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MATH 2413 CALCULUS I Review for test 1, Fall 2004
Date of test
Wednesday, Sept 22 to Monday, Sept 27
Material covered
Chapter 1, sections 1 through 6 Chapter 2, sections 1 through 5
Allowable materials
Calculator (no TI-89 or 92)
Sample problems 1.
The graph of f is drawn below. Sketch the graph of g x f x 2 3 2 1 -3
-2
-1
1
2
3
-1 -2 -3
2.
2 If f x x 2 x 1 , find f 3
3.
Write a formula for a linear function f with slope 1.4, passing through the point 1,3.3
4.
Write a formula for the linear function whose graph is shown: f
5.
The graph of f below can be obtained by transformations of the graph of g x x . Find a formula for f x .
Page 1 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 6.
Evaluate f 1 based on the graph of f below
7.
Evaluate the following based on the graphs of f and g below
f
a.
f + g 0
b.
fg 4
c.
g o f 1
d.
draw a graph of f g
g
8.
Determine whether the relation defined by the graph is a function a.
9.
b.
Federal income tax is calculated an individual’s annual income. Tax for an annual income of up to $63,550 is given by the piecewise function below, where T and x are both in dollars. Use this function to calculate the tax for the incomes given. .10 x T x 600 + .15 x 6000 3637.50 + .27 x 26, 250
0 x 6000 6000 x 26, 250 26, 250 x 63,550
Page 2 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004
10.
a.
An annual income of $5200
b.
An annual income of $22,500
Determine whether the graph represents an odd function, an even function, or neither. a.
11.
b.
Refer to the graph of f below: a.
What is the domain of f?
b.
Approximate the zeros of f.
c.
Over what intervals is f increasing? decreasing?
-2
12.
-1
1
2
3
Complete the table if f is an even function x f(x)
13.
c.
-2 5
-1 3
0 2
1
2
Complete the graph of f below if f is an odd function
Page 3 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 14.
Find the inverse of the function f ( x) x + 1
15.
Sketch the graph of the inverse of the function below, on the same grid:
16.
17.
18.
For each of the following pairs of functions, state whether they are inverses of each other. a.
f x
x +1 , g x 2x 1 2
b.
f x
x 1 , g x 2 x + 1 2
Give a numeric representation of f 1 x
-2
-1
0
1
2
f(x)
-10
-4
2
3
5
Determine whether an inverse exists for the function whose graph is drawn. a.
19.
b.
Find a function of the form f x Ca that fits the data exactly x
a.
x f(x)
0 2
1 6
2 18
3 54
4 162
b. Page 4 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004
x f(x) 20.
0 8
1 4
2 2
3 1
4 1/2
State that the function is odd, even, or neither: a.
f x 2x5 1
c.
h x 2 x 3 x
b.
g x 2x 2 + 5
d.
m x x2 + 2
x
21.
22.
1 Sketch a graph of the function f x 4 . Clearly label scale on axes, and indicate 3 intercepts, if any. State the domain a.
f x log 6 2 x 1
c.
h x x 3
b.
g x ex
d.
m x
2
1
x 1 x+2
23. The table gives Revenue for a tire company each year from 1990 to 1995.
Year Revenue ($m) a. b.
c. 24.
1990 25.6
1991 28.3
1992 31.9
1993 34.8
1994 37.1
1995 40.5
Make a scatter plot of the data. Find the regression line. You may use a calculator algorithm or come up with an approximation by hand. Explain what you did. Draw the regression line on your scatter plot.
Match each of the graphs with a function from the list
I
II
III
IV
Page 5 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004
25.
26.
27.
a.
f x ax 3 + bx 2 + cx + d
c.
h x ax, a > 1
b.
g x log a x
d.
m x ax, a 1
Evaluate the logarithms. Approximate with a calculator if necessary. a.
ln e 7
c.
log 3 27
b.
1 log 1000
d.
ln15
e.
log 3 6
Write as a single logarithm and simplify. Do not evaluate with a calculator. a.
