Warm-up for Calculus

Nichols School Mathematics Department
Summer Work Packet
Warm-up for Calculus
Who should complete this packet? Seniors who have completed Functions and will be
taking mathematics course in the fall of 2014.
Due Date: The first day of school
How many of the problems should I do? – ALL OF THEM
How should I organize my work? You should show all work in a separate sheets of looseleaf paper. If a problem requires a graph, then you should use graph paper. Keep your
materials in a standard 3-Ring binder. You will continue to use this ring binder in your math
class next fall.)
How will my teacher know that I’ve done the work? –Your teacher will collect your
notebook on the first day of school. Your teacher may choose to QUIZ or TEST you on this
material if he or she feels it is necessary – BE PREPARED!
How well should I know this material when I return? – You should recognize that you’ve
seen this material before, and you should also be able to answer questions like the ones in
this packet. If the material is revisited in your next class, it will only be for a brief amount of
time – your teacher will assume that all you need is a quick refresher.
Note from your teachers:
We feel that this summer work will truly help you succeed this year. We understand that
summer is a time for relaxation and fun, but it is imperative that you spend some time
before you return reviewing your materials. This packet is mandatory, and you must treat it
as you would any other extremely important homework assignment. You will be held
accountable for this material. We also highly suggest that you do a bit of it at a time in the
weeks leading up to school – don’t leave it for the last day!!!
Nichols School Mathematics Department
What if I get stuck? - You should check out additional study materials. Consult a standard
precalculus textbook. Find a study buddy or classmate to help you remember the material.
Consult the following websites for hints and examples:
http://coolmath.com/
http://www.math.ucdavis.edu/~marx/precalculus.html
http://www.brightstorm.com/math/precalculus
http://www.khanacademy.org
Nichols School Mathematics Department
Warm-up for Calculus
Instructions:
▪ Complete the problems on loose-leaf paper in pencil. If a problem requires a graph, please
use graph paper and a ruler.
▪ At the top of each page of your work, write the topic name and the page number of the
worksheet. When you start a new worksheet, start a new sheet of paper.
▪ Complete all the problems carefully. Show enough work to indicate your method of
solution.
▪ Make sure your work justifies your answer.
▪ Keep the packet and your work in a standard 3-Ring binder.
▪ Place the worksheets in the ring binder in page number order with your completed work
immediately following the worksheet.
Remember, your teacher will collect your work on the first day of school! Late work will
be severely penalized.
You will continue to use your 3-Ring binder for your math class throughout the school year.
Nichols School Mathematics Department
Nichols School Mathematics Department
Quadratic and Polynomial Functions
1. Graph each function using transformations (shifting, stretching, reflecting, etc.) from the
graph y = x 2 . State the transformations needed.
a.
f ( x ) = 2( x + 1)2 − 4
b. g ( x ) = − ( x − 4 )
2
2. For each quadratic function:
i. Determine if the function has a maximum or a minima and explain how you know.
ii. Algebraically, determine the vertex of the graph.
iii. Determine the axis of symmetry of the graph.
iv. Algebraically, determine the intercepts of the graph.
v. Using the information in parts a-d, graph the function by hand. Plot additional points as
needed in order to create an accurate graph.
a.
f ( x ) = 3 x 2 − 12 x + 4
b. g ( x ) = −2 x 2 − x + 4
3. Solve each equation over the complex numbers.
a. 3x 2 − 2 x − 1 = 0
b. x(1 − x ) = 6
c.
x4 + 2x2 − 8 = 0
4. Given the polynomial function P ( x ) = −2( x − 1) 2 ( x + 2) , do the following:
a. Find the x and y intercepts of the graph.
b. Determine whether the graph crosses or touches the x-axis at each x-intercept.
c. Describe the end behavior of the graph, that is, describe what happens to the y-values as
x increases.
d. Using the information in parts a-c, make an accurate sketch of the graph.
5. For the polynomial function g ( x ) = 2 x 3 + 5 x 2 − 28 x − 15 :
a. Determine the maximum number of real zeros that the function may have.
b. List the potential rational zeros.
c. Determine the real zeros of g. Factor g(x) over the real numbers.
6.
Give the equation of a polynomial function with real coefficients of degree 3 that has zeros
of 5, 3 + 4i .
Warm-up for Sr Electives
Page 1
The Algebra of Functions
Questions 1&2: Each function given is a transformation of y = sin x or y = cos x . For each
problem, graph the fundamental function and use it to create the graph of the transformed function.
1. Sketch the graphs of these functions. Determine the amplitude and the period. USE RADIANS
1
1 
a. y = 2 sin  θ 
b y = − cos ( 2θ )
2
3 
2. Sketch the graphs of these functions. Determine the amplitude, period, and any horizontal or
vertical shifts. USE RADIANS
a.
1 
y = 4 + 3sin  θ 
2 
 
