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.