Math 120 Sample questions for Test 1, with answers

Math 120
Sample questions for Test 1, with answers
Not all questions will necessarily be like the ones on the following pages,
but they do give a pretty good idea of the types of questions to expect.
Answers (but not complete solutions) follow the set of questions for each
of the two chapters.
In the sample questions for Chapter 2, do not consider #17.
Also, study the three quizzes for which you were provided detailed
solutions. There will be at least one Test 1 question similar to a question on
each quiz, i.e., at least one question like a Quiz 1 question, at least one like
a Quiz 2 question, etc.
Stewart - Calculus ET 6e Chapter 1 Form A
1.
If f ( x) = x 2 − 2 x + 3 , evaluate the difference quotient
2.
Find the domain of the function.
f ( x) =
3.
f ( a + h) − f ( a )
.
h
5
3
x
( − 1)
Find the range of the function.
h( x) = ln ( x + 6 )
4.
5.
Find an expression for the function y = f (x) whose graph is the bottom half of the parabola x + (8 − y ) 2 = 0 .
4
A spherical balloon with radius r inches has volume π r 3 . Find a function that represents the amount
3
of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches.
6.
An open rectangular box with volume 3 m 3 has a square base. Express the surface area of the box as a
function S (x) of the length x of a side of the base.
7.
Determine whether f is even, odd, or neither even nor odd.
f ( x) = x3 − x5
8.
In the function f ( x) = 3x + d , what is the value of d, if f (6) = 1 ?
9.
Suppose that the graph of f is given. Describe how the graph of the function y = f ( x − 3) − 3 can be
obtained from the graph of f .
10. Use transformations to sketch the graph of the function.
y = − sin 2 x
11. If the point (7, 3) is on the graph of an even function, what other point must also be on the graph?
12. Use the table to evaluate the expression ( f D g )(3) .
x
f (x )
g (x)
1
3
6
2
2
5
3
1
2
4
0
3
5
1
4
6
2
6
13. Use the functions below to find a function g such that g D f = h .
If f ( x) = x + 3 and h( x) = 4 x − 4 ,
Stewart - Calculus ET 6e Chapter 1 Form A
14. Jason leaves Detroit at 3:00 P.M. and drives at a constant speed west along I-90. He passes Ann Arbor,
40 mi from Detroit, at 3:30 P.M. The graph of the function of the distance traveled (in miles) in terms
of the time elapsed (in hours) is given below. Find the slope of the function.
15. The monthly cost of driving a car depends on the number of miles driven. Samantha found that in
October it cost her $312.5 to drive 500 mi and in February it cost her $375 to drive 1,000 mi.
Express the monthly cost C as a function of the distance driven d assuming that a linear relationship
gives a suitable model.
16. Find the inverse function of f ( x) =
x +1
.
2x +1
17. If f ( x) = 3 x + ln x , find f −1 (3) .
18. Find the exponential function f ( x ) = Ca x whose graph is given.
19. Find the exact value of the expression.
1·
§
tan ¨ arcsin ¸
2¹
©
20. Solve each equation for x.
(a) ln x = 2
x
(b) ee = 3
ANSWER KEY
Stewart - Calculus ET 6e Chapter 1 Form A
1.
2a + h − 2
2.
{ x | x ≠ 1/ 3}
3.
(−∞, ∞)
4.
y =8− − x
5.
6.
4
π (3r 2 + 3r + 1)
3
12
S ( x) = x 2 +
x
7.
odd
8.
−17
9.
Shift the graph 3 units to the right and 3 units down.
10.
11. (-7, 3)
12. 2
13. 4 x − 16
14. 80
15. C = 0.125d + 250
−1
16. f ( x) = −
x −1
2x −1
17. f −1 (3) = 1
18. f ( x) = 2 (1/ 3)
19.
x
3 /3
20. x = e 2 , x = ln ( ln 3)
Stewart - Calculus ET 6e Chapter 2 Form B
(
)
1.
The point P(4, 2) lies on the curve y = x . If Q is the point x, x , use your calculator to find the slope of the
secant line PQ (correct to six decimal places) for the value of x = 4.01.
2.
The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + t 2 / 4 , where t is
measured in seconds. Find the average velocity over the time period [1, 1.5] .
3.
Find the limit, if it exists.
lim
x→4
4.
4− x
| 4− x|
Find f ′(a) .
f ( x) = 2 + x − 5 x 2
5. Find an equation of the tangent line to curve y = x 3 − 2 x at the point (2, 4) .
6.
x− 2
at the given numbers (correct to six decimal places). Use the results to
x−2
guess the value of the limit lim f ( x).
Evaluate the function f ( x) = 2
x→ 2
x
f (x)
1.6
1.8
1.9
1.99
1.999
2.4
2.2
2.1
2.01
2.001
Limit
7.
The graph of f is given. State the numbers at which f is not differentiable.
Stewart - Calculus ET 6e Chapter 2 Form B
8.
Evaluate the limit.
§3·
lim x 9 cos¨ ¸
©x¹
x →0
9.
If 1 ≤ f ( x) ≤ x 2 + 6 x + 6 for all x find the limit.
lim f ( x)
x → −1
10. Evaluate the limit.
(
lim 7 x 2 + 6 x + 8
x →5
)
11. Evaluate the limit.
lim
(2 + x )−1 − 2−1
x→0
x
12. If an arrow is shot upward on the moon, with a velocity of 70 m/s its height (in meters) after t seconds is given by
H (t ) = 70t − 0.99t 2 . With what velocity will the arrow hit the moon?
13. The cost (in dollars) of producing x units of a certain commodity is C ( x) = 4,336 + 13x + 0.08 x 2 . Find the
average rate of change with respect to x when the production level is changed from x = 101 to x = 103.
­ −x
°
14. Let f ( x) = °® 3 − x
°
2
°¯( 3 − x )
if
if
x<0
0≤ x<3
if
x>3
Evaluate each limit, if it exists.
lim f ( x)
x → 0+
b.) lim− f ( x)
x →0
15. If f and g are continuous functions with f (3) = 3 and lim [3 f ( x) − g ( x)] = 3 , find g (3).
x →3
16. Evaluate the limit.
lim x + 9
x → −9
17. Find a number δ such that if | x − 2| < δ , then |4 x − 8| < ε , where ε = 0.1 .
Stewart - Calculus ET 6e Chapter 2 Form B
18. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an
hour, then Torricelli's Law gives the volume of water remaining in the tank after t minutes as
2
t ·
§
V (t ) = 100,000¨1 − ¸ , 0 ≤ t ≤ 60
© 65 ¹
Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to
t) as a function of t.
19. If g ( x) = 3 − 5 x , find the domain of g ′( x) .
20. For the function f whose graph is shown, state the following.
lim f ( x)
x → −4
ANSWER KEY
Stewart - Calculus ET 6e Chapter 2 Form B
1.
0.249844
2.
2.625 m/s
3.
Limit does not exist
4.
1 − 10a
5.
y = 10 x − 16
6.
(1.6, 0.7465), (1.8, 0.7257), (1.9, 0.7161), (1.99, 0.7079), (1.999, 0.707195), (2.4, 0.674899),
(2.2, 0.690261), (2.1, 0.698482), (2.01, 0.706225), (2.001, 0.707018), Limit = 0.707107
7.
− 3, 0, 1
8.
0
9.
1
10. 213
11. -1/4
12. -70
13. 29.32
14. a.) 3 b.) 0
15. 6
16. 0
17. δ = 0.025
− 200000 §
t ·
¨1 − ¸
65
© 65 ¹
18.
y=
19.
( −∞, 3 / 5 )
20.
−∞