Mercer County College       MAT151            Prof. Porter Calculus 1   PRACTICE EXAM #3                    

  1. Given the function:

a.)      Find all critical values


b.)     Show how to use the second derivative test or the first derivative test to determine if the critical values are a maximum or minimum.



c.)      Find the absolute maximum and minimum values on the interval  [-2,2]




  1. Given the function  find the following values.

a.)      Vertical Asymptotes


b.)     Horizontal Asymptotes


c.)      Interval where f(x) increases.


d.)     Interval where f(x) is concave up.



  1. The sales S of a certain book is found to follow the formula  where p is the price asked. The revenue is a product of the price and the amount sold.

a.) Give the Revenue as a function of the price asked.


            b.)     What price will yield the maximum revenue?


c.)      What price(s) will yield the minimum revenue?



  1. Given the function:

Find the zero using Newton's method starting with the value x = 50

Be sure to list the first 5 iteration:

  1. Suppose that during a trip, the distance traveled away from home s miles at time t hours is modeled by the function  


What is the velocity at t = 2?





What is the acceleration at t = 2?





Verify that there must be a time between t = 0 and t = 2 where the speed is 8 mph.





At what time did that occur?







  1. An open rectangular box is created by cutting square corners from a sheet of cardboard that is 12 by 15 inches.


a.)      Find the maximum volume possible








b.)     What would be the largest volume possible if you were restricted to cutting squares that are between 2.5 and 3 inches?








1.           Use the data points to plot a linear regression:

(1900,3.5), (1950,4.5),(2000,6)

2.           After performing a quadratic regression, you obtain the formula y(x) = x(x-2). Find the area under the curve between 0 and 2 in the following manners:

              a.) Estimate the area with 4 rectangles of equal width.

b.) Estimate the area with 4 rectangles using x1=.5, x2=1 x3=1.25 and x4=2 and Δx1=1, Δx2=.25, Δx3=.25, Δx4=.5  Draw a picture of this approximation.

c.) If and show how you might find the exact area as a limit as n goes to infinity of a sum of equally spaced rectangles of width 2/n

              d.) Verify this by finding a definite integral.

              e.) Check this value in the calculator

3.          Evaluate the integrals:


4. Evaluate the integral using a geometric approach

5.  Explain what is wrong with this integral.

6. Evaluate the integral using substitution.

              What is the average value of   over the interval [-2,3]?

7. If the acceleration a(t) = -10 and v(0) = 30, find the velocity v(t).

What is the displacement from t = 0 to t = 4?

What is the distance traveled from t = 0 to t = 4?

8. Sketch the curve y = 1/t and shade the region under the curve whose area is ln3.

If ln a = 2, find a.

Give an integral whose area is 2.