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PREPARATION FOR CALCULUS

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

: Joseph Petrillo, Alfred University :

Xiuhong Du and Juan Marin, Alfred University

: :

7/09, 1/10 Calculus (Early Trans.), 8th ed., Anton/Bivens/Davis, Wiley (2005). Wolfram’s Math World (http://mathworld.wolfram.com/).

Preparation for Calculus – Worksheets

Real Numbers Worksheet 1. Solve each equation for x. (a)

x 2 2 x

0

(b)

2x x 4

0

(c)

2

0

x 3

2. Perform the following operations and simplify where possible. (a) 1 3

(b)

2 9

(c) x

2

5 = 6

1 1 x

=

1

Preparation for Calculus – Worksheets (d)

(e)

3

2 2

x

x 1

2 x 1 = x 3 x

x

2

2

(f)

=

3

5 = 10

3 (g) 5x 3 2

1

(h) x

h h

1 x =

2

Preparation for Calculus – Worksheets 3. Use the properties of exponents to simplify. Rewrite any negative exponents. (a) ( 8)

2

=

3

(b) (27 1 )

1

3

=

(c) 4 x 3 x 2 =

(d)

( 2x 2 ) 3 = (3x 4 ) 2

 x( x 1) 1   (e)  3  x 2 

2

=

3

Preparation for Calculus – Worksheets 4. Solve each equation for x. (a) 2 x2

7 11

(b) 2 x3

21 5

(c) 5 x2

8

2 x2

5

5. Calculate or simplify each of the following. (a)

n n!

(b) 9! 7!

(c)

( n 1)! n!

(d) (n 1)! (n 1)!

(e) (2n 2)! (2n)!

0

1

2

3

4

5

6

7

4

5

Preparation for Calculus – Worksheets

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 . It is centered at the origin and has radius 1. Solve this equation for y to show how the unit circle can be expressed as two separate functions. (In general, the equation of a circle of radius r and center at the origin is x2 + y2 = r2.)

2. (a) The function y is the

x

x

1

2

is the

, and the functiony

. Discuss the domain and sketch the graphs.

3

x

x

1

3

Preparation for Calculus – Worksheets 1 x 1 the graphs of y x

(b) The function y

1

x is the 1 x and y

. Discuss the domain and sketch 1 x2

2 x .

. [The functions (c) Any function of the form y x r , where r is real, is a from parts (a) and (b) are power functions.] Sketch the graphs of the y x 3. y x 2 , and the y x , the

(d) Sketch the graph of the

y

|x|.

6

Preparation for Calculus – Worksheets 3. Let f(x) = 3x(x – 2)(x + 1)2. (a) Find the x-intercepts of f.

(b) Find the y-intercept of f.

(c) Find the intervals of positive and negative of f.

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 + y2 = 8 and the line y = 4 – x. That is, solve the system of equations x2 + y2 = 8 y=4–x

7

Preparation for Calculus – Worksheets 5. (a) Write y

1 3

1 3x 2

as a composition of three functions f, g, and h.

(b) Find formulas for the compositions f  g and g  f given that f(x) = 1 – x2 and g(x) = x 3 .

6. Find the difference quotient of each function. (a) f(x) = 6

(b) f(x) = 3x – 7

8

Preparation for Calculus – Worksheets (c) f(x) = x2

(d) f(x) = x3

(e) f(x) =

1 x

(f) f(x) = 4x2 + 3x – 9

9

10

Preparation for Calculus – Worksheets 7. Sketch the graphs of y y

2

x 4

x, y

x 4 , y

x 4 , y

2

x 4 , and

3 .

8. (a) Sketch the graph of y = 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 so that an inverse can be found.

(b) Sketch the graph of y = 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.

11

Preparation for Calculus – Worksheets Functions in applications 9. For a given outside temperature T in degrees Fahrenheit, the wind chill temperature (WCT) index is the equivalent temperature that exposed skin would feel with a wind speed of v miles per hour. WCT

T , if 0 v 3  0. 16 35 .74 0.6215T 35 .75v

0.4275Tv 0.16 ,

if v

3

Find the WCT to the nearest degree if T = 20  F and v = 15 mi/h.

