ap calculus summer packet - atlanta public schools calculus summer packet name: _____ polynomial...

AP Calculus Summer Packet Name: _______________________

Instructor: Mr. Andrew Nichols

Email questions to [email protected]

Help Session: Monday, July 31, 3:30 – 5:30

Due Date: Tuesday, August 1

Prerequisites Quiz: Thursday, August 3

Instructions: Work on this packet over the summer. This packet is due on the first day of class

and will be graded for completion and accuracy. This is your first graded assignment of the fall

semester in AP Calculus. You must SHOW YOUR WORK to receive credit. You must

complete this packet and pass the Prerequisites Quiz to keep your position in AP Calculus. If

you fail to complete this packet or fail the Prerequisite Quiz, I will recommend that you drop AP

Calculus. The answers to all of these problems are attached. CHECK YOUR ANSWERS.

Strategies for Success:

1) Start working on this packet NOW while you have the support of your PreCalculus teacher.

2) Review the entire packet first and identify problems that you don’t recognize or have

struggled with in the past.

3) Identify a study group that you can work with over the summer either in person or online.

4) Ask a more experienced student for help.

5) Use internet resources like Khan Academy, CoolMath, and YouTube.

6) Email Mr. Nichols if you get stuck and need help.

7) Don’t procrastinate. This packet will take longer than you expect to complete and putting it

off only makes it more stressful.

Submission: Write the answers to problems 1 – 7 in this packet. Work all other problems

on your own paper. Organize your work and answers so that they can be easily graded.

When you hand in your work, remove problems 1 – 7 from the packet and staple them to

your work and answers for the remaining problems.

“We are what we repeatedly do. Excellence, therefore, is not an act, it is a habit. Who are you?

What do you want to be? Look at the habits of your heart, your mind, and your behavior. You!

Me! We are what we repeatedly do. Where our attention goes, we go!” - Unknown

“Success is a lousy teacher. It seduces smart people into thinking they can’t lose.” – Bill Gates

AP Calculus

Summer Packet

mailto:[email protected]


SKILLS NEEDED FOR CALCULUS A working knowledge of these topics is important for success in AP Calculus. Items marked

with and asterisk (*) are vital. The practice problems at the end of this packet cover these topics.

I. Algebra A. *Exponents (operations with integer, fractional, and negative exponents)

B. *Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes,

grouping)

C. Rationalizing (numerator and denominator)

D. *Simplifying rational expressions

E. *Solving algebraic equations and inequalities (linear, quadratic, higher order using

synthetic division, rational, radical, and absolute value equations)

F. Simultaneous equations

II. Graphing and Functions A. *Lines (intercepts, slopes, write equations using point-slope and slope intercept,

parallel, perpendicular, distance and midpoint formulas)

B. Conic Sections (circle, parabola, ellipse, and hyperbola)

C. *Functions (definition, notation, domain, range, inverse, composition)

D. *Basic shapes and transformations of the following functions (absolute value,

rational, root, higher order curves, log, ln, exponential, trigonometric. piece-wise,

inverse functions)

E. Tests for symmetry: odd, even

III. Geometry

A. Pythagorean Theorem

B. Area Formulas (Circle, polygons, surface area of solids)

C. Volume formulas

D. Similar Triangles

IV. Logarithmic and Exponential Functions

A. *Simplify Expressions (Use laws of logarithms and exponents)

B. *Solve exponential and logarithmic equations (include ln as well as log)

C. *Sketch graphs

D. *Inverses

* V. Trigonometry

A. *Unit Circle (definition of functions, angles in radians and degrees)

B. Use of Pythagorean Identities and formulas to simplify expressions

C. and prove identities

D. *Solve trigonometric equations

E. *Inverse Trigonometric functions

F. Right triangle trigonometry

G. *Graphs

VI. Limits

A. Concept of a limit

B. Find limits as x approaches a number and as x approaches ∞


Toolkit of Functions In Calculus it is expected that students know the basic shape of many functions and be able to

graph their transformations without the assistance of a calculator.

(1) For each function below, accurately sketch its graph and give its domain and range.

