Problem Points Sore
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M161, Test 3, Spring 2004
NAME:
SECTION:
INSTRUCTOR:
You may use alulators. No formulas an be
stored in your alulators.
Total
100
Error Bounds. Suppose jf 00(x)j K for a x b. If ET is the error in the Trapezoidal Rule, then
b a)
jET j K (12
or it may be written as jET j K x12(b a) :
n
3
2
2
Suppose jf (x)j K for a x b. If ES is the error involved in using Simpson's Rules, then
(b a) or it may be written as jES j K x (b a) :
jES j K180
n
180
(4)
5
4
4
1
1. (a) Sketh the urve of the polar equation r = 4 sin(2).
2
0
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
−4
−3
−2
−1
0
x
1
2
3
(b) Find the area enlosed in one loop of the lemnisate r = 4 sin(2).
2
2
4
5
2. (a) Sketh the urve of the parametri equations x = t(t 3), y = 3(t 3), 2 t 2 and indiate with
an arrow the diretion in whih the urve is traed as the parameter t inreases.
2
2
4
2
0
−2
−4
−6
−8
−10
−3
−2
−1
0
x
1
2
3
(b) Find the points on the urve given in part (a) where the tangent line is horizontal or vertial|and draw
these tangents on your plot of the urve.
() Set up the integral to determine the length of the urve given in part (a).
3
3. (a) Find the Cartesian equation form of the polar urve r os() + r sin() = 1.
(b) Find the polar equation for the Cartesian equation x + (y 3) = 9. Write your result in the form
r = f (), i.e. solve for r.
2
4
2
Z
4. (a) Give an upper bound for the error when the trapezoidal rule is used to approximate the integral
1 dx with n = 4, 20 and 40.
1
3
x2
Error Bound n=4 Error Bound n=20 Error Bound n=40
Whih of the above n's will approximate the integral with an error of less than 10 ?
3
Z
(b) Give an upper bound for the error when Simpson's rule is used to approximate the integral
with n = 4, 20 and 40.
Error Bound n=4 Error Bound n=20 Error Bound n=40
Whih of the above n's will approximate the integral with an error of less than 10 ?
3
5
3
1
1
x2
dx
5. Derive the trapezoidal rule formula for approximating the integral
plot of f given below that illustrates your derivation.
y
y=f(x)
a
b
x
6
Z
b
a
( ) . Inlude a drawing on the
f x dx
6. For what values of x does the power series
7. Find the Taylor series of f (x) = x
3
X1 ( 1)
n
n=1
2x + 4 at a = 2.
7
1
x2n
n
2
1
1 onverge.
8. (a) Find the Taylor series expansion for f (x) = ln(x) at a = 4. Write the result using summation notation.
(b) Find the rst four terms of the Taylor series expansion for f (x) = x ln(x) at a = 4.
2
8
9. Assume that the funtion y an be written as a power series
y = a +a x +a x +a x +a x + Use this series to nd a solution to the initial value problem (dierential equation)
y y 0 = 3 with y (0) = 4:
Write your solution for y as a power series and then determine the funtion that is represented by the series
(you should be able to see whih funtion it is by the form of the power series).
0
1
2
2
9
3
3
4
4