calculus Worksheets

7/30/2019 calculus Worksheets

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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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(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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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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(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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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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(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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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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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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(c) f(x) =x2

(d) f(x) =x3

(e) f(x) =x

1

(f) f(x) = 4x2 + 3x 9


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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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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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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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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



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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


7. 44 =xy is a quartic function and has degree 4. Factor the function using the difference

of squares formula.


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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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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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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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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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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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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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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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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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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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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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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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3. Most calculators do not have a key to evaluate 15log2 . Use the change of base formula to

convert to base e first.


(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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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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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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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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(c)2

132

3

+=

x

xy

(d)693

22

2

+

=

xx

xxy


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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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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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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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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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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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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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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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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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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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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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(c)xx

xxxg

42

9710)(

3

2

+

+=

(d)xx

xxxh

42

9710)(

2

3

+

+=


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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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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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Continuous Functions Worksheet

1. Find constants c and dthat make the piecewise functionfcontinuous everywhere.

2

211

,

,,

2

2)(

21

2


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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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(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.

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