Constant: y c

D: ___________________

R: ___________________

Linear: y x

D: ____________________

R: ____________________

Quadratic: 2y x

D: ___________________

R: ___________________

Cubic: 3y x

D: ___________________

R: ___________________

Square Root: y x

D: ___________________

R: ___________________

Cube Root: 3y x

D: ___________________

R: ___________________

Reciprocal: 1y x

D: ___________________

R: ___________________

21y x

D: ___________________

R: ___________________

2 3y x

D: ___________________

R: ___________________

3 2y x

D: ___________________

R: ___________________

21 1y x

D: ___________________

R: ___________________

Absolute Value: y x

D: ___________________

R: ___________________


Greatest Integer:𝑦 = ⟦𝑥⟧

D: ___________________

R: ___________________

Semicircle: 21y x

D: ____________________

R: ____________________

Sine: siny x

D: ___________________

R: ___________________

Cosine: cosy x

D: ___________________

R: ___________________

Tangent: tany x

D: ___________________

R: ___________________

Exponential: xy a , 1a

D: ___________________

R: ___________________

Exponential: xy a , 0 1a

D: ___________________

R: ___________________

Natural Exponential xy e

D: ___________________

R: ___________________

Logarithmic

logby x , 1b

D: ___________________

R: ___________________

Logarithmic

logby x , 0 1b

D: ___________________

R: ___________________

Natural Log

lny x

D: ___________________

R: ___________________


Polynomial Functions A function P is called a polynomial if -1 2

1 2 1 0( ) ...n n

n nP x a x a x a x a x a

Where n is a nonnegative integer and the numbers 0 1, , na a a are constants.

Even degree Odd degree

Leading coefficient sign Leading coefficient sign

Positive Negative Positive Negative

Number of roots equals the degree of the polynomial.

Number of x intercepts is less than or equal to the degree.

Number of “bends” is less than or equal to (degree – 1).

Roots with even multiplicity “bounce” off of the x-axis.

Roots with odd multiplicity cross the x-axis.

Rational Functions

Locating Asymptotes

(2) Use the following rational functions to investigate asymptotes. Before you begin,

(a) Factor the numerator and denominator of each function,

(b) Determine the domain for each, and

(c) Graph the function on your calculator in the standard viewing window and

sketch the graph on the coordinate axis.

2( )

1

xf x

x

D: ________________

2

2

9( )

6

xg x

x x

D: ________________

3

2( )

1

x xh x

x

D: ___________________

4 2

3 2

2( )

x xk x

x x

D: ________________

AP Calculus Summer Packet Name: __________________

**In the notes below “→” means “approaches.” So 2x is read “x approaches 2.”

The statement y means that y grows without bound or y “goes to positive infinity.”

Vertical Asymptotes Each domain restriction is the location of either

(a) a point discontinuity (i.e. hole in the graph) or

(b) a vertical asymptote (a vertical line that, as the function approaches the line, y ).

(3) Classify each domain restriction as either a vertical asymptote or point

discontinuity by examining the graph of the function:

( )f x

Point discontinuities at x ________

Vertical asymptotes at x _________

( )g x



( )h x



( )k x



(4) How could you tell the difference between a point discontinuity and a vertical

asymptote with out looking at the graph and just examining the equation of the

function? Write your answer in the space provided below.

____________________________________________________________________________________________________________

____________________________________________________________________________________________________________

Horizontal Asymptotes The end behavior of a function y f x determines its horizontal asymptotes (a

horizontal line that the function approaches as x ).

(a) If a function has infinite end behavior, then it has no horizontal asymptotes, but

(b) If y c as x , then y c is a horizontal asymptote for ( )y f x .

(5) Determine the end behavior of each function and write the equations of any

horizontal asymptotes.

In the function f x :

As x , y ____

As x , y ____

Horizontal asymptotes: y _______

In the function g x :

As x , y ____

As x , y ____


In the function h x :

As x , y ____

As x , y ____


In the function k x :

As x , y ____

As x , y ____



Summary of Horizontal Asymptotes for Rational Functions

Let ( )

( )( )

p xf x

q x be a rational function, i.e., ( )p x and ( )q x are polynomials.

