Calculus Summer Assignment.docx - Home - Abraham …achs.roselleschools.org/UserFiles/Servers/Server_31540… · · 2017-06-15MATHACHS CALCULUS SUMMER ASSIGNMENT JUNE , 2017

CALCULUS SUMMER ASSIGNMENT JUNE, 2018 Page 1

Calculus Summer Assignment

The purpose of this assignment is to have you practice the mathematical skills necessary to be successful in Calculus. All of the skills covered in this packet are from Pre-Calculus, Algebra 2, and Algebra 1. The material covered is from our district approved, Pre-calculus, textbook. If you need to, you may use reference materials to refresh your memory (old notes, textbooks, online resources, etc.). While graphing calculators will be used during a few tests and quizzes, the majority of in class assessments are non-calculator. You are encouraged to learn how to be calculator-independent. At the end of this page, there are links to some suggested online calculators.

Calculus is a fast-paced course that is taught at the college level to prepare you for AP Calculus. There is a lot of material in the curriculum that must be covered before the end of the year. Therefore, we cannot spend a lot of class time re-teaching prerequisite skills. This is why you have this packet. Spend some time with it and make sure you are clear on everything covered in this packet so that you will be successful in Calculus. Of course, you are always welcomed to seek help from your teacher if necessary.

For assistance with the packet you may contact me at [email protected]. Emails may take a few days during summer for a response. Please be specific in your email for what you need assistance with, include the section and the question number as well. For each question in the packet, there is an example in the separate tutorial packet. The tutorial packet is posted online on our school’s website. Reference to the corresponding section is listed under each question. Calculators Links

Online Calculator

https://www.desmos.com/calculator

https://mathway.com/graph

Emulator for Download

https://wabbit.codeplex.com/

http://lpg.ticalc.org/prj_tilem/download.html

Good Luck!

mailto:[email protected]

https://www.desmos.com/calculator

https://mathway.com/graph

https://wabbit.codeplex.com/

http://lpg.ticalc.org/prj_tilem/download.html


1) Write an equation of a line through the point (a) parallel to the given line and (b) perpendicular to the given line: (Tutorial Section I)

a) Point: (1, 5) Line: 6x – 2y = 8

b) Point: (– 2, 2) Line: 3x + 5y = 8

2) Find the slope and y-intercept of the line:

(Tutorial Section I)

a) 2x – 3y = 12

b) 4x + y = 1

c) y = 2

d) x = – 51

3) Find the equation of a line in slope intercept form:


a) Contains (1,2) and (– 2,4)

b) Contains (2,1) and m=4


c) Contains (1,7) and m=0

d) Contains (6,5) and m is undefined

4) Write the equation of the following lines in point slope form:


a) Contains (– 1,4) and (3,8)

b) Passes through (6,2) and had a y–intercept of 5

5) Sketch a graph of the equation:

(Tutorial Section I) a) y = –3x + 2 b) x = 4 – y


c) y –1 = 3x + 12

d) x = 7

6) Find the intercepts for each of the following.

(Tutorial Section II)

a) 3 2y x= − + b) 3 2y x= +

c) 2

2

3

(3 1)

x xy

x

+=

+

d) 2 3 4y x x= −

7) Find the point(s) of intersection of the graphs for the given equations.

(Tutorial Section III)

a) x + y = 8

4x - y = 7

ìíî

b) x2 + y = 6

x + y = 4

ìíî


c)

x2 - 4y2 - 20x - 64y -172 = 0

16x2 + 4y2 - 320x + 64y +1600 = 0

ìíï

îï

8) Let f(x) = 3x + 2 and g(x) = 1 + x2 find each of the following:

(Tutorial Section IV) a) f(g(0))=

b) g(g(2))=

c) g(f(x))= d) f(g(x))=

9) If 𝒇(𝒙) = 𝒙𝟐, 𝒈(𝒙) = 𝟐𝒙 − 𝟏. 𝒂𝒏𝒅 𝒉(𝒙) = 𝟐𝒙, 𝒇𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈:

(Tutorial Section IV) a) f(h(-1))=

10) For f(x) = 8x – 3, find

(Tutorial Section V)

a) 𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ=


11) For f(x) = x2 , find. (Tutorial Section IV)

