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SAMPLE MIDTERM 1, MAT 141 10/11/99
 1.
 The graph of a different the function f is given in
each of the figures below. For each graph
sketch the corresponding function g indicated below on the same axes.
For A,
g(x) = f(x)2.
For B,
g(x) = f( x3).
For C,
g(x) =f(x).
1#1
2#2
3#3
2#2
4#4
 2.
 Place the letter corresponding to the correct answer in the box
next to each question.
 (a)

5#5
The equation of the line passing through (0,2) and (3,1) is
(a)
6#6(b)
7#7(c)
8#8
(d)
9#9(e)
10#10
(f) none of these.
 (b)

5#5Suppose f and g are given by the following tables. What is f(g(2))?
x 
0 
1 
2 
3 
4 
f(x) 
2 
3 
1 
2 
4 
g(x) 
1 
3 
2 
4 
0 
(a) 0
(b) 1(c) 2(d) 3(e) 4(f) it is undefined.
 (c)

5#5Suppose that for all B>0 there is a C>0 so that x > C implies
f(x) > B. Then
(a)
11#11(b)
12#12(c)
13#13(d)
14#14
(e)
15#15.
(f) none of these.
 (d)

5#5Consider the right triangle on the left below. What is
16#16?
(a)
17#17(b)
18#18(c)
19#19(d)
20#20(e)
21#21(f) none of these.
22#22
 (e)

5#5The derivative of xh(x^{2}) is
(a)
1 + 2x h'(x^{2})
(b)
h'(x^{2}) 2x(c)
2x + xh'(x^{2})(d)
xh(x^{2}) + x^{2} h'(x)(e)
h(x^{2}) + 2x^{2} h'(x)
(f) none of these.
 (f)

5#5
The derivative of
f(x) = x^{2} + x^{3} at x= 2 is
(a) 12(b) 13(c) 14(d) 15(e) 16(f) none of these.
 (g)

5#5The natural domain of
23#23is
(a) all real numbers
(b) x> 0(c) x< 5(d)
24#24
or 0 < x(e) 25#25
or x> 5(f) none of these.
 (h)

5#5Suppose
f(1) = 3.4 and
f(1.1) = 3.6. Then the best estimate for
f'(1) is
(a) 3.5
(b) 3.4
(c) 2.0
(d) 20
(e) .2
(f) .002
 (i)

5#5A ball dropped from rest takes 3 seconds to hit the ground. From what
height was it droped (in feet)?
(a) 48
(b) 90
(c) 144
(d) 256
(e) 288
(f) none of these
 (j)

5#5What is the limit of
26#26
as
27#27?
(a) 0(b) 28#28(c) 1 (d) 2(e) 29#29(f) the limit fails to exist
 3.
 For each of the following functions, find the derivative function.
 (a)

x^{10} + x^{1/2}
 (b)
 30#30
 (c)

31#31
 (d)
 32#32
 (e)

33#33
 4.
 Prove by induction that
34#34.
 5.
 What are the following limits (you do not need to
justify your answer),
35#35
Using these, the definition of derivative and addition law for cosines,
36#36
prove that
37#37.
 6.
 Suppose f satisfies the following two conditions for all real
values of x and y.
 (a)

f(x+y) = f(x) f(y)
 (b)

f(x) = 1 + x g(x) where
38#38.
Show that f is differentiable at every point and that
f'(x) = f(x).
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Chris Bishop
19991011