Assignment 3 (8%)

Assignment 3 is graded with a total of 100 marks and it contributes 8 percent towards your course grade.

1. Find the derivative using the differentiation rules. You need not simplify your answer.

b. y = (x4  + 3)e−

c. s = ln(tan x2 ) + tan(ln x2 )

d. g = 4√sec ()

e. f = (x4  + 3) ln ()

2. If a tank holds 50,000 litres of water which drains from the  bottom of the tank in 40 minutes, then the volume of water, V,

which remains in the tank after t minutes is given by

V(t) = 50000(1 − )2      0 ≤ t ≤ 40

a. Find the rate at which water is draining from the tank after

10 minutes and after 20 minutes.

b. At what time is the water flowing the fastest? At which time is it flowing the slowest?

3.

a. For the function f(x) =  at which point(s) does the tangent line have a slope of -1? Give exact answers.

b. Find an equation of the tangent line(s) at the point(s) in a. Give exact answers.

4.

a. Suppose that f(X) = g(X 3 ) + eg(3) where g′ and g′′ exist. Find f′ and f′′ in terms of g′ and g′′ .

b. Find f", when f(X) = e  2   + 5 .

5. Differentiate using implicit differentiation:

a. √2X2  − 3y = X 3y 2  + 4. Do not square both sides.

b. X 2 cos(3y) − y2 sin(X 2) = 5

6. Differentiate using the logarithmic differentiation:

3

a. y =  (42 −3+1)7 (6+2)2

1

b. y = (ln X)x

7. Suppose that the cost, C in dollars, of producing x units of a certain item is C(X) = 920 + 3X − 0.03X2  + 0.0006X3 .

a. Find the marginal cost function

b. Find C′ (100) and explain its meaning

c. Compare C′ (100) with the cost of producing the 101st item.