Let degree of ( )n p x and degree of ( )m q x .

Let leading coefficient of ( )a p x and leading coefficient of ( )b q x .

(a) If n m , then ( )f x has the horizontal asymptote 0y .

(b) If n m , then ( )f x has the horizontal asymptote a

yb

.

(c) If n m , then ( )f x has no horizontal asymptote.

(6) Make a detailed sketch of the graph of3

3

2 8( )

2

x xf x

x x

without the use of your

graphing calculator. Attach your sketch to back of this packet.

Trigonometric Formulas

Arc Length of a circle: L = r or L = d

r360

2

Area of a sector of a circle: Area = 21

2r or Area =

dr

360

2

Solving parts of a triangle:

Law of Sines: a

A

b

B

c

Csin sin sin

Law of Cosines: a b c bc A2 2 2 2 cos

b a c ac B2 2 2 2 cos

c a b ab C2 2 2 2 cos

Area of a Triangle:

Area = 1

2bc sinA or Area =

1

2ac sinB or Area =

1

2ab sinC

Hero's formula: Area = s s a s b s c( )( )( ) , where 1

2s a b c


Trigonometric Identities

You are expected to memorize the Reciprocal, Quotient, and Pythagorean Identities.

Reciprocal Identities:

csc A = 1

sin A sec

cosA

A

1 cot A =

1

tanA

Quotient Identities: tansin

cosA

A

A cot A =

cos

sin

A

A

Pythagorean Identities:

sin2A + cos2A = 1 tan2A + 1 = sec2A 1 + cot2A = csc2A

You do not need to memomize any of the following identities, but you do need to know how to use

them.

Sum and Difference Identities:

sin(A + B) = sinA cosB+cosA sinB sin(A – B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB

tan (A + B) = tan tan

tan tan

A B

A B

1 tan (A – B) =

tan tan

tan tan

A B

A B

1

Double Angle Identities:

sin(2A) = 2sinA cosA tan(2A) = 2

1 2

tan

tan

A

A

cos(2A) = cos2A - sin2A cos(2A) = 2cos2A – 1 cos(2A) = 1 – 2sin2A

Half Angle Identities:

Geometric Area Formulas: You are expected to memorize these formulas.

Area of a trapezoid: 1 2

1

2A h b b Area of a triangle:

1

2A bh

Area if a circle: 2A r Circumference of a circle: 2 or C r C d

1 costan

2 1 cos

A A

A

1 cos

cos2 2

A A

1 cossin

2 2

A A


Unit Circle You are expected to memorize the First Quadrant of the Unit Circle and be able to extend

that knowledge to the other quadrants as needed.

The sixteen points marked on the Unit Circle below correspond to quadrantal angles, special angles,

and their multiples.

(7) Label each point with

(a) the corresponding positive angle measure between 0° and 360°,

(b) positive radian measure between 0 and 2π, and

(c) its (x,y)-coordinates.

(8) For an angle θ in standard position whose terminal side intersects the Unit Circle at (x,y),

sin θ = cos θ = tan θ =

csc θ = sec θ = cot θ =


THE REMAINING PROBLEMS IN THIS PACKET SHOULD BE WORKED ON YOUR OWN

PAPER AND ATTACHED TO THE PACKET WHEN HANDED IN.

(9) Recall the properties of exponents:

(i) b1 = b

(ii) If n > 1, nb b b b b , n times.

(iii) 0 1b

(iv) m n m na a a

(v) , where 0

m m

m

a ab

b b

(vi) , where 0

m m

m

a ab

b b

(vii) If 0b , 1n

nb

b

(viii) 1

, where 0n

n m

m m n

aa a

a a

(ix) If 0 and 1b n , 1

nnb b .

