Download June 2013

Dominion Math Department
June 2013
Dear Calculus Student,
Over the summer it would be beneficial for you to periodically refresh your memory
on some math basics before beginning class in August. With this in mind, your first problem
set is attached. As with the other math courses at DHS, problem sets (POW’s and AP Sets) in
Calculus will be regularly assigned each marking period in order to help you review
concepts and skills which you have previously learned but which need periodic refreshing.
This problem set will be due no later than Friday, September 6th. It will be graded for
correctness and there will be a quiz at the end of the 2nd week of school that will cover all of
this prerequisite material. I will require each of you to enroll in my VISION course. You
may do so by selecting my name: HS-DHS-Misanin, Course: AP Calc and Password: AP
Calc. Please make sure you enter a current email so you can receive updates from me over
the summer. There is a chance I will have a second extra packet of practice for you later in
the summer.
To encourage you to complete this during the summer, there will be two “free”
review sessions during the week preceding the start of school so watch VISION to learn the
exact dates and times as we get closer to the beginning of the school year. I will go over any
questions you have on the packets at that time. You may only come if the packets are done
(or a very good try) before you get there. Calculus is a demanding course and will require
an average of approximately 90 minutes of homework time for every 90-minute block.
Getting ahead during the summer, before the first week of school will be advantageous.
AP Calculus is intended for serious math students who plan to take higher-level
mathematics and/or science courses in college. The curriculum, pace, and rigor of the AP
Calculus AB course is determined by College Board guidelines. All enrolled students are
required to prepare to take the AP exam, so it is taught as if EVERYONE is planning on
taking the exam. Next year the exam is on Wednesday, May 7th, 2014 and will require the
use of a graphing calculator.
If you do not already own a graphing calculator, you should plan to purchase one for
the course (a TI-84 is recommended if you do not already have one) and become familiar
with it over the summer. In the event that your family’s economic situation makes this
difficult, let me know as soon as possible. Be sure that you learn how to graph with ease,
including polynomial, rational, trigonometric, exponential, and logarithmic functions.
Familiarity with the menus on your particular calculator will also be invaluable in a few
months.
The AP Calculus course requires you to think mathematically, something some of you
may never have done before. Do not be discouraged in the beginning if it takes you awhile
to think mathematically. By the time May 2014 rolls around, you will be ready. Throughout
the course, I will Verbally NAG you. Verbally NAG is a way to remember the four ways that
the College Board expects you to think mathematically.
Verbally – through written words and thoughts spoken out loud
Numerical – being able to understand charts, tables, etc and use them to answer
questions
Algebraic – solving equations and word problems using algebra skills
Graphical – comprehending and interpreting graphs and using them to answer
questions
Have a restful summer and be ready to maintain a fast pace next year. The
curriculum must be covered in full at least three weeks before the exam to allow for ample
review opportunities. After the AP exam, the class will investigate additional math topics
through projects and labs.
Sincerely,
Mrs. Misanin
[email protected]
AP Calculus AB
Keep these
notes
in your calculus
notebook.
NAME: ____________________
2013 Summer Assignment
Per: ______
I. Line Review:
It is very important that you can write equations of lines given:
1) two points on the line
2) a point on the line and the line’s slope (m).
The most frequently used form of a line in this class will be
point-slope form of a linear equation. You should know the following:
 Slope of a line going through
Slope (m) =
 x1, y1 
 x2 , y2  .
and
rise
y
y  y1

 2
.
run
x
x2  x1
x1
 x2 
 Remember, vertical lines have an undefined slope while
horizontal lines have a slope of zero.
 Point-slope form of a line going through
and having slope of m.
y  y1 
x1, y1 
m  x  x1 
 Parallel lines have the same slope.
 Perpendicular lines have slopes that are negative reciprocals
of each other.
 Slope-intercept form of a line with slope m and y-intercept b
y = mx + b
EX 1]
Find the slope of the line that goes through (4,-3) and (2,5).
m 
y
5  (3)
8


