 is tangent to the graph of f.