π 
b. y = 3 − 2 cos  2  θ −  
4 
 
3. Shown at right is the graph of the function
π 

y = 6sin  3( x − )  . Determine possible values of the
6 

coordinates of A, B, & C.
4.
Give the equation of a sinusoid (sine or cosine function) that meets the conditions:
a. has a maximum at (20,1) and a minimum at (25,-5)
b. has an amplitude of 3, a period of 180, and passes through the point (25,2)
Warm-up for Sr Electives
Page 2
The Algebra of Functions
5. Identify the function whose graph is shown. Note any attributes of the graph that determined
your answer, for example, period, asymptotes, etc.
a.
y
4
3
2
1
x
-5π/3
-3π/2
-4π/3
-7π/6
-π
-5π/6
-2π/3
-π/2
-π/3
-π/6
π/6
π/3
π/2
2π/3
5π/6
π
7π/6
4π/3
-1
-2
-3
-4
y
4
3
2
1
x
-π/2
-π/4
π/4
π/2
-1
-2
-3
-4
b.
Warm-up for Sr Electives
Page 3
The Algebra of Functions
1.
a)
b)
c)
d)
Given, = 3 − 4 and = √ − 1 , find the following and state its domain
+
−
/
2. Given the graphs of and graph + 3. Given ℎ and knowing ℎ is formed by and , name and and say
whether ℎ = + , ℎ = − , ℎ = , or ℎ = .
2x − 5
x+2
a. ℎ = b. h( x ) =
d. ℎ = e. ℎ = sin + cos Warm-up for Sr Electives
c. ℎ = 4 − 10
f. ℎ = Page 4
Exponential and Logarithmic Functions
1. Find the unknown value without using a calculator.
a. 3x = 343
b. log b 16 = 2
c. log 5 x = 4
2. Solve each equation. Give an exact answer when possible. When using a calculator, round your
answer to three decimal places.
a. 5 x+ 2 = 125
b. log ( x + 9 ) = 2
e. 7 x +3 = 2 x
f. log 2 ( x − 4 ) + log 2 ( x + 4 ) = 3
c. 8 − 2e − x = 4
(
)
d. log x 2 + 3 = log ( x + 6 )
3. Given log a 2 ≈ 0.301 and log a 3 ≈ 0.477 , find each of the following:
a. log a 6
b. log a 3
2
c. log a  
3
d. log a 36
4. Use transformations to graph each function: identify the “base” function, then describe any
shifts, stretches, etc. Sketch an accurate graph of the function. Determine the domain and give
the equation of any asymptotes.
a.
f ( x) = 4 x +1 − 2
b. g ( x ) = 1 − log 5 ( x − 2 )
5. A 50 mg sample of a radioactive substance decays to 34 mg after 30 days.
a. What is the half-life of the substance?
b. How long will it take for there to be 2 mg remaining?
6. The average cost of college at 4-year private colleges was $19,710 in 2003-2004. This was a
6% increase from the previous year.
a. Assuming cost continues to increase at a rate of 6% per year, write a function to model the
cost as a function of time. Let t = 0 represent 2003-2004.
b. Use your model from part a) to determine the cost of college at a 4 year private institution in
2013-2014.
c. If a college savings plans offers an annual rate of 5% compounded weekly, how much
would Mrs. Kuhns need to invest in 2003 in order to pay for 1 year of college in 2013?
Warm-up for Sr Electives
Page 5
Rational Functions and Function Algebra