2 ln 3 + ln 4 ln 6
b.
log12 2 log 2 log 3
29.
e
2ln x 1
b.
log 10 x 102
Solve the equations. Give a simplified answer and a decimal approximation for each. a.
e 2 x 1 5
c.
5e x +1 100
b.
33 p1 25
d.
ln x 4
Krypton-85 is a radioactive isotope of Krypton, with a half-life of 10 years. a.
b.
30.
ln x + 1 + ln x 1 2 ln x
Simplify the expression a.
28.
c.
If 12 grams of Krypton-85 leak into a laboratory, give an equation for the amount of Krypton that will be present after t years. How much will be present after 25 years?
The estimated population of the city of Austin is given by the equation P 650, 000e.07 t where t is the number of years from now. a.
What will the population be in 10 years?
b.
How long will it take the population to reach 1 million? Page 6 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 31.
The height of an object t seconds after it is thrown into the air is described by the graph below.
Height (meters)
Time (seconds)
32.
a.
What is the average velocity for the first second after the ball is thrown? Draw the secant line whose slope is this average velocity.
b.
What is the instantaneous velocity one second after the ball is thrown? Draw the tangent line whose slope is this instantaneous velocity.
The temperature in degrees Fahrenheit in Austin at each hour of a given day in March is �t � described by the function F t 56.6 + 18.2sin � p � where t is the number of hours past �12 � 10:00 AM. a.
What is the average rate of change of the temperature over the following timer intervals? i. from 11:00 AM and 2:00 PM? ii. from 11:00 AM to 12:00 noon
b.
What is the instantaneous rate of change at 11:00 AM
� 1 � 33. For the piecewise-defined function f x � 2 x � � x2 a. Sketch a graph of f b.
x0 0�x2 x �2
At what values of x is f discontinuous?
Page 7 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 34.
Evaluate the following limits if they exist, based on the graph of f drawn below.
3 2 1
a.
b.
f. 35.
lim f x
c.
lim f x
lim x 2 f x
d.
lim f x
x � 2
x � 2 +
x� 2
e.
x� 0
Over what intervals is f continuous?
Evaluate the limits if they exist a.
lim
x2 4 x2
d.
b.
lim
x 1 x 1
e.
c.
4 x3 x lim x� � 3 2 x 3
x� 2
x� 1
f.
x 1 x2 + 1
lim+
x 4 x
g.
lim
lim
x 4x
h.
lim e x
x� 4
x� 4
2 x3 + 1 lim x � � x 2 2
i.
j.
36.
lim � �2 + f x � �
x � 2
x� �
2
x� �
lim
x � 0
2x x
lim sin x x� �
e x sin x and prove your result. Use the squeeze theorem to evaluate xlim � �
Page 8 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 ANSWERS:
Page 9 of 11
MATH 2413, CALCULUS I Review for test 1, Fall 2004
b. funct -1 1 2 -1 io -2 n -3 2 1
-4 -3 -2
2.
a. $520
3.
3
b. no i n v 14. e 2 1 f x x 1 r s , e x 1
11.
15.
a. [ 1,3]
c. 4
2
c. even
a. 5
3
d.
8.
a. not a f u n ct
b. x 1 a n 16. d x2 a. yes c.
19.
a. f x
b. f x
incre b. as no in 17. g: 1, 0-10 -4
20.
2
-1 0 & -2 2,3 18. a. d yes, e a
a. neit h e r b. eve n
b. odd
7.
