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[answered] Math 140 Midterm Exam Professor: C Libis NAME _____________


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Math 140 Midterm Exam

 

Professor: Dr. C Libis NAME _____________________________ INSTRUCTIONS The exam is worth 100 points. There are 20 problems (each worth 5 points). This exam is open book and open notes with unlimited time. This means you may refer to

 

your text book, notes, and online classroom materials. You may take as much time as you

 

wish provided you submit your exam no later than the due date posted in our course

 

syllabus. To be fair to others, late exams will not be accepted. You must show your work to receive full credit. If you do not show your work, you

 

may earn only partial or no credit at the discretion of the instructor. Emailed exams cannot be accepted as they crash my system (thank you for your

 

cooperation and understanding on this one!) If you have any questions, please fee free to send me a message.

 

Best wishes! (1) _____ (11) _____ (2) _____ (12) _____ (3) _____ (13) _____ (4) _____ (14) _____ (5) _____ (15) _____ (6) _____ (16) _____ (7) _____ (17) _____ (8) _____ (18) _____ (9) _____ (19) _____ (10) _____ (20) _____ 1) ______________ Find a function f such that it is both even and odd. You must prove your

 

choice is correct in order to receive full credit.

 

2) _______

 

and Y(0,5).

 

(a)

 

- 2/5 Find the slope of

 

(b) - 5/2 (c) OM if M is the midpoint of XY with points O (0,0), X(2,0),

 

2/5 (d) 5/2 (e) none of the above 3) _______ If f(x) = ax - b and g(x) = cx - d, then when is ( f g )( x ) ( g f )( x ) ? You must

 

show your work in order to receive credit.

 

(a)

 

f(a) = g(c)

 

(b)

 

f(b) = g(d)

 

(c)

 

f(d) = g(b)

 

(d)

 

f(c) = g(d)

 

(e)

 

none of the above

 

4) _______ Find an equation in slope-intercept form (where possible) for the line with yintercept - 2 and perpendicular to x + 8y = - 9.

 

1

 

x 1 (b)

 

8 (a) y (c) y = - 8x - 2 (d) y = 8x - 2

 

1

 

y x 2

 

8 (e) none of the above 5) _______ The information in the chart gives the salary of a person for the stated years.

 

Model the data with a linear function using the points (1, 24800) and (3, 26600).

 

Year, x

 

1990, 0

 

1991, 1

 

1992, 2

 

1993, 3

 

1994, 4

 

Salary, y

 

$23,500

 

$24,800

 

$25,200

 

$26,600

 

$27,200

 

(a)

 

y = - 1,148x + 23,900

 

(b)

 

y = 28.4x + 23,900

 

(c)

 

y = 900x

 

(d)

 

y = 900x + 23,900

 

(e)

 

none of the above

 

6) _______ f ( x ) 9 and lim g ( x ) 8 . Find lim 3 f ( x ) 7 g ( x ) .

 

Let lim

 

x 8

 

x 8

 

x 8

 

9 g( x) (a) (b) -4 8 (c) 83/17 (d) - 29/17 (e) none of the above 7)

 

Use the properties of limits to help decide whether the limit exists. If the limit exists, find

 

its value.

 

lim x 16 (b) 4 (c) 1/4 (d) x 4

 

x 16 (a) 1/8 0 (e) 8) Find all values x = a where the function is discontinuous. none of the above 6x 6 if x 0

 

f ( x ) 2 x 6x 6 if x 0

 

(a) a = 6 (b) Nowhere (c) a = -6 (d) a = 0 (e) none of the above 9) Find the average rate of change for the function over the given interval.

 

y = x2 + 7x between x = 1 and x = 9 (a) 18 (b) 136/9 (c) 16 (d) 17 (e) none of the above 10) ______ Which statement is the converse of "If the sum of two angles is 180, then the

 

angles are supplementary"?

 

(a)

 

If two angles are supplementary, then their sum is 180.

 

(b)

 

If the sum of two angles is not 180, then the angles are not supplementary.

 

(c)

 

If two angles are not supplementary, then their sum is not 180.

 

(d)

 

If the sum of two angles is not 180, then the angles are supplementary.

 

(e)

 

none of the above.

 

1

 

and g ( x ) x 1 , determine the values of f g (3) and g f (3) .