12) Find the domain and range for each function give your answer using interval notation:

(Tutorial Section IV)

a) 2( ) 9h x x= − b) h(x) = sin x

c) 2( )

2 3f x

x=

+ d)

2 1, 0( )

2 2, 0

x xf x

x x

+ =

+

e) 𝑦 =3𝑥−2

4𝑥+1 f) 𝑦 =

𝑥2−4

2𝑥+4

g) 𝑦 =𝑥2−5𝑥−6

𝑥2−3𝑥−18

a) 𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ=


13) Test for symmetry with respect to each axis and the origin. (Tutorial Section V)

a) 𝑦 = 𝑥√𝑥 + 2

b) 𝑦 = |6 − 𝑥|

c) 𝑦 =𝑥

𝑥3+1 d) 𝑦 = 𝑥2 − 𝑥

14) Show work to determine if the relation is even, odd, or neither:

(Tutorial Section V)

a) 𝑓(𝑥) = 2𝑥2 − 7 b) 𝑓(𝑥) = −4𝑥3 − 2𝑥

c) 𝑓(𝑥) = 4𝑥2 − 4𝑥 + 4


15) For each function below list all holes, vertical asymptotes and x-intercepts: (Tutorial Section VI)

a)

f (x) =(x - 3)(x + 2)

(x - 3)(2x +1) b)

y =

x2 -1

2x2 + x -1

c) 82

3212)(

2

23

−−

+−=

xx

xxxxf d)

23

149)(

2

2

++

+−=

xx

xxxg

16) Find the inverse for each function.

(Tutorial Section VII)

a) f (x) = 2x +1

b) f (x) =

x2

3


17) Prove f and g are inverses of each other. (Tutorial Section VII)

a)

f (x) =x3

2g(x) = 2x3

b) f (x) = 9 - x2, x ³ 0 g(x) = 9 - x

18) Solve each equation:

(Tutorial Section VIII)

a) 7𝑥2 − 3𝑥 = 0

b) 4𝑥(𝑥 − 2) − 5𝑥(𝑥 − 1) = 2

c) 𝑥2 + 6𝑥 + 4 = 0

d) 2𝑥2 − 3𝑥 + 3 = 0

e) 2𝑥2 − (𝑥 + 2)(𝑥 − 3) = 12 f) 𝑥 +1

𝑥=

13

6


19) Solve each inequality:


a) 𝑥2 − 16 > 0 b) 𝑥2 + 6𝑥 − 16 > 0

c) 𝑥2 − 3𝑥 ≥ 10

d) 2𝑥2 + 4𝑥 ≤ 3

e) 𝑥3 + 4𝑥2 − 𝑥 ≥ 4 f) 2 sin2 𝑥 ≥ sin 𝑥, 0 ≤ 𝑥 < 2𝜋

20) Solve for x:


a) 2

3−

5

6=

1

𝑥

b) 𝑥 +6

𝑥= 5


c) 𝑥+1

3−

𝑥−1

2= 1

d) 5

2

3

1

15=+

x

x

21) Write the following absolute value expressions as piecewise expressions (by remove the absolute value):

(Tutorial Section IX) a) 𝑦 = |2𝑥 − 4|

b) 𝑦 = |6 + 2𝑥| + 1

c) 𝑦 = |4𝑥 + 1| + 2𝑥 − 3


22) Solve the following absolute value inequalities: (Tutorial Section IX)

a) |𝑥 − 3| > 12 b) |𝑥 − 3| ≤ 4

c) |10𝑥 + 8| > 2 d) |3𝑥 − 4| > −2

e) |𝑥 − 6| > −8

23) Write without fractional exponents:

(Tutorial Section X)

a) 3/12xy = b)

4/12 )16()( xxf =

c) 4/33/127 xy = d) 9

1

2 =

e) 641

3 =

f) 82

3 =

g) 272

3 =


24) Write with positive exponents: (Tutorial Section X)

a) 32)( −= xxf b)

22 )4( −= xy

c) 𝑦 = (−2

𝑥−4)−2

d) 𝑓(𝑥) =(𝑥−3)−2

(2𝑥+1)−3

25) Factor then simplify:

