Exam 2 Review Math 142

1. You should check that f is continuous at x=0.

a) Find the average rate of change of f(x) from x= -1 to x=0.

b) Find the instantaneous rate of change at x= -1.

c) Does f have a derivative at x=0? If yes, find it. If no, why not?

2. Find the equation of the tangent line at (2, f(2)) for each.

a)

3.

a) Is f continuous at x=1? b) Is f differentiable at x=1?

4. Compute the derivative. In c and d, find where the tangent line is horizontal.

a) b) c)

d) e) f)

g) (Rewrite using log rules first)

5. The demand equation for a certain commodity is x=1000-5p. The cost equation for the same commodity is C(x)=24x+2000.

a) Find the demand price as a function of x and determine the domain.

b) Find the revenue function and the domain of the revenue function.

c) Find the profit equation, P(x) and find the marginal profit function.

d) Approximate the change in profit from production level x= 100 to x=101.

e) Find the marginal average profit and the quantity at which average profit is a maximum.

6. The demand equation for a certain commodity is The cost equation for the same commodity is C(x)=2000+4x.

a) Find the revenue function and the marginal revenue.

c) What is the approximate change in revenue from x=1225 to x=1226?.

Note:

d) Find the profit function.

e) At what quantity is profit a maximum?

7. The derivative of f(x) is .

a) On what intervals is f(x) increasing?

b) On what intervals is f(x) decreasing?

c) Where does f have a local max?

d) Where does f have a local min?

8. The derivative of a function is

a) On what interval(s) is f increasing?

b) Locate the x value of any local extremum and tell what type(max or min) of extremum f has there.

c) Find f”(x)

d) On what interval(s) is f concave up?

9. Demand for a product is x = f(p)=

a) Find the elasticity function, E(p).

b) For what values of p is demand inelastic?

c) Approximate the change in demand if price increases from 2 to 2.25. Will this cause an increase or a decrease in revenue?

d) Repeat c if p increases from 6 to 6.5.

10. f(u) =ln(u). u is a function of x. u(0)=5 and u’(0)=4. Find at x=0.

11.

a) At what x value is the tangent line horizontal?

b) At what x value is the tangent line vertical?

12. Find the derivative of each. Assume g(x) is differentiable.

a) b) c)

Key

1. a) -1/4 b) -1/3 c) yes, f’(0)= -1/4

2. a) y=4(x-2)+4 b) y= c) y=

3. yes, f’(1)=3

4. a)

b) -1<x<1

c) Horizontal tangent at 0, 1,-1,1/3,-1/3, defined for all x

d) Horizontal tangents at 0, 4, -4, 1, -1,

e)

f)

g)

5. p = 200-0.2x b) R(x)=

c) P(x)= P’(x) = -0.4x + 176

d) 136

e) is 0 when x=100

6 a) R(x)=

b)

c) P(x)=

d) x=940.444

7.a) b) (1, 3) and (5, ) c) local max at x=1 and at x=5

d) local min at x=3

8. a) Horizontal tangent at x=0, undefined at x=3 and x= -3, increasing on (0,

b) x=0 local min.

c)

d) ( -3, 3) no inflection points.

9. a) 0.04p b) p<5 c) Demand will decrease by about 2%, revenue will increase.

d) Demand will decrease by about 12%, revenue will decrease.

10. 0.8

11. a) x=5/2 b) x=3 and x = 2

12. a) 2xg(x) +xg’(x) b) c)