 4
x
2  4
2
m = -4
EX 2]
Using the point-slope equation, write an equation for
3
the line through the point (2,3) with slope
.
2
y  y1  m  x  x1 
y  3 
EX 3]
Note: No need to write in slope-intercept form, unless you would prefer.
Write an equation for the line through (-2,-1) and (3,4).
1st
Find the slope of theline.
m 
EX 4]
3
 x  2
2
2nd Use one point on theline and the slope.
y
4  (1)
5


1
x
3  (2)
5
y  4  1( x  3)
Write an equation for the line through (-1,2) that is
a) parallel, and b) perpendicular to the line L: y = 3x – 4.
mL  3
a) m  3, (1, 2)
1
, (1, 2)
3
1
y  2 
 x  (1) 
3
1
y  2 
 x  1
3
b) m 
y  2  3  x  (1) 
y  2  3( x  1)
EX 5]
Find the slope and y-intercept of the line
Graph the line.
How?
Rewrite 8x + 5y = 20
in y = mx + b form.
5 y   8 x  20
y 
8
x  4
5
8x + 5y = 20.
So, m 
8
5
b  4
y
x
Advanced Placement Calculus
2013 Summer Assignment
NAME ____________________
Date __________ PER _____
Line Problems
In #1-14, write an equation for the specified line.
____________________
1)
through (1,-6) with slope 3
(Use point-slope form.)
____________________
2)
through (-1,2) with slope
1
2
(Use point-slope form.)
____________________
3)
the vertical line through (0,-3)
____________________
4)
through (-3,6) and (1,-2)
(Use point-slope form.)
____________________
5)
the horizontal line through (0,2)
(Use point-slope form.)
____________________
6)
through (3,3) and (-2,5)
(Use point-slope form.)
____________________
7)
with slope –3 and y-intercept 3
____________________
8)
through (3,1) and parallel to 2x – y = -2
(Use point-slope form.)
____________________
9)
through (4,-12) and parallel to
4x + 3y = 12
(Use point-slope form.)
____________________ 10)
through (-2,-3) and perpendicular to
3x – 5y = 1
(Use point-slope form.)
y
____________________ 11)
x
____________________ 12)
with x-intercept 3 and y-intercept –5
____________________ 13)
the line y = f(x), where f has the
following values:
x
-2 2
f(x) 4 2
____________________ 14)
4
1
through (4,-2) with x-intercept –3
II. Function Notation:
Throughout this class, we will be using function notation.
EX] f(x) = 3x – 4
The “(x)” represents the inputs of the function more formally
called the domain and the independent variable.
The “f(x)” represents the outputs of the function more formally
called the range and the dependent variable.
If I want to input a ‘3’ into the function, I write f(3) = 3(3) – 4.
So, with a domain (input) of 3 we get a range (output) of 5.
It will be important to be able to identify the domain and range for
a given function. When asking for the domain of a function, I want
to know for which x-values the function is defined. It is often
easier to find where the function is undefined.
Function
Y = x2
Domain (x)
(-,)
Range (y)
[0, )
(-, 0)  (0, )
(-, 0)  (0, )
x
[0, )
[0, )
y =
4  x
(-, 4]
[0, )
y =
1  x2
[-1,1]
[0,1]
Y =
Y =
NOTE:
1
x
‘[’ means include the endpoint in the domain.
‘(’ means the endpoint is not included in the domain.
 f ( x) 
 .
You must be able to deal with rational functions 
 g ( x) 

A rational function will be undefined when
the denominator (g(x)) is equal to zero.

The zeros of a rational function (where the graph crosses the
x-axis) are where the numerator (f(x)) is equal to zero and
the denominator is not equal to zero.
EX 1]
Find the domain and real zeros of the given functions.
x  2
x3
a)
b) g ( x)  2
f ( x)  2
x  3x  2
x 1
Domain
Zero(s)
x2 – 1 = 0
(x + 1)(x – 1) = 0
x =  1
so, {x: x  , x  1}
c)
x3 = 0
x = 0
(0,0)
x2 + 9 = 0

so, {x: x  }
Zero(s)
Domain
0
2
2/3
0)
x + 2 = 0
x = -2
(-2,0)
x2  9
j ( x) 
2x  1
d)
3x – 2 =
3x =
x =
(2/3,
Zero(s)
x2 – 3x + 2 = 0
(x – 2)(x – 1) = 0
x = 1, 2
so, {x: x  , x  1,2}
3x  2
h( x )  2
x  9
Domain
Domain
2x - 1
2x
x
so, {x: x 
= 0
= 1
= 1/2
, x  1/2}
Zero(s)
x2 - 9 = 0
(x + 3)(x – 3) = 0
x =  3
(-3,0) and (3,0)
Note how I set up the appropriate equations
(numerator = 0 for zeros; denominator = 0 for undefined values).
Then, I factored to solve equations.