10. The Surface area S and the volume V of a spherical balloon can be viewed as functions of the radius r of the balloon. That is, S (r)

4 r2

and

V (r)

4 3

r 3.

Find the surface area and volume of a spherical balloon with a 3-inch radius. Explain your answers in terms of the units involved.

12

Preparation for Calculus – Worksheets

Polynomials Worksheet 1. y

2 is a

and has degree 0. It’s graph is a

.

(a) Find the y-intercept.

(b) Find the x-intercept(s).

2. y

4 x 7 is a

and has degree 1. It’s graph is a

and y-intercept ______ . Find the x-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.

with slope ______

Preparation for Calculus – Worksheets 4.

y

16 x2

32 x 48

is a

and has degree 2. It’s graph is a

13 .

(a) Find the y-intercept.

(b) Find the x-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.

y

x3 2 x2

4 x 8

is a

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.

Preparation for Calculus – Worksheets 6.

y

x3 2 x2 5 x 6

is a

14

.

(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.

4 7. y x 4 is a of squares formula.

and has degree 4. Factor the function using the difference

Preparation for Calculus – Worksheets 8. y 9 x 4 81x 3 144 x 2 quadratic formula.)

is a

15

. Find the roots. (You will eventually need the

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.

Preparation for Calculus – Worksheets 11. Even though some functions are not polynomials, we can use similar techniques. (a) Factor the function f ( x)

6x

(b) Factor the function g ( x)

2x

1

3x

3

2

3

4

4x

3

1 3

. (Factor out the smallest power of x.)

, and then find its domain and intercepts.

16

Preparation for Calculus – Worksheets Polynomials in applications: 12. Let s( )t dropped.

16 t2 144 be the position in feet of a falling object t seconds after it was

(a) Find the height from which the object was dropped.

(b) At what time did the object hit the ground?

13. Let v(t) 9.8t 24.5 be the velocity in meters per second of a moving object t seconds 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?

17

Preparation for Calculus – Worksheets

Trigonometry Worksheet 1. (a) Convert 75  and 225 to radians.

(b) Convert

15

and

7 to degrees. 9

2. Fill in the table from memory. cos θ

θ 0

0

6

30

4

45

3

60 

2

sin θ

tan θ







90 180 

3 2 5 6

150

4 3

240



270



3. Given that tan θ = 3, find the exact values of the remaining five trigonometric functions of θ. [Hint: draw the appropriate triangle.]

4. Find the cosine, sine, and tangent of θ. (a)

(b) 5

7

θ

θ 2

4

18

Preparation for Calculus – Worksheets 5. (a) Find all values of θ between 0 and 2π (in radians) such that 4sin2θ – 2 = 0.

(b) Find all values of θ between 0 and 2π (in radians) such that sin θ = cos θ.

6. Find the difference quotient of f(x) = sin x.

19

Preparation for Calculus – Worksheets Trigonometry in applications: 7. A 10-foot ladder leans against a house and makes an angle of 60  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 radar station 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

20

Preparation for Calculus – Worksheets

Trigonometric Functions Worksheet 1. Find the amplitude, period, frequency, and phase shift. Then sketch a graph showing at least two periods. (a) y

(b) y

3 cos( 4 x)

2 sin( 3x

2

) 2

[Notice the extra vertical shift.]

21

Preparation for Calculus – Worksheets   2. (a) cos 1  3   2   

__________

(b) sin 1 (1)

__________

(c) tan 1 (1)

__________

(d) sin 1 

1  2

__________

3. Find θ. (a) 2 θ 1

(b) 50 θ 20

22

Preparation for Calculus – Worksheets 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 y(t )

0.5 cos(1.3t ) meters,

t seconds 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 V 120 2 volts (V) at a frequency of 60 hertz (Hz). Write an equation that expresses V as a function of the time t, assuming that V = 0 if t = 0. [Note: 1 Hz = 1 cycle per second.]

23

Preparation for Calculus – Worksheets 6. A soccer player kicks a ball with an initial speed of 14 m/s at an angle θ with the horizontal see the figure below. The ball lands 18 m down the field. If air resistance is neglected, then the ball will have a parabolic trajectory and the horizontal range R will be given by 2 R v sin 2 g

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 θ, to the nearest degree, at which the ball could have been kicked.

