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( x-3).

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(x2) is
(a) 1 + 2x h'(x2) (b) h'(x2) 2x(c) 2x + xh'(x2)(d) xh(x2) + x2 h'(x)(e) h(x2) + 2x2 h'(x) (f) none of these.

(f)
5#5 The derivative of f(x) = x2 + x3 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)
x10 + x1/2
(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).