(x) If 0b and m and n are integers

with n >1, m

mn m nnb b b

Simplify and write without using negative exponents:

(a) 3

3 2xy

(b)

1

2 4

5xy

x y

(c)

133 3

11 2

3 3

8 2

4

x yz x

x yz

(10) Express using a single radical and positive integer powers: 1 4 3 4 1 46 x y

(11) Express using positive rational exponents and without radicals: 410 2 25 x y z

(12) Expand each expression and combine like terms:

(a) 2

a b (b) 2

a b (c) 2

2 1x y

(13) Factor completely

(a) 29 3 3x x xy y Hint: use grouping

(b) 664 1x Hint: factor as a difference of squares first, then as the sum and difference of cubes.

(c) 4 242 35 28x x

(d) 5 3 1

2 2 215 2 24x x x Hint: first factor out the GCF: 1

2x .

(e) 1 2 33 2x x x Hint: first factor out the GCF: 3x .

(14) Rationalize the denominator: 3

1 2

x

x

(15) Rationalize the numerator: 1 1x

x

(16) Simplify:

3 2

4

1 2 3 1

1

x x x

x

(17) Use Rational Root Theorem and synthetic division to help factor the following polynomial functions

completely. State all of the factors and roots.

(a) 3 24 6p x x x x (b) 3 26 17 16 7q x x x x


(18) Use the Rational Root Theorem to explain why 3

2 cannot be a root of 5 34 5f x x cx dx ,

where c and d are integers.

(19) Explain why 4 27 5g x x x x must have a root in the interval 0,1 . Examine the graph and

use the signs of 0g and 1g to justify your answer.

(20) Solve the following equations and inequalities algebraically. Check your solutions by graphing.

(a) 2

3 4x

(b) 5

03

x

x

(c) 3 23 14 5 0x x x

(d) 1

xx

(e) 2 9

01

x

x

(f) 1 4

01 6x x

(g) 2 1 1 4x

(21) Solve each system of equations algebraically and then check your answer by graphing.

(a) 2

1 0

5

x y

y x

(b)

2

2

4 3

6 9

x x y

x x y

(22) Write the equation for the line described by the given information.

(a) Passes through the point 2, 1 and has slope 1

3 .

(b) Passes through the point 4, 3 and is perpendicular to 3 2 4x y .

(c) Passes through 1,2 and is parallel to 3

15

y x .

(23) A small-appliance manufacturer find that it costs $9,000 to produce 1,000 toaster ovens per week

and $12,000 to produce 1,500 toasters per week.

(a) Express the cost as a linear function of the number of toaster ovens produced and sketch its

graph.

(b) What is the slope of the graph and what does it mean in the context of this problem?

(c) What is the y-intercept of the graph and what does it mean in the context of this problem?

(24) Find the domain and range of each function.

Recall that domain restrictions may result from any of the following

(i) Denominator must be 0

(ii) Argument (input) for a log must be > 0

(iii) Radicand (input) for an even root must be > 0

For range restrictions, use algebraic reasoning and, if all else fails, your graphing calculator.

(a) 3

2y

x

(b) log 3y x

(c) 4 2 2y x x

(d) 2 3y x

(e) 5y x (f)

2

1

1

xy

x


(25) Sketch the graph of the piecewise function f x given below on the interval 3,3 . What is the

range of f on this domain?

if 0

1 if 1 0

2 if 1

x x

f x x

x x

(26) Use the functions f, g, and h defined below to compute each inverse or composition.

2 3 2f x x x 4 3g x x lnh x x 4w x x

(a) 1g x

(b) 1h x

(c) 1w x for 4x

(d) ( )f g x

(e) (1)h g f

(27) A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60

cm/second.

(a) Express the radius r of this circle as a function of time t (in seconds).

(b) If A is the area of the circle as a function of the radius, find A r and explain its meaning.

(28) Does 23 9y x have an inverse function? Justify your answer.

(29) Let 2f x x , g x x , and 4h x then find

(a) f g x

(b) f g h x

(30) Let 4s x x and 2t x x then find the domain and range of s t x .

(31) Express the function 1

F xx x

as the composition of three functions f, g, & h so that

F x f g h x .

(32) Use your Toolkit of Functions and your knowledge of transformations to sketch the graphs of the

following functions without using your calculator (except to check your answer).