(Tutorial Section X)

a) f (x) = 4x-3 +2x -18x-2 b) f (x) = 5x2(x -2)-1/2 + (x -2)1/23x

c) f (x) = 6x(2x -1)-1 - 4(2x -1)


26) Simplify rational expressions: (Tutorial Section X)

a) f (x) =(4x2 )3

2x

b) y =(2x +1)(x -3)2

(x -3)4(2x -1)

c) f (x) =16x2 -8x +1

4x2 +3x -1

d) y =x2 - 25

x2 -10x + 25

27) Simplify:

(Tutorial Section XI)

a) 2

3+

4

5= b)

2

3−

4

5=

c) 2

3∙

4

5= d)

2

34

5

=


e) 𝑥+3

𝑥2−9= f)

𝑥2+10𝑥+21

𝑥2+8𝑥+15=

g) 1

4+

1

𝑥

𝑥+4= h)

1

𝑥+1−

1

𝑥=

i) 𝑥−

1

𝑥

𝑥+1

𝑥

= j) 1

𝑥+4

1

𝑥−2

=

k) 𝑥

𝑥−1

2

= l)

9 3x

x

+ −=

m)

x h x

h

+ −=


28) Express as a single logarithm:

(Tutorial Section XII) a) 3 ln 𝑥 + 2 ln 𝑦 − 4 ln 𝑧 = b) 3 ln 𝑥 = 1

c) Solve for x: 𝑒𝑥−3 = 7

29) Graph two complete periods of the function.

(Tutorial Section XIV)

a)

c)

c)

d) f (x) = -cos x -p

4

æ

èçö

ø÷

f (x) = 5sin x f (x) = sin2x

f (x) = cos x - 3


e) f (x) = cos(2x +p )-2

f) f x( ) = 2sin x +p( ) -1

g) f x( ) = - tanx

2+

p

2

æ

èç

ö

ø÷


30) Perform the following transformations based on the given diagram.

(Tutorial Section XV)

a) ( ) ( ) 12 −−= xfxh

b)

c)

( ) ( ) 21 ++−= xfxk

( ) ( ) 2+−= xfxj


31) Identify the parent functions of f, describe the sequence of transformations from f to h. (Tutorial Section XVI)

a) b)

32) Graph the following circle:

(Tutorial Section XVII)

a) (x - 3)2 + (y + 1)2 = 4

33) Answer the following questions:

(Tutorial Section XVII)

a) Find the standard form, center, and radius of the following circles: x2 + y2 – 4x + 8y – 5 = 0

b) Give the equation of the circle whose center is (5, -3) and goes through (2, 5)

34) Factor the following:

(Tutorial Section XVIII)

a) 27x3 +125 b) 5129 −x

( ) ( ) 132++−= xxh ( ) 7−= xxh


35) Complete the following table:

(Tutorial Section XX)

a)

θ in Degrees (Radians) Cosθ Sinθ Quadrant

0°

30°

45°

60°

90°

120°

135°

150°

180°

210°

225°

240°

270°

300°

315°

330°

360°


36) Find all solutions to the equations. You should not need a calculator. (Hint: one of these has NO

solution.)

(Tutorial Section XX)

a) 4cos2 x - 4cosx = -1 b) 2sin2 x +3sin x +1= 0

c) sin2 x -2sin x = 0 d) 3sin x = 2cos2 x

e) 2sin2 x +3sin x = 2 f) cos2x +5cosx = 2 (hint: use double angle identity)

g) sin(cosx) =1 h) sin2 x -2sin x -3= 0


37) For each of the following, express the value for “y” in radians. (Tutorial Section XXI)

a) y = arcsin

- 3

2

b) y = arccos -1( )

c) y = arctan(-1)

38) For each of the following give the value without a calculator.

(Tutorial Section XXI)

a)

tan arccos2

3

æ

èçö

ø÷ c)

sec sin-1 12

13

æ

èçö

ø÷

d)

sin arctan12

5

æ

èçö

ø÷ e)

sin sin-1 7

8

æ

èçö

ø÷

Documents

Calculus Summer Assignment.docx - Home - Abraham …achs.roselleschools.org/UserFiles/Servers/Server_31540… · · 2017-06-15MATHACHS CALCULUS SUMMER ASSIGNMENT JUNE , 2017