Another Midterm Preparation Exam

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Math 115 Midterm Review (Spring 2011). Name: Instructor: KUID: ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ. This exam consists of two parts.
Math 115 Midterm Review (Spring 2011)

Name: Instructor: KUID: ——————————————————————————————————————— This exam consists of two parts. Part I consists of multiple choice questions. Part II consists of true or false questions. The three blank pages in the back are scratch paper. Warning: There will be 16 multiple choice questions and 9 true or false questions in the midterm, even though in this review the numbers are di¤erent. ——————————————————————————————————————— Part I. For the questions in Part I, circle clearly the answer that is correct, in case of an approximation, circle an answer that is most nearly correct. Each problem is worth ten (10) points. p 1 x is (1) The domain of the function f (x) = 2 x 4 a) ( 1; 2) [ ( 2; 1] b) ( 1; 2) [ ( 2; 1) c) ( 1; 2] [ ( 2; 1) d) ( 1; 1) e) ( 2; 1]

p 3

(2) Let f (x)= x2 1 ; g(x) = 3x3 + 1: Evaluate h(2) where h = g f: p p p c) 10 d) 3 3 e) 3 7 a) 3 224 b) 9

2

x (3) Find lim 2x4 2 +x 3 x! 2

a)

0 0

b) 1

c) 1

d) 2

e) N one of the above

(4) Let f (x) =

x2 + 2 kx2

for x 1 for x > 1

Find the value of k that will make f continuous on ( 1; 1): 1

a) 2

b) 1

c) 0

3

d) 3

e) N o value of k will work

2

+x +1 (5) Find lim 3x 4 x! 1 3x +x+1

a) 0

b) 1

(6) Find lim

p 3

x!0

a)

2 3

b)

1+x x

c)

p 3

1

d) 1

e) N one of the previous

1

1 p 3 3

c) does not exist

d) 0

e)

1 3

(7) The pilot of a coast guard patrol aircraft on a search mission had just spotted a disabled …shing trawler and decided to go in for a closer look. Flying in a straight line at a constant altitude of 1000f t and at a steady speed of 264f t= sec; the aircraft passed directly over the trawler. How fast was the aircraft receding from the trawler when it was 1500f t from it? a) 196:8 f t= sec

b) 200 f t= sec

c) 264 f t= sec

d)

p

125 264 1500

f t= sec

e) N one of the above

(8) Let f (x)=

8 2 < x :

2x + 3 if x < 1 0 if x = 1 3 x if x > 1

a) lim f (x) does not exist

b) lim f (x) = 0 and lim+ f (x) = 0

c) lim f (x) = 2 and lim+ f (x) = 2

d) lim f (x) = 3 and lim+ f (x) = 3

x!1

x!1

x!1

x!1

x!1

x!1

x!1

e) N one of the above

(9) A division of Krypton Industries manufactures the Superman model microwave oven. The daily cost (in dollars) of producing these microwave ovens is C(x) = 0:0002x3

0:06x2 + 120x + 5000

where x stands for the number of units produced. What is the marginal cost to produce 300 units? 2

a) $138

b) $120

c) $114

d) $400

e) N one of the previous answers

(10) A hot air balloon rises vertically from the ground so that its height after t sec is h = 21 t2 + 12 t f t (0 t 60): What is the velocity of the balloon at the end of 40 sec? a)

80 2

f t= sec

b)

81 3

f t= sec

c) 820 f t= sec

d)

81 2

f t= sec

e) N one of the above

(11) Find the equation of the tangent line to the curve f (x) = x(2x + 1)3 at the point (1; 27): a) y = 81x

27

b) y = 54x

27

c) y = 81x

54

d) y = 81x

27

e) N one of the previous answers

(12) Suppose F (x) = g(f (x)): Find F 0 (2) given that f (2) = 3; f 0 (2) = g(3) = 5 and g 0 (3) = 4: a) 4

b) 12

c)

15

d)

12

3;

e) 5

(13) What is the slope of the tangent line to the curve 4x2 + 9y 2 = 36 at (0; 2): a) x = 2

b) y =

4 x 9

+2

c) y = 2x + 1

d) y = 2

e) y = 4

(14) The distance s (in feet) covered by a car after t sec is given by s=

t3 + 8t2 + 20t

(0

t

6)

The car after 2 32 sec a) is accelerating b) comes to rest c) starts to decelerate d) has zero velocity e) N one of the previous answers

(15) Let y = f (x) = 3x2 from 2 to 1:97:

2x + 6: Find the approximate change in y if x changes

3

a) 0:3

b)

0:03

c)

3

d)

0:3

e) 30

Part II. For each question in Part II, circle the correct answer. Each problem is worth …ve (5) points. (17) If lim f (x)exists, then f is continuous at x = a: x!a

(a) True

(b) False

(18) If f is continuous for all x 6= 0 and f (0) = 0; then lim f (x)= 0. x!a

(a) True

(b) False

(19) The function f (x) = jx + 1j is not di¤erentiable at x = (a) True

2:

(b) False

(20) If the second derivative of f exists at x = a; then f 00 (a) = [f 0 (a)]2 (a) True

(b) False

(21) If f and g are di¤erentiable functions then (f (x) + g(x))0 = f 0 (x) g 0 (x)? (a) True (b) False p (22) If h(x) = h( 3x); then h00 (x) = 3h00 (x): (a) True

(b) False

(23) Every di¤erentiable function is continuous. (a) True

(b) False

4