θ R

24

Preparation for Calculus – Worksheets

Exponential and Logarithmic Functions Worksheet 1. On the set of axes below, sketch and label the graphs of y = bx for bases b = 1, 2, 3, and ½.

1

–1

1

2. Find the exact value of each expression without a calculator. (a) ( 8)

(c) 9

1

2 3

2

(b)

2

8

3

(d) log 2 32

(e) log10 (0.01)

(f) log10 1000

(g) ln e 3

(h) (ln e)3

25

Preparation for Calculus – Worksheets 3. Most calculators do not have a key to evaluate log 2 15 . Use the change of base formula to convert to base e first.

4. Solve each equation for x. (a) log10 (1

x)

2

(b) ln(4 x) 3 ln( x2) ln 2

(c) 5

2x

3

(d) 2 exp(3x ) 7

26

Preparation for Calculus – Worksheets

27

Exponential and logarithmic functions in applications: 5. The “loudness” of a sound can be measured by its I (in watts per square meter), which is related to the energy transmitted by the sound wave—the 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 of β (in decibels dB):  I  10 log  dB 12  10  Damage to the average ear occurs at 90 dB or greater. Find the decibel level of each of the following sounds and state whether it will cause ear damage.

Sound

Intensity I

(a) Jet aircraft from 50 ft

1.0×102 W/m2

(b) Amplified rock music

1.0 W/m2

(c) Garbage disposal

1.0×10–4 W/m2

–5 2 (d) TV mid volume from 10 ft 3.2×10 W/m

Preparation for Calculus – Worksheets

28

6. The equation Q 12e 0.055 t gives the mass Q in grams of radioactive potassium-42 that will remain from some initial quantity after t hours of radioactive decay. (a) How many grams were there initially?

(b) How many grams remain after 4 hours?

(c) What is the 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

 A exp 

1 t

Q   is rewritten as a linear RT

function of 1/T, namely, ln t

a

1 T

b

where a is the slope and b is the (extrapolated) vertical axis intercept. Find the slope a and intercept b in terms of A, Q, and R.

Preparation for Calculus – Worksheets

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)

(a) y

(b) y

intervals of positive and negative

5 x 2

x 4 x2 9

29

Preparation for Calculus – Worksheets

(c) y

3 x3 1 x2 2

(d) y

x2 3x 2

x 2 9x 6

30

31

Preparation for Calculus – Worksheets

2. Perform long division to rewrite the improper rational function y

3x 3 1 . x2 2

Rational functions in applications: 3. According to Coulomb’s law, the magnitude of the electrical force F between two charged particles with charges Q1 and Q2 is inversely proportional to the square of the distance d between them. That is, F kQ 1Q 2 . d2 Describe the force as the particles get closer and closer together.

Describe the force as the particles get further and further apart.

32

Preparation for Calculus – Worksheets

Limits Worksheet 1. Use the graph of y = f(x) to fill in the blanks. y f(x)

6 5 4 3 2 1 –3

–2

–1

0

1

2

3

4

5

6

x

–1 –2 –3

(a) f(–2) ________

(b) f(1) ________

(c) f(3) ________

(d) f(4) ________

x

lim f x( )________ 2

limf (x )________

x 1

x

lim f (x )________ 3

limf x( )________

x 4

x

limf x( )________

limf x( )________

x 1

limf x( )________

x 3

lim f (x )________

x 4

x

2

limf x( )________ 2

limf (x )________

x 1

limf (x )________

x 3

limf x( )________

x 4

33

Preparation for Calculus – Worksheets 2. Use the graph of y = g(x) to fill in the blanks. y 6

g (x )

5 4 3 2 1 –3

–2

–1

0

1

2

3

4

5

6

x

–1 –2 –3

(a) g(–1) ________

(b) g(1) ________

(c) g(2) ________

(d) g(5) ________

x

x

x

x

limg x( )________ 1

limg (x )________ 1

limg (x )________ 2

limg (x )________ 5

x

limg x( )________

limg (x )________

x 1

limg x( )________

x 2

limg x( )________

x 5

x

1

limg x( )________ 1

limg (x )...