1220-90 Exam 1
Name
Fall 2013
K Y
Show all work and include appropriate explanations when space is provided. Correct
Instructions.
answers unaccompanied by work may not receive full credit. Please circle your final answer. The last page
contains some useful identities.
1. (25pts) Find the indicated derivatives. No need to simplify.
(a) (5pts) D(cosh
3
S
Ksk2x(wk)
] (e!)
(b) (5pts) D(e)
e
(ln (x + 5))
1
(c) (5pts) D
3?2
-c)(
4
3
(d) (5pts) D(sin’ (3x + 5))
-
3
5-
—
]
(e) (5pts) D(xco)
(
.et4x
co5X
4AX
StL4X
I
tx
-
1
.--
25
2. (8pts) Let
f(x)
(a) (3pts) Why can we conclude that
f(x)
+ 2x + 1.
=
has an inverse? Hint: Conclude something from
-vvthL.f
(b) (2pts) Fill in the blank: f’(1)
14
aIu)j45
0
=
posthi’-
1 C)
j
f’(x).
f
s
i)
(c) (3pts) Compute (f—’)’(l), i.e. compute the derivative of the inverse function to
f evaluated
at 1.
_j_,
‘1t)
‘())
31+1
3. (l0pts) Evaluate the following. Any answer representing an angle should be given in radians.
(a) cos’()
=
0
(b) sin’(O)
(c) sin (sin’ (1))
(d) cos(siif
1
(i))
(e) sin(tan’
(i))
=
=
=
I
—
41’
/7
-
4. (lopts) A particular radioactive isotope decays from 10 grains to 7 grams in 32 hours. Let A(t) denote
the amount (in grams) of the isotope present after t hours.
(a) (Gpts) Find constants C and k such that A(t)
ii)
3
=
Cekt.
7 )
3
01:-
10—A(o)t
XC-.
=
7
k-(3’-)
32*
(b) (4pts) What is the half-life of this isotope? In other words, how long will it take before half of
the isotope has decayed? No need to simplify.
J
L/.
£,C
-
32J-(Yz-)
2
)
j T
5. (25pts) Evaluate the following indefinite integrs. Remember +C!!
(a) fsin3 x dx
N
-
ItX
=
cX
tLU
—sn X
-
=
(b) Jxe3x dx
I
=
JJ_
x
otV
(c)
3X
Db V
f
1+:e2x
A-
3
dx
L
3
3e
J
1
3X(
(d) fsin2xcos2x dx
C’ 4
(e) fxln(2x) dx
-
x tX
3
$
6. (l2pts) Evaluate the following indefinite integral using rationalizing substitution in three steps.
1
f
1
J 2
/i
x
+
(a) (4pts) First, write as an integral with respect to t by
di
using
=
1 Lft
2
çec-
z.
IC
the_rationalizing substitution x
q
=
tan t.
=
Sec-•L €1
(1)) (4pts) Second, evaluate the integral from part (a). You answer should he in terms oft.
1-
I
—
•t
S/t
-
(c) (1pts) Finally, write you answer to part (b) in terms of x.
Lf
,.
j
7. (lOpts) Use partial fractions to find
f
+2
—
dx
==)
—
-
2..A(x-J+
B1x-).
2-L#q
Ve
(1x-1 -ilIx-t)
4
zz