M161, Midterm 1, Fall 2010 Problem Points Score 1 20

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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.

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