Being able to solve equations by factoring is a really good
skill to know and you should practice it on this review.

Verify your solutions with your calculator.

Composite functions
f g ( x) or f  g, " f of g", we replace the x in
f(x) with g(x) and simplify.
EX 2]
f g (x)
Find the formula for
Then, find f(g(2)).
 
f 2  
if g(x) = x2 and f(x) = x – 7.
f g ( x)  f x 2  x 2  7
f g 2 
EX 3]
2
If f(x) = x2, find
f (4)  4  7   3
f ( x  h)  f ( x )
.
h


x  h   x 2  x 2  2 xh  h 2  x 2  2 xh  h 2  h(2 x  h)  2 x  h
f ( x  h)  f ( x )

h
h
h
h
h
2
Function Problems:
Answer the following questions. Show how you arrived at your answer
(see review sheet for examples).
_______________
Domain
2x  1
.
x2  2
2)
Find the zeros and the domain for
g ( x) 
x  2
.
x 1
3)
Find the zeros and the domain for
h( x ) 
x  3
.
x  2
Find the zeros and the domain for
x2  9
f ( x) 
.
2x  1
_______________
_______________
Domain
f ( x) 
_______________
_______________
Domain
Find the zeros and the domain for
_______________
_______________
Domain
1)
4)
_______________
x-int _________
y-int _________
5)
Find the x-intercepts (zeros) and the
x 2  16
f ( x) 
y-intercepts (x = 0) for
.
x2
_______________
6)
Given
f(x) = 4x2 – 2, find
_______________
7)
Given
h(x) = 10x – 2x2, find h(-2).
_______________
8)
Given
3x  4, x  2
find f(3).
f ( x)   2
 x  1, x  2
_______________
9)
Given
2 x  1, x   2
find f(-6).
f ( x)  
 x  6, x   2
_______________ 10)
Given
f(x) = 2 – x2, find
f(-2).
f ( x  h)  f ( x )
.
h
III. Exponents and Radicals
It is very important that you remember the Rules for Exponents.
Rules for Exponents:
If a > 0 and b > 0, the following hold true for
all real numbers x and y.
1)
ax  ay = ax +
2)
ax
 ax  y
y
a
3)
a 
4)
(ab)x = ax  bx
5)
a
 
b
x y
 
 ay
x
x
y
 a xy
ax
bx

Also, remember how to work with
Rational Exponents:
a 
 a
Negative Exponents:
a2 
1
a2
Zero Exponents:
x
y
a0 = 1,
y
x

a
y
and
x
1
 a2
2
a
a  0
When you are asked to simplify expressions there should be

no parenthesis,

no negative exponents, or

no powers with the same base in the answer.
EX]
9
a 
9
b 
9
ab
Exponent Problems:
#1-6
Simplify the following.
4
1)
 x 3 y 2 


 z 
_______________
2)
 3x 2 y 3 


2 
xw


_______________
3)
3x 2 2 x  5 x 1
_______________
4)
3x y z  xy 
_______________
5)
x2  x3  x4
_______________
6)
 2
 A
_______________
7)
Evaluate
_______________
3
3
2
 
2
3
3




4
2
1
27
1
3
.
3
_______________
8)
Evaluate
 1 2
  .
 64 
IV. Trigonometric Functions

An angle has three parts: an initial ray, a terminal ray, and a vertex (the point of
intersection of the two rays).

Standard position for an angle occurs if the initial ray of the angle coincides with the
positive x-axis and its vertex is at the origin.

Positive angles are measured counterclockwise, and negative angles are measured clockwise.

Coterminal angles have the same terminal ray. EX] -45 is coterminal with 315.