1
a. neith f x 2 x + 3 er f 1 2
0
b. e. m o y 2.97 x + 25.6 s n ln 6 1.63 d d ( ln 3 30. L d. 26. a. i ev 1.3 a. n e m ln 6 R n ill e b. 21. io g Time (seconds) 0 n 7 32. 6 c. o b. 5 4 a. n x 2 1 6.15 3 ln 2 2 y i. x 1 T e -3 -2 -1 1 2 3 4 5 6 7 3.6837 -1 I 27. ar -2 de -3 s gr a. 8 22. ee 1 31. 3 2 s a. x 1 ) a. pe 1 12 x x > r b. 2 m ho x2 et ur b. 28. er 24. � ii. s a. a. 4.3895 c. p 1 II de x 1 + ln 5 er 1.30 { x x �3} 2 gr se b. ee d. c b. IV s 1 �ln 25 o� { x x �2c.} p � + 1n� 1.31 pe 3 �ln 3 � I 23. 2 3x r d ho c. d. a. ur x ln 20 1 2.00 III (ax 1 b. 8 & 25. d. 2 4 4.60 x e 54.60 a. 2 c –7 29. 4 Time (seconds) ) d b. a. b. t 10 e –3 16.7 g K t 12 m c. 2 r et 3 e b. er d. e 2.12 s 2.7 s g p 0 p r er 8 e a se r c
e x i s t s
13.
5.
6.
-1
c. od
i n v e r s e
12.
b. f x 1.4 x + 1.9 -2 $307 5 5 4. f x 510. x 3
b. 0
4
-4
9.
– 4
3
n
Height (meters)
3
c r e a si n g : 0, 2
Height (meters)
4
45
Revenue ($m)
io n
1.
40
35
30
25 1990
1991
1992
1993
1994
1995
MATH 2413, CALCULUS I Review for test 1, Fall 2004
h o u r
e. 4
j. z does e n f. o T [ 4, 2 ] , 2, 2 t , 2, 4] h 33. e e 35. a. x o a. is r 4 t e ( m b. o does s lim e x sin x 0 n x �� ci ot . ll e at xi i st b. o (j disc n u o ) m n p ti 36. di n e x �e x sin x �e x sc u a o o n nt u d in s ui at lim e x 0 lim e x ty x � � x �� x ) . = T 0 c. h –2 34. e d. r a. � e 2 f e. b. o +� 4 r f. c. e � does b n y g. o t 0 t h h. e e 0 x S is q i. t u –2 e d. e 0
Date of test
Wednesday, Sept 22 to Monday, Sept 27
Material covered
Chapter 1, sections 1 through 6 Chapter 2, sections 1 through 5
Allowable materials
Calculator (no TI-89 or 92)
Sample problems 1.
The graph of f is drawn below. Sketch the graph of g x f x 2 3 2 1 -3
-2
-1
1
2
3
-1 -2 -3
2.
2 If f x x 2 x 1 , find f 3
3.
Write a formula for a linear function f with slope 1.4, passing through the point 1,3.3
4.
Write a formula for the linear function whose graph is shown: f
5.
The graph of f below can be obtained by transformations of the graph of g x x . Find a formula for f x .
Page 1 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 6.
Evaluate f 1 based on the graph of f below
7.
Evaluate the following based on the graphs of f and g below
f
a.
f + g 0
b.
fg 4
c.
g o f 1
d.
draw a graph of f g
g
8.
Determine whether the relation defined by the graph is a function a.
9.
b.
Federal income tax is calculated an individual’s annual income. Tax for an annual income of up to $63,550 is given by the piecewise function below, where T and x are both in dollars. Use this function to calculate the tax for the incomes given. .10 x T x 600 + .15 x 6000 3637.50 + .27 x 26, 250
0 x 6000 6000 x 26, 250 26, 250 x 63,550
Page 2 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004
10.
a.
An annual income of $5200
b.
An annual income of $22,500
Determine whether the graph represents an odd function, an even function, or neither. a.
11.
b.
Refer to the graph of f below: a.
What is the domain of f?
b.
Approximate the zeros of f.
c.
Over what intervals is f increasing? decreasing?
-2
12.
-1
1
2
3
Complete the table if f is an even function x f(x)
13.
c.
-2 5
-1 3
0 2
1
2
Complete the graph of f below if f is an odd function
Page 3 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 14.
Find the inverse of the function f ( x) x + 1
15.
Sketch the graph of the inverse of the function below, on the same grid:
16.
17.
