Part I. No Calculators

You may not have a calculator while this part of the test is in your posession. When you are through with this part, hand it in. Then you may work on the rest of the test using a calculator.

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(1.)

6 t + 1

t^{3}+ 2 t^{2}+ t(2.)

ln(x ^{3})

xdx (3.)

1

0(- 3 x + 2) e ^{2 x }dx(4.)

e ^{sin2 x}sinx cosx dx(5.)

x ^{2}

2 x - 3dx (6.)

6 x - 4

x^{2}+ 8 x - 9dx (7.)

- 2

x ^{2}+ x +26

4dx (8.)

sin( e ^{x}) e^{2 x}dx(9.)

3

1e ^{Ùt}

Ùtdt (10.)

1

1 + Ùxdx (11.)

ln( 2 t ) dt (12.)

x sin(w t) dt Part II. Calculators are allowed

- [13.]
Use the midpoint rule with n = 3 to approximate ø
_{-1}^{4}[1/( 1+x)] dx. Do the calculations by hand, write down details. - [14.]
The following table gives values of a function f, whose
concavity does not change in the interval [0,2]:

x 0.0 .25 .50 .75 1.00 1.25 1.50 1.75 2.00 f(x) 0.0 1.492 2.08 2.48 2.75 2.92 3.00 2.98 2.86

We want to estimate ø_{0}^{2}f(x) dx .a) Find the midpoint estimate with 4 subdivisions, MID(4).

b) You can easily calculate that LEFT(4) = 3.915 and RIGHT(4) = 5.345. Find the trapezoid and Simpson's estimates TRAP(4) and SIMP(4).

c) Is f concave up or down? Justify your answer.

d) Explain why your value of SIMP(4) must be within 0.35 of the exact value of the integral. (

*Hint: How far apart are your values for TRAP(4) and MID(4)?*) - [15.]
The numerical approximation of an integral ø
_{a}^{b}f(x) dx with Left, Right, Trap, and Mid, produced the numbers shown below (in some order): It is known that the function f(x) is decreasing and concave up. Choose a suitable method for each number. Give a reason for your choice.6.725, 6.745, 6.756, 6.800 - [16.]
The exact value of an integral ø
_{a}^{b}f(x) dx is 2.50. It is also known that Left(5) = 2.532431 and MID(5) = 2.502215.a) What is the error in each case.

b) Give a reasonable guess for the error corresponding to LEFT(50) and to MID(50).

c) How many decimals you predict will be correct when using MID(50) to calculate the integral?

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Say which of the following integrals are improper.
For those that are improper, determine if they are convergent
or divergent.
(18.)

Ñ

1x

_____

Ù1 + x^{6}

dx (19.)

5

-2x ^{2}

x+1dx (20.)

5

02

t^{2}+ 3 tdt (21.)

Ñ

0t e ^{-t}dt(22.)

Ñ

1t

t^{2}+ 1dt (23.)

1

0t ^{2}sint dt

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