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PREPARATION
FOR
CALCULUS
WORKSHEETS
Second Edition
DIVISION OF MATHEMATICS
ALFRED UNIVERSITY
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Contents
Real Numbers Worksheet 1
Functions and Graphs Worksheet 5
Polynomials Worksheet . 12
Trigonometry Worksheet 18
Trigonometric Functions Worksheet 21
Exponential and Logarithmic Functions Worksheet 25
Rational Functions Worksheet 29
Limits Worksheet 32
Computing Limits Worksheet 36
Limits at Infinity Worksheet 40
Continuous Functions Worksheet 44
Prepared by: Joseph Petrillo, Alfred University
Edited by: Xiuhong Du and Juan Marin, Alfred University
Editions: 7/09, 1/10
References: Calculus (Early Trans.), 8th
ed., Anton/Bivens/Davis, Wiley (2005).
Wolframs Math World(http://mathworld.wolfram.com/).
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Preparation for Calculus Worksheets 1
Real Numbers Worksheet
1. Solve each equation forx.
(a) 022 =
xx
(b) 04
2=
x
x
(c) 03
2=
x
2. Perform the following operations and simplify where possible.
(a) = 23
1
(b)6
5
9
2+ =
(c)x
x
+1
1=
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Preparation for Calculus Worksheets 2
(d)1
2
2
3
+
xx=
(e)3
122
+
+
x
x
x
x=
(f)
103
52
=
(g) =
23
5 3x
(h)h
xhx
11
+ =
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Preparation for Calculus Worksheets 3
3. Use the properties of exponents to simplify. Rewrite any negative exponents.
(a) 32
)8( =
(b) 31
1)27( =
(c) 234 xx =
(d)24
32
)3(
)2(
x
x=
(e)
2
3
1
2
)1(
+
x
xx=
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Preparation for Calculus Worksheets 4
4. Solve each equation forx.
(a) 1172 2 =x
(b) 5212 3 =+x
(c) 5285 22 +=+ xx
5. Calculate or simplify each of the following.
(a)
(b) =!7
!9
(c) =+
!
)!1(
n
n
(d) =+
)!1(
)!1(
n
n
(e) =+
)!2(
)!22(
n
n
n 0 1 2 3 4 5 6 7
n!
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Preparation for Calculus Worksheets 5
Functions and Graphs Worksheet
1. (a) Does the graph of a circle in the Cartesian plane represent a function? Explain.
(b) The circle x2
+y2
= 1 called the unit circle. It is centered at the origin and has radius 1.Solve this equation fory to show how the unit circle can be expressed as two separate
functions.
(In general, the equation of a circle of radius rand center at the origin is x2 +y
2 = r2.)
2. (a) The function 21
xxy == is the square-root function, and the function 31
3 xxy ==
is the cube-root function. Discuss the domain and sketch the graphs.
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Preparation for Calculus Worksheets 6
(b) The function 11 == x
xy is the reciprocal function. Discuss the domain and sketch
the graphs of 11 == x
xy and 2
2
1 == x
xy .
(c) Any function of the form rxy = , where ris real, is a power function. [The functions
from parts (a) and (b) are power functions.] Sketch the graphs of the identity
function xy = , the squaring function 2xy = , and the cubing function 3xy = .
(d) Sketch the graph of the absolute value function y= |x | .
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Preparation for Calculus Worksheets 7
3. Let f(x) = 3x(x 2)(x + 1)2.
(a) Find thex-intercepts off.
(b) Find they-intercept off.
(c) Find the intervals of positive and negative off.
4. (a) Sketch the graphs, and then find the points(s) of intersection, if any, of the lines
3x 4y = 7 and x + 2y = 1. That is, solve the system of equations
3x 4y = 7
x + 2y = 1
(b) Sketch the graphs, and then find the point(s) of intersection, if any, of the circle
x2 +y
2 = 8 and the liney = 4 x. That is, solve the system of equations
x2
+y2
= 8
y = 4 x
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Preparation for Calculus Worksheets 8
5. (a) Write3 231
1
xy
= as a composition of three functionsf, g, and h.
(b) Find formulas for the compositions gf o and fg o given that f(x) = 1 x2
and
g(x) = 3+x .