(a) 2y x

(b) 1

2y

x

(c) 2 4

xy

x

(d) 2 xy

(e) 3sin 26

y x

(f) ln 1 1y x

(g)

2

2

25 0

250, 5

5

0 5

x x

xf x x x

x

x

(33) Identify each function as odd, even, or neither and justify your answer algebraically.

Use these algebraic tests:

(i) f is and even function if and only f x f x for all x.

(ii) f is and odd function if and only f x f x for all x

(a) 3 3f x x x

(b) 4 26 3f x x x

3

2

x xf x

x

(c) sin 2f x x

(d) 2f x x x (e) 2

1 xf x

x


(34) Fill in each blank with either “even” or “odd.”

(a) The product or quotient of even functions is _________.

(b) The product or quotient of odd functions is _________.

(c) The product or quotient of an even and an odd function is __________.

(35) The graph of an equation is symmetric with respect to the

(i) y-axis if the equation is unchanged when replacing x with –x.

(ii) x-axis if the equation is unchanged when replacing y with –y.

(iii) origin if the equation is unchanged when replacing x with –x AND y with –y.

Test each function for symmetry:

(a) 4 2y x x

(b) siny x

(c) cosy x

(d) 2 1x y (e) 2 1

xy

x

(36) Even functions are symmetric with respect to the ________.

Odd functions are symmetric with respect to the ________.

(37) Recall the properties of logarithms:

(i) log 1 0b

(ii) log 1b b

(iii) log n

b b n

(iv) log log logb b bmn m n

(v) log log logb b b

mm n

n

(vi) log logp

b bm p m

(vii) If log log , then b bm n m n

Simplify each logarithmic expression below using the properties of logarithms.

(a) 4

1log

16

(b) 3 3 3

3 1 13log 3 log 81 log

4 3 27

9log 27

(c) 125

1log

5

(d) 45logw w

(e) ln e

(f) ln1

(g) 2ln e

(38) Solve each equation for x:

(a) 6 6log 3 log 4 1x x (b) 2log log100 log1x (c) 13 15x

(39) Under ideal conditions a certain bacteria population is known to double in size every three hours.

Suppose that there are initially 100 bacteria.

(a) What is the size of the bacteria population after 15 hours?

(b) What is the size of the bacteria population after t hours?

(c) How long will it take the bacteria population to grow to 50,000?


50˚

10 x

x

10

70˚ 70˚

(40) The population of a certain species of tree frogs in a limited environment and initial population of

100 can be predicted using 100,000

100 900 tP t

e

.

(a) Use your calculator to graph this function in an appropriate viewing window. Sketch the graph.

(b) Based on the graph, what seems to be the maximum number of frogs this environment can

accommodate?

(c) Estimate how long it takes for the population to reach 900 frogs.

(d) Find the inverse of this function and explain its meaning.

(e) Use the inverse function to find the time required for the population to reach 900. Compare with

the result of part (c).

(41) Simplify each expression using trigonometric identities:

(a) 2 2

2

tan csc 1

csc tan sin

x x

x x x

(b) 21 cos x

(c) 2 2sec tanx x

(42) Verify the identity 2 21 sin 1 tan 1x x .

(43) Solve each equation for 0 2x :

(a) 2cos cos 2x x (b) 2sin 2 3x (c) 2cos sin 1 0x x

(44) Evaluate each expression. Note that 1sin arcsinx x

(a) 1sin 1 (b) 2

arcsin2

(c) arctan1 (d) 1 3sin cos

2

(45) Sketch the graph of each function and state its domain and range:

(a) Arcsine: 1siny x , (b) Arccosine: 1cosy x

(b) Arctangent: 1tany x

(46) Solve for x in each diagram.

(a) (b)

(47) The rollercoaster car shown in diagram below takes 23.5 seconds to go up the 23 degree incline

segment AH and only 2.8 seconds to go down the curved drop from H to C. The car covers

horizontal distances of 180 feet on the incline and 60 feet on the drop.

(a) What is the maximum height of the rollercoaster above point B?

(b) Find the distances AH and HC.

(c) How fast (in ft/sec) does the car go up the incline?

(d) What is the average speed of the car as it goes down the drop?

