Problem Points Score
1
25
2ab
30
circle where you took M160 (or equivalent):
I Please
CSU AP Community College other College
2c
15
Name:
3
30
4
20
5
30
6
30
7
20
X
200
M161, Final, Fall 2007
Section:
Instructor:
Time: 110 minutes. You may not use calculators on this
exam
d
sin(x) = cos(x),
dx
d
csc(x) = − csc(x) cot(x),
dx
d
sinh(x) = cosh(x),
dx
d
1
asin(x) = √
,
dx
1 − x2
d
1
,
acsc(x) = − √
dx
x x2 − 1
d
1
asinh(x) = √
,
dx
1 + x2
sin(2x) = 2 sin(x) cos(x)
d
cos(x) = − sin(x),
dx
d
sec(x) = sec(x) tan(x),
dx
d
cosh(x) = sinh(x),
dx
d
1
acos(x) = − √
,
dx
1 − x2
d
1
,
asec(x) = √
dx
x x2 − 1
d
1
acosh(x) = √
,
2
x −1
Zdx
ln xdx = x ln x − x + C
tan2 (x) + 1 = sec2 (x)
cos2 (x) =
1 + cos(2x)
2
Taylor series of the function f (x) about x = a:
d
tan(x) = sec2 (x),
dx
d
cot(x) = − csc2 (x),
dx
d
tanh(x) = sech2 (x),
dx
d
1
atan(x) =
,
dx
1 + x2
d
1
acot(x) = −
dx
1 + x2
d
1
atanh(x) =
,
1 − x2
Zdx
sec(x)dx = ln(sec(x) + tan(x)) + C
sin2 (x) =
1 − cos(2x)
2
∞
f (a) +
X f (n) (a)
f 00 (a)
f 0 (a)
(x − a) +
(x − a)2 + · · · =
(x − a)n
1!
2!
n!
n=0
Theorem (The Derivative Rule for Inverses) If f has an interval I as domain and f 0 (x)
exists and is never zero on I, then f −1 is differentiable at every point in its domain. The
1
value of (f −1 )0 at a point b = f (a) in the domain of f −1 is given by (f −1 )0 (b) = 0 .
f (a)
1) a) Sketch the polar curves r = 2(1 + cos(θ)) and r = 2(1 − cos(θ)) in the following
diagram:
5
4
3
2
1
0
−5
−4
−3
−2
−1
−1
0
1
2
3
4
5
−2
−3
−4
−5
b) Determine the area enclosed by the two curves in a) (i.e. the overlap of the insides of
both curves).
2) Z Evaluate the following integrals. Show your work
x
√
dx (You may assume that x > 2.)
a)
x−2
Z
b)
1 · ln2 (x)dx (Hint: integrate by parts with 1 as one of the factors).
Z √
c)
x2 − 9
dx
x4
3) Determine, for example by appropriate manipulation of known power series or by calculating the Taylor series about a = 0, power series for the following functions. Your solution
should make
Z clear what steps were performed with series of what functions.
x
2
2
e−t dt (i.e. f (x) is an antiderivative of e−x .)
a) f (x) =
0
b) √
lem.)
1
(Giving the Taylor polynomial of degree 4 will give you full credit for this prob1+x
4) a) Determine a power series for
∞
X
1
in the form
an .
x+5
n=0
b) Determine the end points of the interval of convergence of the power series determined
in a) (i.e. find two points a, b that the power series converges for all a < x < b but does not
converge for x < a or x > b).
5) (You way write complex numbers in cartesian (a + bi) or polar (r · eiφ ) form – either
way is accepted.)
a) Determine all complex numbers z such that z 6 = −1.
b) Determine a complex number z such that ez = i − 1.
6) a) Solve the following initial value problem using power series.
y 0 (x) + y(x) = x,
y(0) = 0
b) Recognize the resulting power series as a function.
7) For each of the following improper integrals determine, with explanation, whether the
integral converges or diverges. (You do not need to calculate the values of convergent integrals.)Z
∞ −x
e
√ dx
a)
x
1
Z
∞
ln(ln(x))dx
b)
ee