6. Find the difference quotient of each function.
(a) f(x) = 6
(b) f(x) = 3x 7
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Preparation for Calculus Worksheets 9
(c) f(x) =x2
(d) f(x) =x3
(e) f(x) =x
1
(f) f(x) = 4x2 + 3x 9
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Preparation for Calculus Worksheets 10
7. Sketch the graphs of xy = , 4+= xy , 4+= xy , 42 += xy , and
342 ++= xy .
8. (a) Sketch the graph ofy = 2x 3. If possible, find a formula for the inverse and sketch its
graph on the same set of axes. If the function is not invertible, then restrict its domain sothat an inverse can be found.
(b) Sketch the graph ofy =x2. If possible, find a formula for the inverse and sketch its graph
on the same set of axes. If the function is not invertible, then restrict its domain so that
an inverse can be found.
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Preparation for Calculus Worksheets 11
Functions in applications
9. For a given outside temperature Tin degrees Fahrenheit, the wind chill temperature
(WCT) index is the equivalent temperature that exposed skin would feel with a wind
speed ofv miles per hour.
>++
=
3if,4275.075.356215.074.35
30if,WCT
16.016.0vTvvT
vT
Find the WCT to the nearest degree ifT= o20 F and v = 15 mi/h.
10. The Surface area Sand the volume Vof a spherical balloon can be viewed as functions
of the radius rof the balloon. That is,
24)( rrS = and 33
4)( rrV = .
Find the surface area and volume of a spherical balloon with a 3-inch radius. Explain
your answers in terms of the units involved.
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Preparation for Calculus Worksheets 12
Polynomials Worksheet
1. 2=y is a constant function and has degree 0. Its graph is a horizontal line.
(a) Find they-intercept.
(b) Find thex-intercept(s).
2. 74 += xy is a linear function and has degree 1. Its graph is a line with slope ______
andy-intercept ______ .
Find thex-intercept(s).
3. A line passes through the points (2, 7) and (4, 2).
(a) Find the slope of the line.
(b) Find the point-slope form of the line.
(c) Find the slope-intercept form of the line.
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Preparation for Calculus Worksheets 13
4. 483216 2 ++= xxy is a quadratic function and has degree 2. Its graph is a parabola.
(a) Find they-intercept.
(b) Find thex-intercept(s). (The equation factors easily.)
(c) Determine the intervals on which the polynomial is positive and the intervals on which
the polynomial is negative.
5. 84223
+++= xxxy is a cubic function and has degree 3.
(a) Find all roots. (Use grouping.)
(b) Determine the intervals on which the polynomial is positive and the intervals on which
the polynomial is negative.
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Preparation for Calculus Worksheets 14
6. 652 23 += xxxy is a cubic function.
(a) Find all zeros. (Use guess-and-check and long division. The remaining quadratic factors
easily.)
(b) Determine the intervals on which the polynomial is positive and the intervals on which
the polynomial is negative.
7. 44 =xy is a quartic function and has degree 4. Factor the function using the difference
of squares formula.
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Preparation for Calculus Worksheets 15
8. 234 144819 xxxy += is a quartic function. Find the roots. (You will eventually need the
quadratic formula.)
9. Solve x2
9x + 16 = 0 by completing the square.
10. Complete the square on y = 2x2
+ 3x 4 to determine the vertex of the graph.
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Preparation for Calculus Worksheets 16
11. Even though some functions are not polynomials, we can use similar techniques.
(a) Factor the function 34
31
36)( xxxf += . (Factor out the smallest power ofx.)
(b) Factor the function 31
32
42)( xxxg +=
, and then find its domain and intercepts.
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Preparation for Calculus Worksheets 17
Polynomials in applications:
12. Let 14416)( 2 += tts be the position in feet of a falling object tseconds after it was
dropped.
(a) Find the height from which the object was dropped.
(b) At what time did the object hit the ground?
13. Let 5.248.9)( += ttv be the velocity in meters per second of a moving object tseconds
after it was thrown straight up into the air.
(a) What was the initial velocity?
(b) At what time did the object reach its maximum height and begin to descend?
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Preparation for Calculus Worksheets 18
Trigonometry Worksheet
1. (a) Convert o75 and o225 to radians.
(b) Convert15
and
9
7to degrees.
