Supplemental Review problems

 

  1. Demand for a product is 150 units per day when the price per unit is $30. Demand decreases by 50 units for each  price increase of $5. Assume that demand and price have a linear relationship.

a)      Find the linear demand equation.

 

 

 

 

 

 

 

 

 

b)      Find the revenue function, put it in vertex form and find the maximum revenue.

 

 

 

 

 

 

 

 

For the same product, fixed costs are $270 and each unit costs an additional $15 to produce.

c)      Find the cost and profit functions and the maximum profit.

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. List the transformations of y=ln(x) which result in f(x)= -2ln(x-4) +7.

 

 

 

 

 

  1. The total U.S. private health expenditure in billions of dollars is shown for 1960 through 2005.

 

Year    1960   1965   1970    1975    1980    1985   1990   1995    2000    2005

 Exp.      27       41        73      130       246      427     696     990    1300   1902

 

Let x be the number of years past 1960 instead of the actual year.

Find the quartic and exponential regression models. Which is a better fit to the existing data? What does each model predict for 2010? 

 

 

 

 

 

  1. Find all asymptotes, horizontal and vertical for each function.

 

a)

 

 

 

 

 

b)

 

 

 

 

  1. Solve for x.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.

 

a) For what value(s) of c does  not exist?

 

 

 

 

 

 

b) At which x value(s) is f not continuous?

 

 

 

 

 

 

c)      At which x value(s) is f not differentiable?

 

 

 

 

 

 

 

 

 

 

7. Answer a), b) and c) as above for

 

 

 

 

 

 

 

 

 

 

 

 

8. f(x) is differentiable except at x= -1 where it has a vertical asymptote.

a)      Find the intervals where f is increasing and where f is decreasing. Find all local extrema.

 

 

 

 

 

 

 

 

 

 

 

 

 

b)      Find f”(x) and all inflection points of f.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9. A profit function is . 

a) Approximate the change in profit if x increases from 100 to 101.

 

 

 

 

 

 

b) Find the average profit function and the marginal average profit function.

 

 

 

 

 

 

10. A store can sell 50 sandwiches per day at the current price of $3.00 per sandwich.

For each decrease of $0.15 in the price, they can sell 10 more sandwiches per day.What should the price be to maximize revenue?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11. An object has velocity  miles per hour.

 

a)      Find the acceleration v’(t). Find the times for maximum and minimum velocity.

 

 

 

 

 

 

 

b)      What is the total distance traveled between the end of the 1st hour and the end of the 4th hour? (1<t<4)  Be able to do this by hand and only use the calculator to check your answer.

 

 

 

 

 

 

 

 

c)      What is the average velocity for this same time period?

 

 

 

 

 

 

 

12.  is the demand quantity if the price per unit is p.

a) Find the elasticity as a function of p. Is demand elastic or inelastic when p=512?

If p increases from 512, will revenue increase or decrease?

 

 

 

 

 

 

 

 

b)      Approximate the % change in demand if price increases from 512 to 520.

 

 

13. Find the antiderivatives.

 

a)

 

 

 

 

 

 

 

 

b)

 

 

 

 

 

 

c)

 

 

 

 

 

 

 

d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14. Find the left and right hand Riemann sums and their average for  and

 [a, b]=[0, 2.5] using 5 equal subintervals. Compare to the actual definite integral.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15. Find the consumers’ surplus and the producers’ at equilibrium for

  and 

Be able to do this by hand and only use fnint to check your answer.

 

 

 

 

 

 

 

 

16. Money is deposited as a continuous income stream with the rate of deposit

 where t is in years.

a)      How much is deposited between t=0 and t=10?

 

 

b)      Find the present value of the income stream for the first 10 years if interest is 6% compounded continuously.

 

c)      Find the future value of the amount in b)

 

 

17. Find the first and second partial derivatives of

Find the critical points and classify them by the 2nd derivative test or state that the test fails.

 

Key:

 

1. a) p= -0.1x+45   b)    Rmax=5062.5

 

2. i) shift right 4 units  ii) expand vertically by a factor of 2 iii) reflect across the x-axis and last of all  iv) shift up 7 units. The y-shift must be done last.

 

3. quartic is a better fit to the existing data because R^2 = .99807 where as the exponential model has r^2=.9877.  The quartic prediction for 2010 is 2565.75 billion and the exponential prediction is 3995.86 billion.

 

4. a) H.A. is y=0,  V.A. x=3 and x= -3  b) H.A. y=1,  V.A. x=2 .  There is only a hole at x=-2, not a V.A.  Be sure to write asymptotes as y=a or x=b.

 

5. x=3.

 

  1. a) at x=2 which is a V.A and at x=3 which is a gap.  b) 2, 3, -2  (hole at -2)

c) 2, 3, -2

 

7. a) 2    b)  -2, 2   c)  -2, 2, 3  (corner at x=3)

 

8. a) decreasing    increasing (-1,2) and (2, no local extrema

b) concave up (2,8)   concave down (

inflections at x=2 and x=8

 

9. a) $30    b)  

 

10. $1.875 or $1.88 rounded

 

11. a) minimum velocity is at t=0 and v(0)=0.  maximum is at t=e-1 and v(e-1)=1/e

b) Find  miles    c) 1.055/3 = .35164

12. a)     E(512)=4/3 so demand is elastic when p=512. Revenue will decrease if p increases from 512.

b)

13. a)      b)     c)

d)

 

14. left sum=3.75  right sum=6.875  average=5.3125    actual integral=5.2083333

 

15. equilibrium is at (624,25)  C.S.=5184   P.S.=1036.8

 

16. a) $11070.14     b) $8242.00 (rounded)   c) $15017.90

 

17. critical points are (0,0), (-3,-18), and (3, 18)     (0,0) is a saddle point and the other two are local maximums.