dear calculus bc student, - key school

Advanced Calculus BC Summer Work Due: 1st Day of School

Dear Calculus BC student,

I hope that you’re all enjoying your first few days of summer! Here’s something that will make it a little more fun! Enclosed you will find a packet of review questions that you should complete before the first day of classes. This packet covers mainly Algebra I, Algebra II, and Precalculus skills. These are skills that we will work on having down pat so that you won’t even have to think about, “how do I solve this problem?” The packet will be graded upon your return to school and will be worth 4 homework assignments (a week’s worth of assignments). Solutions are also attached, but I expect to see full work for each problem.

There will be a quiz covering this material on the second or third day of classes.

Many of you may want to complete this as soon as possible and have it over and done with. If this is the case, please be sure to review your solutions in the days before school starts. The aim of this summer work is to keep your mathematical mind from rusting in the months that you’ll spend away from school. We’ll be hitting the ground running in August and you don’t want to be out of breath. Enjoy!

E-mail any questions: [email protected]

See you all in August!

Mrs. Bowman

If you come across a topic with which you are uncomfortable, you may find the following websites useful:

http://www.khanacademy.org http://www.coolmath.com/ http://www.purplemath.com/

Help is also available in your Precalculus notebook or textbook. Did you save your notes?

The following problems involve the skills that you will be using throughout the year in Calculus BC. Let’s see what you remember! We will answer some questions on the first day of class, but your goal is to answer your own questions before class starts. You can phone a friend, google, use khanacademy.org, use your Precalculus notes/textbook etc. Be honest with yourself and how comfortable you are with these skills. We’ll discuss in August!

http://www.khanacademy.org/

http://www.coolmath.com/

http://www.purplemath.com/

mailto:[email protected]?subject=custom subject


Part 1: Simplifying

Simplify each expression completely.

1.

1 14 4xx

�� 2. Rationalize the numerator:

1x xx

� �

3. 2 5 6

1x xx� ��

4. 2 2( 3) ( 1)

2x

x� � �

�

5. 2 2sin cos

cosx x

x�

6. � �sin 2 cos1 2sinT T

T�

�

7. Expand: � �32 1

log2

x xx

§ ·�¨ ¸¨ ¸�© ¹

8. Condense: � �13ln 1 ln5 2ln

2x x� � �ª º¬ ¼

9. � �42x 10. 13

t

t


11. 3

1 2x

x x�

� �12.

� � � �1 1

2 2 22 2

2

1 1

1

x x xx

��

�

13. 8!6!

14. 10!2! 8!

15. � �1 !

!nn�

16. � �� 2 1 !2 3 !nn��

17. Factor the following expressions completely.

a.) 24 81x � b.) 38 125x �

c.) 3 24 8 25 50x x x� � � d.) 3 1

4 42 20x x��


18. Write the sum using sigma notation.

a.) 2 3 171 x x x x� � � � � b.) 2 4 6 404 4 4 4 4x x x x� � � � �

19. Rewrite the trigonometric expression: 3

tan arccosx

§ ·§ ·¨ ¸¨ ¸© ¹© ¹

as an algebraic expression.

Part II: Evaluating

1. Fill in and memorize the values in the following table. You should be able to come up with thesevalues without having to draw the unit circle each time.

Angle

(degree) Angle (radians) sin(x) cos(x) tan(x)

0

30

45

60

90


2. Find exact values for each of the following trigonometric expressions.

a.) cos2S§ ·¨ ¸© ¹

b.) 11

sin6S§ ·

¨ ¸© ¹

c.) 13

sin4S§ ·�¨ ¸

© ¹

d.) 1 1cos

2� § ·¨ ¸© ¹

e.) 1 2sin

2� § ·

�¨ ¸¨ ¸© ¹

f.) � �1tan 3�

g.) 3

sin arctan4

§ ·¨ ¸© ¹

h.) 5

cot arcsin13

§ ·¨ ¸© ¹

3. Perform the following polynomial division:

a.) � � � �5 4 34 7 1 2x x x x x� � � � y � using synthetic division

b.) � � � �5 4 3 2 32 4 1x x x x x x� � � � � y � using long division.


