Problem
1
Points
20
2
26
3
32
4ab
20
4cd
20
5
8
6
30
7
14
8
20
9
10
Total
200
M161, Final Exam, Fall 2006
Score
NAME:
SECTION:
INSTRUCTOR:
You may not use calculators.
Taylor series of the function f (x) about x = a.
∞
f (a)
f (n) (a)
f (a)
(x − a) +
(x − a)2 + · · · =
(x − a)n
f (a) +
1!
2!
n!
n=0
Taylor polynomial plus error term
f (x) = Tn (x) + Rn (x) =
n
1
j=0
j!
f (j) (a)(x − a)j + (−1)n
1
n!
x
(t − x)n f (n+1) (t) dt
a
Taylor Inequality: If M is a constant such that |f (n+1) (x)| ≤ M for a ≤ x ≤ b, then |Rn (x)| ≤
β
r2 +
Length of a polar Curve: L =
lim
n→∞
ln(n)
= 0,
n
lim
n→∞
√
n
α
n = 1,
dr
dθ
2
dθ
lim x1/n = 1 any x, lim
n→∞
n→∞
Sum of a geometric series: a + ar + ar 2 + · · · + ar n = a
θ
sin =
2
1 − cos θ
2
1+
x
n
n
= ex ,
lim
n→∞
M |x−a|n+1
.
(n+1)!
xn
= 0, any x.
n!
n+1
1−r
1−r
θ
cos =
2
1 + cos θ
2
Error Bounds. Suppose |f (x)| ≤ K for a ≤ x ≤ b. If ET is the error in the Trapezoidal Rule, then
|ET | ≤
K(b − a)3
KΔx2 (b − a)
.
or it may be written as |ET | ≤
2
12n
12
Suppose |f (4) (x)| ≤ K for a ≤ x ≤ b. If ES is the error involved in using Simpson’s Rules, then
|ES | ≤
K(b − a)5
KΔx4 (b − a)
or it may be written as |ES | ≤
.
4
180n
180
cos2 θ =
1 + cos 2θ
2
sin2 θ =
1 − cos 2θ
2
1. (a) Find the 3rd degree Taylor polynomial, T3 (x), that approximates the function f (x) =
ln x about the point a = 1. Show all of your work.
(b) Use the Taylor Inequality to estimate the accuracy of the approximation f (x) ≈ T3 (x)
when x satisfies .9 ≤ x ≤ 1.1.
√
2. Consider the function f (x) = x − 2, x ≥ 2. Answer the following questions, showing all
of your work.
(a) State the domain and range of f .
(b) Without explicitly finding f −1 , explain why f −1 exists. State the domain and range of
f −1 .
(c) Let g = f −1 . Compute g (4). Note that 4 = f (18).
3. Answer the following questions showing your work.
(a) Write the complex number 1 − i in polar form (in the form reiθ , r > 0 and −π ≤ θ ≤ π).
(b) Express eπi in the form a + bi, a and b real.
(c) Simplify
√1
2
+ i √12
60
. (Write as a + bi, a and b real.)
(d) Compute the cube roots of 8. You may leave your answer in polar form.
4. Calculate
the following integrals. You must show your work.
(a) x ln(2x) dx
(b)
1
dx
+6
2x2
(c)
1
0
(d)
√
1
dx
4 − x2
1
dx
+ 4)
x(x2
5. Match the Taylor series with the functions. Place the number (Roman numeral) of
the appropriate series in the center column, to the right of the given function.
A. e−3x
(i)
∞
(−1)n 3n xn
n=0
B.
1
1 − 3x
3x
C. e
(ii)
∞
3n
n!
n=0
(iii)
∞
xn
3n xn
n=0
D.
1
1 + 3x
(iv)
∞
(−1)n
n=0
3n xn
n!
6. Determine whether the series is convergent or divergent. In any case you must justify your
answer and show complete work.
∞
n
(a)
2
n=3 n − n − 4
(b)
(c)
∞
ln(14n)
2
n=1 ln((2n + 1) )
∞
(−1)n
n=1
n9n
10n
7. For what values of x does the power series
work.
∞
(3x)n
converge. You must show all of your
n=0 n + 3
8. Below are the plots of r = 1 + cos(2θ), 0 ≤ θ ≤ 2π and θ = π/6.
2
θ=π/6
1.5
r=1+cos(2θ)
1
y
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
2.5
(a) Find the points of intersection of the two curves given in the plot above.
(b) Find the area inside the curve r = 1 + cos(2θ) and above the curve θ = π/6.
4
1
dx.
2 (x − 1)2
(a) Evaluate the integral given above.
9. Consider the integral
(b) Find an upper bound on the error incurred in estimating the above integral with the
Trapezoidal Rule with n = 16 steps.