180 =  radians
Please, please, please, I beg you to know your unit circle with radian measures. (We will only use
radian measures in calculus.) I’m sure you got familiar with it in your Pre-Calculus class. Remember???

Know these:
sin  =
y
r
csc  =
r
y
cos  =
x
r
sec  =
r
x

y
x
cot  =
x
y
You should know the graphs of sin , cos , and tan , and be able to picture them quickly
from [-2, 2]. (In case you were wondering, the “[-2, 2]” represents the domain values.
The “[ ]” means including endpoints.)

Remember to draw triangles to represent the problem.
tan  =
EX] Determine all six trigonometric functions for the angle whose terminal side occurs at (-3,4).
y
(-3,4)
x

5 (Use Pythagorean Theorem to find the length of this side.)
4

-3
sin  
y
4

r
5
csc 
r
5

y
4
cos 
x
3
 
r
5
sec  
r
5
 
x
3
tan  
y
4
 
x
3
cot  
x
3
 
y
4
As mentioned before, the unit circle might be helpful. Know the trig values of the basic angles.
(Some are listed in the table below.)
You should be quick with the sin x, cos x, and tan x. Complete the tables below.
x
sin x
cos x tan x
x
0
0
1
3
4

6

4

3

2
1
2
3
2
2
2
1
2
0
2
2
3
2
1
sin x
cos x tan x

0
 2
2
-1
3
2
2
-1
0
0
1
2
2
2
2
3
1
2
x
1
45
30
3
60
1
1
Trigonometric Problems
#1 & 2 Determine two coterminal angles (one positive and one negative) for each given angle.
__________
__________ 1)   300 (Express your answer in degrees.)
__________
__________ 2)  
9
 (Express your answer in radians.)
4
#3 & 4 Express each angle in radian measure as multiples of .
__________ 3) -20
__________ 4) 270
#5 & 6 Express each angle in degree measure.
__________ 5)
7
3
__________ 6)
11
6
#7 Determine all
six trigonometric functions for the angle whose terminal side occurs at (-12,-5).
Draw and label a diagram.
y
x
sin   ________
csc  ________
cos   ________
sec  ________
tan   ________
cot   ________
#8 & 9 Determine the quadrant in which  lies.
__________ 8) sin   0 and cos   0
__________ 9) csc  0 and tan   0
#10 & 11 Evaluate each trigonometric function. Give EXACT answers.
__________ 10) If sin  
1
, find tan  .
3
__________ 11) If sec 
13
, find csc  .
5
3

1
13

5
#12-15 Evaluate the sine, cosine, and tangent of each angle without using a calculator.
Give EXACT answers.
12) sin  30