18.
For each of the following pairs of functions, state whether they are inverses of each other. a.
f x
x +1 , g x 2x 1 2
b.
f x
x 1 , g x 2 x + 1 2
Give a numeric representation of f 1 x
-2
-1
0
1
2
f(x)
-10
-4
2
3
5
Determine whether an inverse exists for the function whose graph is drawn. a.
19.
b.
Find a function of the form f x Ca that fits the data exactly x
a.
x f(x)
0 2
1 6
2 18
3 54
4 162
b. Page 4 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004
x f(x) 20.
0 8
1 4
2 2
3 1
4 1/2
State that the function is odd, even, or neither: a.
f x 2x5 1
c.
h x 2 x 3 x
b.
g x 2x 2 + 5
d.
m x x2 + 2
x
21.
22.
1 Sketch a graph of the function f x 4 . Clearly label scale on axes, and indicate 3 intercepts, if any. State the domain a.
f x log 6 2 x 1
c.
h x x 3
b.
g x ex
d.
m x
2
1
x 1 x+2
23. The table gives Revenue for a tire company each year from 1990 to 1995.
Year Revenue ($m) a. b.
c. 24.
1990 25.6
1991 28.3
1992 31.9
1993 34.8
1994 37.1
1995 40.5
Make a scatter plot of the data. Find the regression line. You may use a calculator algorithm or come up with an approximation by hand. Explain what you did. Draw the regression line on your scatter plot.
Match each of the graphs with a function from the list
I
II
III
IV
Page 5 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004
25.
26.
27.
a.
f x ax 3 + bx 2 + cx + d
c.
h x ax, a > 1
b.
g x log a x
d.
m x ax, a 1
Evaluate the logarithms. Approximate with a calculator if necessary. a.
ln e 7
c.
log 3 27
b.
1 log 1000
d.
ln15
e.
log 3 6
Write as a single logarithm and simplify. Do not evaluate with a calculator. a.
2 ln 3 + ln 4 ln 6
b.
log12 2 log 2 log 3
29.
e
2ln x 1
b.
log 10 x 102
Solve the equations. Give a simplified answer and a decimal approximation for each. a.
e 2 x 1 5
c.
5e x +1 100
b.
33 p1 25
d.
ln x 4
Krypton-85 is a radioactive isotope of Krypton, with a half-life of 10 years. a.
b.
30.
ln x + 1 + ln x 1 2 ln x
Simplify the expression a.
28.
c.
If 12 grams of Krypton-85 leak into a laboratory, give an equation for the amount of Krypton that will be present after t years. How much will be present after 25 years?
The estimated population of the city of Austin is given by the equation P 650, 000e.07 t where t is the number of years from now. a.
What will the population be in 10 years?
b.
How long will it take the population to reach 1 million? Page 6 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 31.
The height of an object t seconds after it is thrown into the air is described by the graph below.
Height (meters)
Time (seconds)
32.
a.
What is the average velocity for the first second after the ball is thrown? Draw the secant line whose slope is this average velocity.
b.
What is the instantaneous velocity one second after the ball is thrown? Draw the tangent line whose slope is this instantaneous velocity.
The temperature in degrees Fahrenheit in Austin at each hour of a given day in March is �t � described by the function F t 56.6 + 18.2sin � p � where t is the number of hours past �12 � 10:00 AM. a.
What is the average rate of change of the temperature over the following timer intervals? i. from 11:00 AM and 2:00 PM? ii. from 11:00 AM to 12:00 noon
b.
What is the instantaneous rate of change at 11:00 AM
� 1 � 33. For the piecewise-defined function f x � 2 x � � x2 a. Sketch a graph of f b.
x0 0�x2 x �2
At what values of x is f discontinuous?
Page 7 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 34.
Evaluate the following limits if they exist, based on the graph of f drawn below.
3 2 1
a.
b.
f. 35.
lim f x
c.
lim f x
lim x 2 f x
d.
lim f x
x � 2
x � 2 +
x� 2
e.
x� 0
Over what intervals is f continuous?
