Quiz #7

Name:______________________________

Mathematics 1501
October 14, 1999
Tom Morley

This quiz is one half hour. It is open book and notes,
and caculators are allowed. Show all work and explain your proceedures.

Problem 1 (10 points) . Write down an
(a) Upper
and
(b) Lower Reimann sum for
. Do not evaluate them . Write down sums.
The interval [-1,2] should be divided into n sub-intervals.

Width of intervals is (b-a)/n = (2 - -1)/n = 3/n.

Endpoints are: -1 = -1 + 0 3/n,
-1 + 1 (3/n),
-1 + 2 (3/n),
....,
-1 + (n-1) 3/n,
-1 + n 3/n = 2

Lower sums are at left hand endpoints;

Mathematica can do this sum:

Upper sums are similar:

Again Mathematica can do this sum:

The actual area is:

Triangle below axis is -(1/4) - (1) in width, and 1 + 4 (-1) in hight.

Triangle above axis is 2 - (-1.4) in width, and 1 + 8 in height:

Problem 2 (10 Points) Graph one function f(x) with the all of the following properties

a) f''(x) > 0, for x < -1,
f''(x) > 0, for -1<x<2,
f''(x) <0, for x > 2
b) f'(x) >0 for x <-1,
f'(x) <0 for -1<x<.357,
f'(.357) = 0,
f'> 0 for .357 < x < 2,
f'(x) > 0, for x> 2
c) vertical asymptotes at x = -1, and x=2
d) horizontal asymptotes to y = 1, as x -> plus and
minus infinity.

A closer look at the local min near .35: f(x) is BLUE, f'(x) is RED