(e) Is HC an over- or under- approximation of the true distance traveled

by the car from H to C? 180 ft 60 ft

A

H

B C


(48) Identify the amplitude, period, horizontal shift, and vertical shift of each function compared to its

parent. Sketch the graph of each function on the interval 2 ,2 .

(a) 2sin 2 1y x

(b) 3cos2

y x

(49) Familiarize yourself with the following CALC (2nd TRACE) commands on your TI-84 graphing

calculator:

(i) Root

(ii) Minimum

(iii) Maximum

(iv) Intersect

Given 4 3 22 11 30f x x x x x :

(a) Find an appropriate calculator viewing window that shows all of the important features of the

function. Report Xmin, Xmax, Ymin, and Ymax. Sketch the resulting graph on your paper.

(b) Find the x-coordinates all of the roots of the function accurate to three decimal places.

(c) Find the (x,y)-coordinates of all local maximums accurate to three decimal places.

(d) Find the (x,y)-coordinates of all local minimums accurate to three decimal places.

(e) Find the value of 1f , 2f , 0f , and 0.125f accurate to three decimal places.

(50) Graph the following functions and find their points of intersection using the Intersect command on

your calculator (do not use TRACE). Sketch the graphs on your paper and state your answer

accurate to three decimal places.

(a) 3 2

2

5 7 2

0.2 10

y x x x

y x

(b) sin

2 1

xy e

y x

(c)

2

ln cos

2

y x x

y x


AP Calculus Summer Packet – Answers

You can not copy these answers and receive credit for the summer packet. You must submit complete

solutions, not just answers, to receive credit.

1. All graphs must be neatly sketched with correct domain and range listed. Check your answers on

your graphing calculator.

2. All graphs must be neatly sketched with correct with correct factorizations and domains given.

3. f x has no point discontinuities and has vertical asymptotes at 1x .

g x has a point discontinuities at 3x and has a vertical asymptote at 2x .

h x has point discontinuities at 1x and has no vertical asymptotes.

k x has a point discontinuities at 0x and has a vertical asymptote at 1x .

4. Hint: compare the factors of the numerator and denominator.

5. In f x , as x , 0y , so 0y is a horizontal asymptote.

In g x , as x , 1y , so 1y is a horizontal asymptote.

In h x and k x , as x , y and as x , y , so neither has any horizontal

asymptotes.

6. Neatly sketch the graph. Be sure to label your axes and indicate the positions of all x-intercepts,

vertical, and horizontal asymptotes. Check your graph on your calculator.

7. Label the Unit Circle as instructed. You must be able to reproduce at least the first quadrant

from memory.

8. sin y , cos x , tan y x , csc 1 y , sec 1 x , cot x y . You should memorize

these formulas.

9. a) 3 3

3

8x y

b) 31

5xy

c) 11 3 4 34x y z

10. 34 6x y

11. 2 2 5 1 2x y z

12. a) 2 22a ab b

b) 2 22a ab b

c) 2 4 2 2 22x y x y x

13. a) 3 3 1 9 1 3 1 3x y x x y x

b) 2 22 1 2 1 4 2 1 4 2 1x x x x x x

2 264 1 2 1 2 1 2 1 4 1 2 1 4x x x x x x

c) 2 2 27 3 4 2 1 42 4 3 2 2 2 2x x x x x

13. d) 1 2 1 23 4 5 6 15 4 3 6 5x x x x x x

e) 3 1 2x x x


14. 1 2x

15. 1

1 1x

16.

2

2

1

1

x x

x

17. a) 2 3 1p x x x x

b) 6 7 2 1 3 1q x x x x

18. 3 2 can not be a rational root of f x since 3 is not a factor of 5.