Evaluate the limits if they exist a.
lim
x2 4 x2
d.
b.
lim
x 1 x 1
e.
c.
4 x3 x lim x� � 3 2 x 3
x� 2
x� 1
f.
x 1 x2 + 1
lim+
x 4 x
g.
lim
lim
x 4x
h.
lim e x
x� 4
x� 4
2 x3 + 1 lim x � � x 2 2
i.
j.
36.
lim � �2 + f x � �
x � 2
x� �
2
x� �
lim
x � 0
2x x
lim sin x x� �
e x sin x and prove your result. Use the squeeze theorem to evaluate xlim � �
Page 8 of 11
MATH 2413 CALCULUS I Review for test 1, Fall 2004 ANSWERS:
Page 9 of 11
MATH 2413, CALCULUS I Review for test 1, Fall 2004
b. funct -1 1 2 -1 io -2 n -3 2 1
-4 -3 -2
2.
a. $520
3.
3
b. no i n v 14. e 2 1 f x x 1 r s , e x 1
11.
15.
a. [ 1,3]
c. 4
2
c. even
a. 5
3
d.
8.
a. not a f u n ct
b. x 1 a n 16. d x2 a. yes c.
19.
a. f x
b. f x
incre b. as no in 17. g: 1, 0-10 -4
20.
2
-1 0 & -2 2,3 18. a. d yes, e a
a. neit h e r b. eve n
b. odd
7.
1
a. neith f x 2 x + 3 er f 1 2
0
b. e. m o y 2.97 x + 25.6 s n ln 6 1.63 d d ( ln 3 30. L d. 26. a. i ev 1.3 a. n e m ln 6 R n ill e b. 21. io g Time (seconds) 0 n 7 32. 6 c. o b. 5 4 a. n x 2 1 6.15 3 ln 2 2 y i. x 1 T e -3 -2 -1 1 2 3 4 5 6 7 3.6837 -1 I 27. ar -2 de -3 s gr a. 8 22. ee 1 31. 3 2 s a. x 1 ) a. pe 1 12 x x > r b. 2 m ho x2 et ur b. 28. er 24. � ii. s a. a. 4.3895 c. p 1 II de x 1 + ln 5 er 1.30 { x x �3} 2 gr se b. ee d. c b. IV s 1 �ln 25 o� { x x �2c.} p � + 1n� 1.31 pe 3 �ln 3 � I 23. 2 3x r d ho c. d. a. ur x ln 20 1 2.00 III (ax 1 b. 8 & 25. d. 2 4 4.60 x e 54.60 a. 2 c –7 29. 4 Time (seconds) ) d b. a. b. t 10 e –3 16.7 g K t 12 m c. 2 r et 3 e b. er d. e 2.12 s 2.7 s g p 0 p r er 8 e a se r c
e x i s t s
13.
5.
6.
-1
c. od
i n v e r s e
12.
b. f x 1.4 x + 1.9 -2 $307 5 5 4. f x 510. x 3
b. 0
4
-4
9.
– 4
3
n
Height (meters)
3
c r e a si n g : 0, 2
Height (meters)
4
45
Revenue ($m)
io n
1.
40
35
30
25 1990
1991
1992
1993
1994
1995
MATH 2413, CALCULUS I Review for test 1, Fall 2004
h o u r
e. 4
j. z does e n f. o T [ 4, 2 ] , 2, 2 t , 2, 4] h 33. e e 35. a. x o a. is r 4 t e ( m b. o does s lim e x sin x 0 n x �� ci ot . ll e at xi i st b. o (j disc n u o ) m n p ti 36. di n e x �e x sin x �e x sc u a o o n nt u d in s ui at lim e x 0 lim e x ty x � � x �� x ) . = T 0 c. h –2 34. e d. r a. � e 2 f e. b. o +� 4 r f. c. e � does b n y g. o t 0 t h h. e e 0 x S is q i. t u –2 e d. e 0
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