1. Given f ( x ) =
a.
f g ( x)
x+2
and g ( x ) = 2 x + 5 , find:
x−2
b. g f ( −2)
2. Find the inverse of each function.
2
a. f ( x ) =
3x − 5
3.
b.
c.
f g ( −2)
f ( x ) = e5 x + 2
If the point (3,-5) is on the graph of the function g ( x) , what point must be on the graph of
g −1 ( x) , the inverse function ?
4. For each rational function, identify: i) domain ii) the equations of any asymptotes iii) x &
y-intercepts . Use this information (along with any other helpful points) to create an accurate
graph of the function.
3
x−2
a. f ( x ) =
b. g ( x ) = 2
4− x
x + 2x − 3
5. Give the equation of a rational function that has vertical asymptotes at x = 3 & x = −3 and
also has a horizontal asymptote at y = 4 .
6. Each of the following graphs passes through the point P = (1, 2) . Match the correct function
to each graph. Identify a characteristic that enables you to determine which graph.
Warm-up for Sr Electives
a.
y = 2x
b.
y = 2 sin(30 x ) + 1
c.
y = log 2 x
d.
y=
x+3
x +1
e.
y=
1
3
x+
2
2
f.
y = x2 + 4 x + 5
Page 6
Complex Rational Expressions
1. Find all of the numbers that must be excluded from the domain of each rational expression
a.
!
b.
x+7
x 2 − 49
b.
x 2 − 8 x + 16
3x − 4
c.
x −1
x + 11x + 10
2
2. Simplify each rational expression
a.
3x − 9
2
x − 6x + 9
c.
x 2 + 12 x + 36
x 2 − 36
3. Multiply or divide as indicated. Simplify as much as possible.
a.
x − 2 2x + 6
•
3x + 9 2 x − 4
b.
x + 5 4 x + 20
÷
7
9
x 2 + x − 12 x 2 + 5 x + 6
x+3
c. 2
• 2
÷ 2
x + x − 30 x − 2 x − 3 x + 7 x + 6
4. Add or subtract as indicated. Simplify as much as possible.
a.
3
6
+
x+4 x+5
b.
3
5x
−
5 x + 2 25 x 2 − 4
c.
6 x 2 + 17 x − 40
3
5x
+
−
2
x + x − 20
x−4 x+5
5. Simplify each complex rational expression
x
−1
a. 3
x −3
Warm-up for Sr Electives
1 1
+
x y
b.
x+ y
3
4
−
c. x − 2 x + 2
7
2
x −4
1
1
− 2
2
( x + h)
x
d.
h
Page 7
Trigonometric ID’s, Equations, and Radian Measure
1. For each angle on the unit circle shown:
a) fill in the angles (in degrees and radians)
b) give the coordinates of terminal points P
2. Give the exact value of each expression without using a calculator.

3
a. cos −1  −

 2 
1
b. sin −1  
2
c. sec(45°)
d. cot (180° )
e. cos−1 0
f.
tan ( 315° )
3. Verify each identity. Remember to work only on one side of the equation.
tan θ − cot θ
a. sin θ tan θ + cos θ = sec θ
b.
= 1 − 2 cos2 θ
tan θ + cot θ
c. 4cos2 θ + 3sin 2 θ = 3 + cos2 θ
d. 1 −
cos 2 θ
= sin θ
1 + sin θ
4. Solve each equation on the interval 0 ≤ θ < 2π
a. 4sin 2 θ − 3 = 0
b. 2cos2 θ + cos θ = 1
c. sin ( 3θ ) + 1 = 0
d. tan θ + 3 = 0
5. Use sum and difference ID’s to find the exact value of each expression.
 5π 
a. cos 255°
b. cos  
 12 
c. sin( −15°)
Warm-up for Sr Electives
 5π 
d. sin 

 8 
Page 8

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