M161, Midterm 1, Fall 2010
Problem Points Score
1 20
Name:
2 15
Section:
Instructor:
Time: 75 minutes. You may not use calculators or other electronic devices on this exam
3
4
5
X
20
25
20
100 d sin( x ) = cos( x ) , dx d dx asin( x ) =
√
1
1
− x
2
, d dx acsc( x ) = sin(2 tan
2 x
−
) = 2 sin(
1 x
√ x
2 x ) cos(
( x ) + 1 = sec
2
( x )
− x )
1
, d cos( x ) = dx d dx acos( x ) = d dx
Z asec( x ) = ln xdx cos
2
( x ) =
=
− sin( x ) ,
− x
√
1 x
2 x ln x
−
1
√
1
−
− x
1 + cos(2 x )
2 x
2
1
,
+
,
C d tan( x ) = sec
2
( x ) , dx d dx atan( x ) =
1
1 + x
2
, d dx
Z sec( x ) = sec( x ) tan( x ) , sec( x ) dx = ln
 sec( x ) + tan( x )

+ C sin
2
( x ) =
1
− cos(2 x )
2
Theorem (The Derivative Rule for Inverses) If exists and is never zero on I , then f
−
1 f has an interval I as domain and f
0
( x ) is differentiable at every point in its domain. The value of ( f point a =
− 1 f
−
1
)
0
( b at a point
): ( f
−
1
)
0 b in the domain of
1
( b ) = f
0
( f
−
1
( b ))
.
f
− 1 is the reciprocal of the value of f
0 at the
1. These are short answer questions. Put your answer in the box. No work outside the box will be graded.
(a) What does asin(
√
3 / 2) equal?
(b) Suppose that y = g ( x ) is a differentiable function that has an inverse. Suppose that the graph of g ( x ) passes through the origin with slope 2. What is the slope of the inverse function g
−
1
( x ) at the origin?
(c) What is the derivative of ln(5x)
− ln(7x)?
(d) What is the domain of the function atan(3 x )?
2. The first list contains 7 statements (5 correct and 2 false). The second list contains 5 reasons (labeled A
−
E ) why the correct statements are true. In the box next to each false statement, write the letter F . Match each correct statement with the reason why it is true, and write the letter in the box.
Statements: ln(e x
) = x.
ln(e) = 1.
d dx
(ln(x)) =
1 x
.
ln(x) is one
− to
− one.
ln(
1 x
) =
1 ln(x)
.
ln(x) =
Z
1 x dt
.
t ln(x) is concave up.
Reasons:
(A) by the Definition of the Natural Logarithm Function.
(B) by the Definition of the Number e .
(C) by the Fundamental Theorem of Calculus.
(D) because 1 /x is positive for all x > 0.
(E) by the Definition of the Natural Exponential Function.
3. Evaluate the following integrals. Show all your work.
(a)
Z sin
3
( x )
· cos
2
( x ) dx .
(b)
Z
√
6 x dx
−
3 x
2
.
4. Let f ( x ) = e
2 x
−
2
(a) State the domain and range of f ( x ).
(b) Explain why f ( x ) has an inverse, without computing what the inverse is.
(c) Find a formula for f
−
1
( x ), and state the domain and range of f
−
1
( x )
(d) Graph f ( x ) and f
−
1
( x ).
5. The following differential equation fulfills the initial value condition y (1) =
−
1 / 3: e
− x
3
+1 dy dx
= 3 x
2 y
2
.
(a) Show that the differential equation is separable.
(b) Solve the differential equation. In other words, solve for y an integration constant C .
as a function of x and
(c) Determine the integration constant C that fulfills the condition y (1) =
−
1 / 3.