1220-90 Exam 1
Summer 2014
Name
Instructions.
Show all work and include appropriate explanations when space is provided. Correct
answers unaccompanied by work may not receive full credit. Please circle your final answer. The last page
contains some useful identities.
1. (15 pts) Find the indicated derivatives. No riced to simplify.
(a) (5pts) D(ln (tan))
J
.ec.
—
(b) (5pts)
Dx(eshl (7x))
p’t
(7y)
c-cS (7
.
7
(c) (5pts) Ds(xsx)
e
2. (lOpts) Find the indicated antiderivatives. Pemeher +C!!
(a) (5pts)fexcosh(ex) dx
Js /)
SLJi
tt
()
SlL()
-
(b) (5pts)
12
dx
J
J
-
1
C
3. (4pts) Suppose the function
f
is one-to-one. Listed below are a few values of
f
and its derivative:
x f(x) f’(a’)
For example, f(0)
(f’)’(2)
=
1, f’(l)
0
1
1
2
3
4
2
4
5
4, etc. Fill in the blanks:
-J
4- Ci)
=
4. (4pts) Find a formula for the inverse of the function
-1y+-S
-
-(ji)
f(x)
=
2x ± 5, x > —1.
jx—+
(J
+.
5. (l0pts) Evaluate the following. Any answer representing an angle should be given in radians.
(a) sinl(4)
‘L
(b) cos’(O)
=
=
(c) sin (sin’
(i))
=
2.. (d) sin’ (sin (37r))
2—
(e) cos(sin’
—
(i))
6. (l0pts) A 100 gram sample of a radioactive isotope was found to only contain 73 grams of radioactive
material after 17 days. Let A(t) denote the amount of radioactive material (in grams) after t days. So
A(0) = 100 and A(17) = 73.
(a) (6pts) Find constants C and k such that A(t)
3
=
3
k-
=
Cekt. No need to simplify.
loo
4LL)
73
(ii)
Qi)
-
(b) (1pts) Find the half-life of the isotope; that is, the amount of time (in days) that it takes for half
of the isotope to decay. No need to simplify.
, i13,
-
‘DO
1.
13/)
_
7. (25pts) Evaluate the following indefinite integrals. Remember -i-C!!
(a) fcos3xsin2x dx
—
&L6’5LI1)L
tf
3
cc
LX
t
C
x
3
(b) fxe3x dx
L
y
4
xe
—_—
(c)
f
sin x
dx
—
Hint: Use integration by parts with u
sin’ x and dv
=
_
xs’
ç
xst
)<
aCLC
Jsecxtan3x
dx
-2
-7.-
4-
—
xs
(d)
dx.
.
C-
X
X
Je
1
) CL
t’-€
e.
Jft)u
(e)
f
3
3
—C
S—Sc t C.
-
2 x sin
cos
2 x dx
4J(i co(z)) (1
-c(2))
f(i
os
-
4fC±
(i
co5()
8. (l2pts) Evaluate the following indefinite integral usin rationalizing substitution in three steps.
(a) (4pts) First, write as an integral with respect tot by using the rationalizing substitutionx
=
taut.
X&ZLI
I
cAi:
‘c- ::
(b) (4pts) Second, evaluate the integral from part (a). You answer should be, in terms of t. Hint:
and sect =
‘Write the integrand in terms of sint and cost using the identities tant =
(1-
j
LL
J
(3)
( )
L5L-
j
—
fL
-
-
x
(_-ct
-
x
-
(&-C
>
9. (1Opts Use partial fractions to find
f
—
—
10
dx.
7
2-
x=i
77k
_3
X-
/0
S
-7
1L
_L
3
(c) (4pts) Finally, write you answer to part (b) in terms of x.
-—
-
_J—\
J
4
¶;,t