 

x

 

2

 

2

 

1

 

f g (3) f g (3) and g f (3) and g f (3) 2

 

(b)

 

3

 

2

 

3

 

1

 

1

 

1

 

f g (3) 2 and g f (3) f g (3) and g f (3) (d)

 

3

 

2

 

2 11. ______

 

(a)

 

(c)

 

(e) none of the above 12) ______

 

(a) If f ( x ) 1

 

4 2h f (1 h ) f (1)

 

1

 

, then the expression

 

can be simplified to

 

x 1

 

h

 

h

 

1

 

1

 

(b)

 

(c)

 

(d)

 

(e)

 

none of the

 

h 1

 

2h 1

 

4 If f ( x ) above

 

13) ______ Let f ( x ) 2x3 x 2 2x 1

 

. Compute f(-1), f(0), and f(1). Which of the following

 

2x 1 statements is true?

 

(a)

 

The Intermediate Value Theorem implies that f(x) = 0 has a solution in [-1, 0].

 

(b)

 

The Intermediate Value Theorem implies that f(x) = 0 has a solution in [0, 1].

 

(c)

 

The Intermediate Value Theorem implies that f(x) = 0 has two solutions in [-1, 1].

 

(d)

 

The Intermediate Value Theorem does not apply since f is not continuous at x = 1/2.

 

(e)

 

none of the above.

 

14) ______ What conditions, when present, are sufficient to conclude that a function f(x) has

 

a limit as x approaches some value of a?

 

(a)

 

The limit of f(x) as x?a from the left exists, the limit of f(x) as x?a from the right

 

exists, and these two limits are the same.

 

(b)

 

Either the limit of f(x) as x?a from the left exists or the limit of f(x) as x?a from the

 

right exists.

 

(c)

 

The limit of f(x) as x?a from the left exists, the limit of f(x) as x?a from the right

 

exists, and at least one of these limits is the same as f(a).

 

(d)

 

f(a) exists, the limit of f(x) as x?a from the left exists, and the limit of f(x) as x?a from

 

the right exists.

 

(e)

 

none of the above. 15) ______ f ( x) L

 

Select the correct statement for the definition of the limit: lim

 

x t (a)

 

if given any number ? > 0, there exists a number ? > 0, such that for all x,

 

0 < |x - t| < ? implies |f(x) - L| < ?.

 

(b)

 

if given any number ? > 0, there exists a number ? > 0, such that for all x,

 

0 < |x - t| < ? implies |f(x) - L| < ?.

 

(c)

 

if given any number ? > 0, there exists a number ? > 0, such that for all x,

 

0 < |x - t| < ? implies |f(x) - L| > ?.

 

(d)

 

if given a number ? > 0, there exists a number ? > 0, such that for all x,

 

0 < |x - t| < ? implies |f(x) - L| > ?.

 

(e)

 

none of the above.

 

16) ______ A function f(x), a point c, the limit of f(x) as x approaches c, and a positive

 

number ? are given. Find a number ? > 0 such that for all x, 0 < |x - c| < ? ? |f(x) - L| < ?.

 

f(x) = -2x - 6, L = -12, c = 3, and ? = 0.01

 

(a)

 

0.005 (b)

 

0.01 (c)

 

0.0025

 

(d)

 

-0.003333

 

(e) none of the above

 

17) ______ The statement "the limit of a constant times a function is the constant times the

 

limit" follows from a combination of two fundamental limit principles. What are they?

 

(a)

 

The limit of a product is the product of the limits, and a constant is continuous.

 

(b)

 

The limit of a constant is the constant, and the limit of a product is the product of the

 

limits.

 

(c)

 

The limit of a function is a constant times a limit, and the limit of a constant is the

 

constant.

 

(d)

 

The limit of a product is the product of the limits, and the limit of a quotient is the

 

quotient of the limits.

 

(e)

 

none of the above.

 

18) ______

 

(a)

 

1/ e2 The slope of the line tangent to the graph of y = lnx2 at x = e2 is

 

(b)

 

2/ e2 (c)

 

4/ e2 (d)

 

1/ e4 (e)

 

4/ e4 19) ______

 

(a)

 

0 If F(x) = x sin x, then find F ' (3/2).

 

(b)

 

1

 

(c)

 

-1

 

(d)

 

3/2 20) ______ If H ( x ) x 3 x 2 , which of the following is H ''(2)? (a) (b) 31/4 (e) - 3/2 1

 

x 39/4 (c) 79/8 (d) 81/8 (e) 41/4

 


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