4. Calculate the volume of each:

a.) Outer radius: 5 cm, Inner radius: 2 cm, height: 10 cm

b.) Cone with height: 8 in, Base Circumference of 10S in

5. Evaluate the following sums:

a.) � �5

2

0

2 1n

n n

� �¦

b.) If 0

12

2

n

n

f

§ · ¨ ¸© ¹

¦ , find 2

12

n

n

f

§ ·¨ ¸© ¹

¦ .

c.) If 5

2 1284

3 81

n

n

f

§ · ¨ ¸© ¹

¦ , find 2

24

3

n

n

f

§ ·¨ ¸© ¹

¦ .


Part III: Graphing

1. Sketch each of the following functions and state their domain and range. Sketch any asymptotes. You should have these functions and their characteristics memorized.

y x y x

2y x y x

3y x 3y x

1yx

2

1yx


xy e � �lny x

� �siny x � �cosy x

� �tany x � �arctany x


2. Sketch the following graphs on the axes provided.

a.) x

yx

b.) 29y x �

c.) 2

2 5

( ) 4

5

xf x x

x

�°

�®° �¯

33 2

5

xx

x

� �� d �t

d.) 2 4y x �

y

x

y

x

y

x

y

x


3. Given the graph of f(x), sketch the following:

a.) � �1f x � b.) � �f x� c.) ( )f x

4. Identify any asymptotes (vertical, horizontal, or slant) of each of the following functions:

a.) 2

( )3

f xx

�

b.) � ��

2

2 1 2( )

1x x

f xx� �

�

5. Sketch the following polar functions.

a.) 5r b.) 3ST


c.) 2cosr T d.) 1 2sinr T �

Part IV: Solving

1. Solve for the indicated variable.

a.) A P nrP � for P b.) 2 2x yd y xd� � for d

2. Solve for x.

a.) 24 1 10xe � � b.) � �ln 1

45x �


c.) � �log log 6 2x x� � d.) .5

10050

1 2 xe�

�

e.) 2sin 1 0T � on > @0,2S f.) � �cos 1 2sin 0T T� on > @0,3S

g.) 2tan 2 3T h.) 3 5 1x� �

Part V: Applications

1. Calculate the average rate of change of f on the given interval.

a.) the graph of f is given. Find the average rate of change of f on the interval > @4,2�


b.) 2( ) 5 2 3f x x x � � on > @2,6

c.) 2( ) 5 2 3f x x x � � on > @,a a h�

2.

3.


4. Using a graphing calculator, find any relative extrema (min’s and max’s) of the polynomial: 4 3 2( 3 0) 3 5 6 1x xx xf x� � � � .

5. Determine from the graph whether f has a minimum on the interval. Write yes or no for each graph.

6. Calculate the area of the shaded region.


7. A curve is traced by a point � �,P x y which moves such that its distance from the point � �1,1A � is

three times its distance from the point � �2, 1B � . Determine the equation of the curve.

8. Find an exponential model of the form bxy ae that passes through the points � �2,3 and � �5,7 . Use

the model to predict the y-value when 6x .

9. For each function, find the difference quotient: ( ) ( )f x h f x

h� �

. Simplify as much as you can.

a.) 3( ) 4f x x x � b.) ( ) 2 3 1f x x � �

c.) 2

( )1

f xx

�

d.) ( ) sinf x x (HINT: Use addition identity for sine)


10. Write the partial fraction decomposition of:

a.) 2

1( )

1f x

x

�b.)

2

3

12 12( )

4x xf xx x� �

�

11. Complete the square for the following quadratic expressions. Write in the form: � �2a x h k� �

a.) 2 6x x� b.) 22 18x x� c.) 210x x�

12. Find the nth term of the sequence starting at 0n

a.) 3,7,11,15, b.) 2,6,18,54,

c.) 3 5 7

1, , , ,2 6 24


13.