2. Fill in the table from memory.
3. Given that tan = 3, find the exact values of the remaining five trigonometric functions
of. [Hint: draw the appropriate triangle.]
4. Find the cosine, sine, and tangent of.
(a) (b)
5 7
2 4
cos sin tan o00 = o30
6=
o454=
o603=
o902=
o180= o270
23=
o1506
5=
o2403
4=
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Preparation for Calculus Worksheets 19
5. (a) Find all values ofbetween 0 and 2(in radians) such that 4sin2 2 = 0.
(b) Find all values ofbetween 0 and 2(in radians) such that sin = cos .
6. Find the difference quotient off(x) = sinx.
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Preparation for Calculus Worksheets 20
Trigonometry in applications:
7. A 10-foot ladder leans against a house and makes an angle of o60 with level ground.
How far is the top of the ladder above the ground? How far is the bottom of the ladder
from the base of the house?
8. An airplane flies over a radar station and then a checkpoint 1 mile away, both located on
level ground. At the moment the angle of elevation of the airplane above the radarstation is 50 and the angle between the station and checkpoint is 30, find the distance
between the airplane and the checkpoint using the Law of Sines, and then find the
distance between the airplane and the radar station using the Law of Cosines.
30
50 100
1 mi
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Preparation for Calculus Worksheets 21
Trigonometric Functions Worksheet
1. Find the amplitude, period, frequency, and phase shift. Then sketch a graph showing at
least two periods.
(a) )4cos(3 xy =
(b) 2)3sin(22++=
xy [Notice the extra vertical shift.]
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Preparation for Calculus Worksheets 22
2. (a) =
2
3cos 1 __________
(b)=
)1(sin
1
__________
(c) = )1(tan 1 __________
(d) =
2
1sin 1 __________
3. Find .
(a)
2
1
(b)
50
20
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Preparation for Calculus Worksheets 23
Trigonometric functions in applications:
4. Suppose a mass is attached to a hanging spring and is allowed to come to rest at its
equilibrium position. The mass is pulled 0.5 meters below equilibrium and is released
at time t= 0. Assume the mass vibrates up and down with position given by
)3.1cos(5.0)( tty = meters,
tseconds after release.
(a) Find the amplitude, period, and frequency of the vibration.
(b) Find the position of the mass after 3 seconds.
5. In the United States, a standard electrical outlet supplies sinusoidal electrical current
with a maximum voltage of 2120=V volts (V) at a frequency of 60 hertz (Hz).Write an equation that expresses Vas a function of the time t, assuming that V= 0 if
t= 0. [Note: 1 Hz = 1 cycle per second.]
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Preparation for Calculus Worksheets 24
6. A soccer player kicks a ball with an initial speed of 14 m/s at an angle with the
horizontal see the figure below. The ball lands 18 m down the field. If air resistance isneglected, then the ball will have a parabolic trajectory and the horizontal rangeR will
be given by
2sin2
g
vR =
where v is the initial speed of the ball and g = 9.8 m/s2 is the acceleration due to
gravity. Approximate two values of, to the nearest degree, at which the ball could
have been kicked.
R
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Preparation for Calculus Worksheets 25
Exponential and Logarithmic Functions Worksheet
1. On the set of axes below, sketch and label the graphs ofy = bx for bases b = 1, 2, 3, and .
1
1 1
2. Find the exact value of each expression without a calculator.
(a) = 32
)8( (b) = 32
8
(c) =
21
9 (d) =32log2
(e) =)01.0(log10 (f) =1000log10
(g) =3ln e (h) =3)(lne
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Preparation for Calculus Worksheets 26
3. Most calculators do not have a key to evaluate 15log2 . Use the change of base formula to
convert to base e first.
4. Solve each equation forx.
(a) 2)1(log10 =+x
(b) 2ln)ln(3)4ln( 2 = xx
(c) 35 2 = x
(d) 7)3exp(2 =x
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Preparation for Calculus Worksheets 27
Exponential and logarithmic functions in applications:
5. The loudness of a sound can be measured by its intensityI(in watts per square meter),
which is related to the energy transmitted by the sound wavethe greater the intensity, the
greater the transmitted energy, and the louder the sound is perceived by the human ear. Since
intensity units vary over an enormous range, we measure loudness in terms ofsound level(in decibels dB):
=
1210log10
I dB
Damage to the average ear occurs at 90 dB or greater. Find the decibel level of each of thefollowing sounds and state whether it will cause ear damage.
