quiz/test

Unit 2 Exam Name ________________________ Math 331 Spring 2015 DIRECTIONS: Show as much work as possible within each question. This is the only way I can give partial credit if you happen to make a mistake. TI-89’s are wonderful calculators, but they don’t show me if you know anything about calculus! Therefore, do not expect to receive full credit for a question if little or no work is shown. You may have 2 hours maximum to complete the exam. Show all work on the exam itself, you should not use any outside paper, notes, etc.

1. (8 points each) Use calculus to determine the interval where the functions are decreasing a. 4 316( ) 9f x x x+= −

b. 1/ 7 8/ 7( ) xf x x +=

2. (8 points each) Determine the intervals where the functions are concave up and concave down a. 4 316( ) 9f x x x+= −

b. ( )2 3( ) ln xf x +=

3. (12 points) An apartment complex owner currently has 50 units rented at $400 each month. For each increase in price of $25 per month, there will be 2 less tenants.

a. Complete the following table: Tenants Price charged/month Revenue

50 400 20000 48 600

b. Use calculus to determine the maximum revenue which can be earned each month for the

owner.

4. (8 points) The height of a ball thrown into the air is described by a certain function ( )f x . When the ball reaches its maximum height,

a. Is '( )f x positive, negative, or zero? Explain your reasoning.

b. Is "( )f x positive, negative, or zero? Explain your reasoning.

5. (8 points) Determine the absolute maximum and absolute minimum values for the equation 2( ) 8 lnf x x x= − on the interval [1, 4] .

6. (8 points) Use implicit differentiation to calculate dy dx

for 2 36x xy y+ − = .

7. (8 points) Use differentials to approximate 97

8. (8 points) Instead of plugging in a number to a very difficult equation, we sometimes plug the value into the equation of the tangent line instead, because lines are easier to work with. Let’s say we want to plug in 3.2x = to 2 3 23 4 40x y xy+ + = . Instead, write the equation of the tangent line for

2 3 23 4 40x y xy+ + = through the point ( )3,1 and then plug in 3.2 to the tangent line equation to get a value for y.

9. (8 points) In a certain town, sociologists have determined that crime rates are influenced by temperature by the equation ( )20.1 60 100C T= − + where C is the crimes per month and T is the temperature (in Fahrenheit). If the average temperature for May was 70° and was changing at 8 degrees per month, determine the rate of change of crimes per month.

10. (8 points) For the function 5( ) xf x e= , calculate the 12th derivative. (calculate and simplify the first few derivatives, then look for a pattern to predict the 12th derivative.)