14. A particle is moving along a diagonal line. Its speed in both the horizontal and vertical directions are given. Calculate the speed of the particle in the direction in which it is traveling.

15. Write a formula for the area of a sector of a circle of radius r with given angle T : A = ____________

Find the area of the sector shown here:

16. Convert the following coordinates to either rectangular or polar form.

a.) � �5,9 o polar coordinates b.) 7,6S§ ·o¨ ¸

© ¹rectangular coordinates


ANSWER KEY

Part I: Simplifying

1.� �

14 4x

��

2.� �

1

1x x x�

� �3. 6x� 4. 4x� 5. sec x 6. cosT�

7. � � � �2log 3log 1 log 2x x x� � � � 8. � �3

2

5 1ln

xx

§ ·�¨ ¸¨ ¸© ¹

9. 416x 10. 2

3t

11.� ��

2 2 61 2

x xx x

� ��

12. � �

32 2

1

1 x� 13. 56 14. 45 15. 1n� 16.

� �� 1

2 3 2 2n n� �

17. a.) � �� 2 9 2 9x x� � b.) � �� 22 5 4 10 25x x x� � � c.) � �� 2 2 5 2 5x x x� � �

d.) � �1

42 10x x� � 18. a.)17

0

n

n

x ¦ b.)

202

0

4 n

n

x ¦ 19.

2 93x �

Part II: Evaluating

1.

2. a.) 0 b.) 12

� c.) 2

2d.)

3S

e.) 4S

� f.) 3S

g.) 35

h.) 125

3. a.) 4 3 2 896 13 26 45

2x x x x

x� � � � �

�b.)

22

3

31

1xx xx�

� � ��

4. a.) 3210 cmS b.) 32003

inS 5.

a.) 79 b.) 12

c.) 163


Part III: Graphing

1.


2. a.) b.)

c.) d.)


3.

4. a.). . : 3. . : 0v a xh a y

b.)

. . : 1. . : 2: 1,1

v a xh a yD x

�

z �

5.


Part IV: Solving

1. a.)1APnr

�

b.) 2

2x ydx y

�

�2. a.) 2.811x | b.) 20 1x e �

c.) 60099

x d.) 1.386x | e.) 7 11

,6 6S ST f.)

5 3 13 5 17, , , , , ,

6 2 6 2 6 2 6S S S S S S ST

g.) ,3

n n ZST S � � h.) 4

23

x� �

Part IV: Applications

1a.) 13

� b.) 42 c.) 10 5 2a h� � 2. a.)rtx th

� b.) 2 2

rtxr h

�

3. a.) 14S

� b.) 4P r rS � c.) 294mS

d.) 100 5km e.) 30T

4. rel min: ( 0.87) 4.69, (1.094) 10.949f f� , rel. max: (0.526) 11.566f

5. a.) yes, b.) no, a.) no, b.) yes 6. 9

242S

� 7. 2 28 8 38 20 43 0x y x y� � � �

8. 0.2821.705 xy e ; (6) 9.284y | 9. a.) 2 23 3 4x xh h� � � b.) 2 3 2 3x h x

h� � � �

c.) � ��

21 1x h x�

� � �d.)

sin cos( ) sin( )cos sinx h h x xh

� �10. a.)

1 1 12 1 1x x§ ·�¨ ¸� �© ¹

b.) 3 5 1

2 2x x x�

� ��

11. a.) � �23 9x � � b.) 29 81

22 2

x§ ·� �¨ ¸© ¹

c.) � �25 25x� � �

12. a.) 4 3na n � b.) � �2 3 nna c.)

� �2 1

1 !nnan�

�

13. a.) 2 3 2y x x � �

b.) � � � �6 31 1y x x � � � c) 2 2 1x y� 14. 130ms 15. 212

A r T , 25A

16. a.) � �106,1.064 b.) 7 3 7

,2 2

§ ·¨ ¸¨ ¸© ¹

dear calculus bc student, - key school

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