MAT112 Calculus II Pretest #1

Prof. Porter


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 v0 = 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.