# 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