19. Since 0 0g , 1 0g , and g is a continuous function, 0g x for some x in 0,1 .

20. a) 5x or 1x ; in interval notation: , 5 1,

b) 5x

c) 1 3, 0, or 5x

d) 1x or 0 1x ; in interval notation: , 1 0,1

e) 3 0x or 3x ; in interval notation: 3,0 3,

f) 2x

g) 5 8 3 8x ; in interval notation: 5 8, 3 8

21. a) 2, 1 or 3,4

b) 2, 1 or 3,0

22. a) 1 1 3 2y x or 1 3 1 3y x

b) 3 2 3 4y x or 2 3 17 3y x

c) 2 3 5 1y x or 3 5 13 5y x

23. a) 6 3000y x ; sketch the graph neatly being sure to label and scale the axes.

b) The slope, $6 per toaster oven, is the cost of each additional toaster produced.

c) The y-intercept, $3000, is the base cost of operating the factory even if no toasters are

produced.

24. Domain Range

a) ,2 2, ,0 0,

b) 3, ,

c) , 2,

d) 3 2, 0,

e) , 0,

f) 1,1 1, , 1 3 0,

25. Neatly sketch the graph of each piece of the function on its restricted domain. Be sure to scale your

axes. Domain: ,


26. a) 1 1 4 3g x x

b) 1 xh x e

c) 1 2 4w x x for 4x

d) 216 12 2f g x x x

e) 1 ln5 1.609h g f

27. a) 60r t t

b) 23600A r t t which is the area of the circle as a function of time.

28. No, its graph does not pass the Horizontal Line Test. OR No, because, 12y both when 1x and

when 1x (for example).

29. a) 2f g x x b) 8f g h x

30. : 2,2D , : 0,2R

31. F x f g h x where 1f x x , g x x , and h x x x

32. Sketch each graph neatly. Be sure to scale your axes carefully. Check your answers on your

graphing calculator.

33. a) odd ?) odd d) neither f) even

b) even c) odd e) odd

34. a) even b) even c) odd

35. Each function is only symmetric to the:

a) y-axis b) origin c) y-axis d) x-axis e) y-axis

36. y-axis; origin

37. a) 2 b) 1 ?) 3 2 c) 1 3 d) 45 e) 1 f) 0 g)2

38. a) 1x or 6x

b) 10x

c) ln15

1 1.465ln 3

x

39. a) 3200 bacteria

b) 3100 2tP t

c) 3ln500

26.897 hoursln 2

t

40. a) Sketch a copy of the graph on your paper. Be sure to label and scale the axes carefully.

b) Approximately 1000 frogs

c) A little less than 4.5 years.

d) 100,000

ln 100 900t PP

or simplified a bit:

900ln

100,000 100

Pt P

P

e) 900 4.394t years which is a little less than 4.5 years as predicted in (c).

41. a) 1 b) 2sin x c) 1

42. Hint: use the Pythagorean identities to show that the left-hand side simplifies to 1.

43. a) x b) 6, 3,7 6,or 4 3x c) 3 2x

44. a) 2 b) 4 c) 4 d) 1 2


45. Check your graphs on your graphing calculator. Be sure to label your axes.

Domain Range

1siny x 1,1 2, 2

1cosy x 1,1 0,

1tany x , 1,1

46. a) Note: x is the length of the leg opposite the 50 angle. 7.660x

b) 14.619x

47. Note: the path of the roller coaster is along the line segment and then along a curved path from H to

C. This was inadvertently omitted from the diagram.

a) 76.405 feet

b) 97.148 feet

c) 8.321 feet per second

d) 34.696 feet per second

e) Under approximation

48. (a) (b)

Amplitude 2 3

Period 4

Horz. Shift 0 2

Vert. Shift 1 0

49. a) Answers will vary, but should use a window that shows all of the major features of the function:

two local minimums, one local maximum, four x-intercepts, and both positive and negative end

behaviors. Sketch and label your graph carefully on your paper.

b) x-intercepts at 1.5, 0, 2, and 5x

c) 1.067,20.101

d) 0.890, 18.483 and 3.948, 88.155

e) 1 18f , 2 0f , 0 0f , 0.125 3.713f

50. Include neat, labeled sketches of each system with your answers.

a) 5.772,16.663 , 0.787,10.124 , and 1.760,10.620

b) 1.818,2.637

c) 1.432,0.050 and 0.135, 1.982

ap calculus summer packet - atlanta public schools calculus summer packet name: _____ polynomial...

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