Sound IntensityI
(a) Jet aircraft from 50 ft 1.0102
W/m2
(b) Amplified rock music 1.0 W/m2
(c) Garbage disposal 1.0104
W/m2
(d) TV mid volume from 10 ft 3.2105
W/m2
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Preparation for Calculus Worksheets 28
6. The equation teQ 055.012 = gives the mass Q in grams of radioactive potassium-42 that will
remain from some initial quantity after thours of radioactive decay.
(a) How many grams were there initially?
(b) How many grams remain after 4 hours?
(c) What is the half-life of potassium-42. That is, how long will it take to reduce the amount
of radioactive potassium-42 to half of the initial amount?
7. In thermodynamics, an equation or the form
=
RT
QA
texp
1is rewritten as a linear
function of 1/T, namely,
bT
at +=1
ln
where a is the slope and b is the (extrapolated) vertical axis intercept. Find the slope a andintercept b in terms ofA, Q, andR.
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Preparation for Calculus Worksheets 29
Rational Functions Worksheet
1. For each rational function, find the following, if possible.
(i) y-intercepts
(ii) x-intercepts
(iii) holes
(iv) vertical asymptotes
(v) intervals of positive and negative
(a)2
5
=x
y
(b)9
42+
+=
x
xy
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Preparation for Calculus Worksheets 30
(c)2
132
3
+=
x
xy
(d)693
22
2
+
=
xx
xxy
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Preparation for Calculus Worksheets 31
2. Perform long division to rewrite the improper rational function2
132
3
+=
x
xy .
Rational functions in applications:
3. According to Coulombs law, the magnitude of the electrical force Fbetween two charged
particles with charges Q1 and Q2 is inversely proportional to the square of the distance dbetween them. That is,
2
21
d
QkQF= .
Describe the force as the particles get closer and closer together.
Describe the force as the particles get further and further apart.
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Preparation for Calculus Worksheets 32
Limits Worksheet
1. Use the graph ofy =f(x) to fill in the blanks.
y
(a) f(2) ________ )(lim2
xfx
________ )(lim2
xfx
+
________ )(lim2
xfx
________
(b) f(1) ________ )(lim1
xfx
________ )(lim1
xfx +
________ )(lim1
xfx
________
(c) f(3) ________ )(lim3
xfx
________ )(lim3
xfx +
________ )(lim3
xfx
________
(d) f(4) ________ )(lim4
xfx
________ )(lim4
xfx +
________ )(lim4
xfx
________
6 f(x)
5
4
3
2
1
3 2 1 0 1 2 3 4 5 6
1
2
3
x
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Preparation for Calculus Worksheets 33
2. Use the graph ofy = g(x) to fill in the blanks.
y
x
(a) g(1) ________ )(lim1
xgx
________ )(lim1
xgx +
________ )(lim1
xgx
________
(b) g(1) ________ )(lim1
xgx
________ )(lim1
xgx +
________ )(lim1
xgx
________
(c) g(2) ________ )(lim2
xgx
________ )(lim2
xgx +
________ )(lim2
xgx
________
(d) g(5) ________ )(lim5
xgx
________ )(lim5
xgx
+
________ )(lim5
xgx
________
6 g(x)
5
4
3
2
1
3 2 1 0 1 2 3 4 5 6
1
2
3
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Preparation for Calculus Worksheets 34
3. Numerically analyze the following limits.
(a)1
1lim
2
1
x
x
x
?From the left side From the right side
(b)xx
1lim
0
?From the left side From the right side
(c)22 )2(
1lim
xx
?
From the left side From the right side
(d)x
x
x
sinlim
0
?
From the left side From the right side
x 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.1
112
xx
x 0.1 0.01 0.001 0.0001 0 0.0001 0.001 0.01 0.1
x1
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
2)2(
1
x
x 0.1 0.01 0.001 0.0001 0 0.0001 0.001 0.01 0.1
xxsin
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Preparation for Calculus Worksheets 35
Limits in applications:
4. In the special theory of relativity the mass m of a moving object is a function m = m(v) of
the objects speed v. In the figure, c denotes the speed of light.