 _____
cos  30

 _____
tan  30

 _____
  
13) sin 
  _____
 6 
  
cos 
  _____
 6 
  
tan 
  _____
 6 
14) sin  750
cos  750
tan  750

 _____
 10 
15) sin 
  _____
 3 

 _____
 10 
cos 
  _____
 3 

 _____
 10 
tan 
  _____
 3 
#16 & 17 Determine the period and amplitude of each function.
Period = _____ 16) y 
3
x
cos  
2
2
Amp = _____
x
Period = _____ 17) y   2sin  
3
Amp = _____
Period = _____ #18 Find the period of the function y  7 tan  2 x  .
V. Graphs of Common Functions
Here are the graphs of very common functions.
Know the characteristics of these graphs:
Increasing vs Decreasing;
Domain;
Range; etc…
Be able to sketch these basic functions!
Area and Volume:
b1
Area Formulas:
w
h
b
A 
s
l
1
bh
2
h
r
s
b2
A   r2
A  s2
A  lw
A 
1
h  b1  b2 
2
Volume Formulas:
r
h
h
r
h
w
r
l
V   r 2h
V  lwh
V 
1 2
r h
3
V 
4 3
r
3
#1-7 Find the EXACT area of each of the following shaded figures. Show formulas and work for each on the answer sheet.
y
1)
2) 4
3
2
1
y
3)
6
3
2
1
x
3
2
1
x
-1
-1
5
5
y
5)
6
6
x
-1
y
4)
6)
3
2
1
3
y
y
7)
4
3
2
1
3
2
1
x
x
x
-1
-1
-1
5
5
5
#8-10 Find the EXACT volume of each solid. Show formulas and work for each on the answer sheet.
8)
9)
6
7
5
4
Do NOT make  a decimal!
10)
24
Area and Volume: Answer Sheet
Show all formulas and work to receive full credit.
Do NOT make  a decimal!
1) A = ____________________
2) A = ____________________
3) A = ____________________
4) A = ____________________
5) A = ____________________
6) A = ____________________
7) A = ____________________
8) V = ____________________
9) V = ____________________
10) V = ____________________
VI. Logarithmic Functions
Logarithmic Function – For x  0 , a  0 , and a  1 , y  log a x iff x  a y .
f ( x)  log a x is called the logarithmic function with base a.
NOTE:
A logarithm is an EXPONENT!
b) Write log 32 4 
EX] a) Write log 4 64  3 in exponential form.
2
5
4  64
32  4
3
EX] a) Write 7 2 
log 7
2
in exponential form.
5
1
in logarithmic form.
49
b) Write e x  17 in exponential form.
1
 2
49
log e 17  x
NOTE: x  ln 17
Common Logarithmic Function – the logarithmic function with base “10”.
Natural Logarithmic Function – the function defined by
f ( x)  log e x  ln x , x  0 .
Properties of Logarithms
1) log a 1  0
2) log a a  1
Properties of Natural Logarithms
1) ln1  0
2) ln e  1
3)
log a a x  x
3)
4)
If log a x  log a y, then x  y.
4)
EX]
ln e x  x
If ln x  ln y, then x  y.
Evaluate each expression without using a calculator.
a) log 2 32
b) ln e3
Let log 2 32  x .
Let ln e3  x .
(Go exponential!)
(Go exponential!)
2 x  32
x
5
x  3
(Get like bases.)
2  2
 x  5
e x  e3
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1 then, log a x is given by
log b x
log a x 
.
log b a
Properties of Logarithms
Let a be a positive number such that a  1, and let n be a real number.
If u and v are positive real numbers, the following properties are true.
1.
loga  uv   loga u  loga v
1.
ln (uv)  ln u  ln v
2.
u
log a    log a u  log a v
v
2.
u
ln    ln u  ln v
v
3.
log a u n  n log a u
3.
ln u n  n ln u
EX] Expand each logarithm.
a) ln
3x  5
7
b) log10 5x 3 y
1
= ln  3x  5 2  ln 7
=
= log 5  log x3  log y
1
ln  3 x  5   ln 7
2
= log 5  3log x  log y
EX] Write each expression as the logarithm of a single quantity.
a) ln x  3ln ( x  1)
= ln x  ln  x  1
= ln
x
x
 1
1
1
ln x 2 
ln y 3
3
2
b)
3
=
ln x
= ln
3
(Condense each logarithm.)
2
3
x
y
 ln y
2
3
3
2
 ln
3
2
3
EX] Find the EXACT value of the logarithm without using a calculator.
a)
log 5 75  log 5 3
75
 log 5
 log 5 25
3
 5x  25
5  5
x
2
 log5 75  log5 3  2
b)
ln e4.5
e x  e4.5
 x  4.5
x2
y3
3
x2
 ln
y y
Logarithmic Function Problems
#1-6 Solve for x. Show all work. Round answers to the nearest thousandth when necessary.
__________ 1) 2xex - 9ex = 0
__________ 2) log 0.08429 = x
__________ 3) logx 81 = 2
__________ 4) log9 x = -3
__________ 5) ln x = 6.5
__________ 6) logx 2401 = 4
#7-11 Express each in condensed form and simplify.
______________________________ 7) log x + 4 log x2 - 3 log x3
______________________________ 8) 2 ln
______________________________ 9)
y -
1
ln y4 + ln 2y
2
1
(logb x - logb y)
4
______________________________ 10) 3 log (x + y) - (log x + log y)
______________________________ 11) 3 ln e + 4 ln e2 - 2 ln e
#12-15 Express each in expanded form, if possible.
_________________________ 12) logb x3 y
_________________________ 14) ln
x3
4 y
_________________________ 13) logb
x2
5 yz
_________________________ 15) logb (x - y)