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
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5
X
20
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20
100 d sin( x ) = cos( x ) , dx d dx asin( x ) = √
1
1 − x 2 d dx sin(2 tan
, acsc( x ) = −
1 x
√ x 2 − 1
2 x ) = 2 sin( x ) cos(
( x ) + 1 = sec 2 ( x ) x )
, d d cos( x ) = − sin( x ) , dx d dx acos( x ) = −
1
√
1 − x 2
, dx
Z asec( x ) = x
√ x 2 − 1
, ln xdx = x ln x − x + C cos 2 ( x ) =
1 + cos(2 x )
2
1 d tan( x ) = sec
2
( x ) , dx d dx atan( x ) =
1
1 + x 2
, d sec( x ) = sec( x ) tan( x ) , dx
Z sec( x ) dx = ln | sec( x ) + tan( x ) | + C sin
2
( x ) =
1 − cos(2 x )
2
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 value of ( f point a = f
− 1
)
0
− 1 at a point b in the domain of f
− 1
( b ): ( f
− 1
)
0
( b ) = f 0 ( f
1
− 1 ( b ))
.
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 x
1 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 dx
√
6 x − 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 as a function of x and an integration constant C .
(c) Determine the integration constant C that fulfills the condition y (1) = − 1 / 3.