Mass m
m0Speed v
c
(a) What is the physical interpretation ofm0?
(b) What is )(lim vmcv
? What is the physical significance of this limit?
5. In the special theory of relativity the length l of a narrow rod moving longitudinally is afunction l = l(v) of the rods speed v. In the figure, c denotes the speed of light.
Length l
l0
Speed vc
(a) What is the physical interpretation ofl0?
(b) What is )(lim vlcv
? What is the physical significance of this limit?
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Preparation for Calculus Worksheets 36
Computing Limits Worksheet
Compute the following limits.
1. 5lim2x
2. )53(lim 232
++
xxx
3. xx
coslim6
4.3
4lim
2
3 +
x
xx
x
5.2
23lim
2
1 +
+
x
xx
x
6. xx
tanlim2
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Preparation for Calculus Worksheets 37
7.82
3lim
24
xx
x
xSign Chart:
N
D
N/D
4
8.12
6lim
2
2
1 ++
+
xx
xx
xSign Chart:
N
D
N/D
1
9.1
1lim
4
1
x
x
x
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Preparation for Calculus Worksheets 38
10.6
44lim
2
2
2 +
+
xx
xx
x
11.2510
103lim
2
2
5 +
xx
xx
xSign Chart:
N
D
N/D5
12.3
9lim
9
x
x
x
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Preparation for Calculus Worksheets 39
13. )(lim2
tft
, where
>
=
2,54
2,1)(
tt
tttf .
14. Find the derivative of each function.
(a) xxf =)(
(b) 23)( 2 += xxxf
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Preparation for Calculus Worksheets 40
Limits at Infinity Worksheet
1. Investigate the end behavior of each of the following functions.
(a)4
4
132xxy
+=
(b)3
3
1
32)(
x
xxf
+=
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Preparation for Calculus Worksheets 41
(c)xx
xxxg
42
9710)(
3
2
+
+=
(d)xx
xxxh
42
9710)(
2
3
+
+=
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Preparation for Calculus Worksheets 42
2. Find any horizontal asymptotes of65
23 2
+=
x
xxy .
3. Evaluate each limit.
(a)3
1
2
2
81
32lim
+
+
x
xx
x
(b)9
limx
ex
x
(c) 9limxe
x
x
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Preparation for Calculus Worksheets 43
Limits at infinity in applications:
4. Let T=f(t) denote the temperature of a baked potato tminutes after it has been removed
from a hot oven. The accompanying figure shows the temperature versus time curve for
the potato, where ris the temperature of the room.
T
400
T=f(t)
Temp (oF)
r
t
Time (min)
(a) What is the physical significance of )(lim0
tft
+
?
(b) What is the physical significance of )(lim tft +
?
5. Suppose that the speed v (in ft/s) of a sky diver tseconds after leaping from a plane is
given by the equation v(t) = 190 (1 e0.168t).
(a) Show that the graph ofv(t) has a horizontal asymptote v = c for some constant c.
(b) What is the physical significance of the constant c in part (b)?
(c) Sketch a graph showing v versus t.
v
t
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Preparation for Calculus Worksheets 44
Continuous Functions Worksheet
1. Find constants c and dthat make the piecewise functionfcontinuous everywhere.
2
211
,
,,
2
2)(
21
2
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Preparation for Calculus Worksheets 45
3. (a) Use the Squeezing Theorem to show that 1sin
lim0
= x
x
x.
Hint: In the unit circle, consider the areas of the sector and the two triangles determined
by the anglex (in radians) shown below.
Area of large triangle Area of sector Area of small triangle
x
1
(b) Use part (a) and the identity sin2x = 1 cos2x = (1 cosx)(1 + cosx) to show that
0cos1
lim0
=
x
x
x.
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Preparation for Calculus Worksheets 46
(c) In a previous example and on a previous worksheet we calculated the difference quotients
of sinx and cosx, respectively, as follows:
=
+
h
x
h
x
h
xhx cosh1sin
sinhcos
sin)sin(
=
+
hx
hx
h
xhx sinhsin
cosh1cos
cos)cos(
Use the limits from parts (a) and (b) to calculate the derivatives of sinx and cosx.