Name: __________________________________
Due Date: August 17, 2015
Greetings and Salad Dressings AP Calculus Students,
This summer packet must be completed and handed in on the first day of class. It will be counted as a 50-point
major grade. Ten (10) points will be deducted for each day that it’s late.
You must complete the entire assignment showing sufficient evidence of effort in mathematical reasoning, use of
computational skills, understanding of concepts, and communication of appropriate mathematical processes and terms.
Show all your work or explain how you arrived at the solution, circle your answers, label when necessary. Use a
graphing calculator only where specified. However, please do use a calculator to check all answers where possible.
Finally, please notice that I’ve provided a formula sheet in the back of this packet for you to use.
Have a great summer!
Mr. D.
7
x9
5
x6
1)
Simplify
. Express your answer using a single radical.
2).
Factor completely.
3)
Use a graphing calculator to determine the domain, range and the zeros of: f ( x)  13  20 x  x 2  3x 4 .
6 x3  17 x 2  5 x
Domain: __________________ write using interval notation
4)
Range:
__________________ write using interval notation
Zeros:
__________________
Find the equation of the line through ( 2, 7) and (3,5) in point-slope form.
5) Convert the function
f ( x)  5 x  4
6) Given the function f ( x) 
2 x2  x  2
x2  4
to a piecewise function.
(No Calculator, except for the range in part e)
a) Write the equation of the horizontal asymptote.
b) Write the equation of any vertical asymptotes.
c) Find the x and y coordinate of any holes.
d) Find the x and y-intercepts
e) Find the domain and range.
f) Find a few key points and graph the function above.
7) Find the domain of the following functions without a calculator.
a) f ( x)   x 2  6 x  4
b) f ( x)   3  x  5
c) f ( x)   log 2 ( x  4)  1
4 x 1
d) f ( x)  e
8)
Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence
used for the three sides is 240 ft. Let x be the length of two sides perpendicular to the wall as shown. Write
an equation of area A of the enclosure as a function of the length x of the rectangular area as shown in the
above figure. The find value(s) of x for which the area is 5500 ft 2
?
x
x
Existing wall
9)
Let f ( x)  x  3 and g ( x)  x 2  1 . Compute ( g f )( x) , state its domain in interval notation.
( g f )( x) : _______________________
Domain:
_______________________
3x  7
. Find f 1 ( x) , the inverse of f ( x)
x2
10)
Let f ( x) 
11)
Find an equation for the parabola whose vertex is (2, 5) and passes through (4, 7) . Express your
answer in the standard form for a quadratic.
12)
Which of the following could represent a complete graph of f ( x)  ax  x3 , where a is a real number?
A.
B.
C.
D.
13)
Find a degree 3 polynomial with zeros -2, 1, and 5 and going through the point (0, 3) .
14)
The graph of y  2  a x 3 for a  1 is best represented by which graph?
A.
15)
B.
C.
D.
Describe the transformations that can be used to transform the graph of f ( x)  ln x to a graph of
f ( x)  4 ln( x  2)  3 . Graph each function in the designated area below.
f ( x)  ln x
f ( x)  4 ln( x  2)  3
__________________________________________________________________________________________
16)
17)
18)
The number of elk after t years in a state park is modeled by the function P (t ) 
1216
.
1  75e 0.03t
a)
What was the initial population of elk?
b)
When will the number of elk by 750?
c)
Use your graphing calculator to determine the maximum number of elk possible in the park?
Arturo invests $2700 in a savings account that pay 9% interest, compounded continuously. When will
his balance reach $4550? Recall F = P * e r t
Use Trig identities to simplify  csc( x)  tan( x)  sin( x)cos( x)
A.
sin( x)  cos 2 ( x)
B.
cos( x)  sin 2 ( x)
C.
sin 2 ( x)  cos( x)
D.
cos 2 ( x)  sin( x)
19)
Find the exact value of each without the use of a calculator. Think unit circle.
sin  3  
 3
cos  
 2
 2
csc 
 3
 
cot   
2



 5
tan  
 6






20)
The endpoints of the diameter of a circle are (-3, 5) and (1, 1). Write the equation of the circle.
21)
Solve the inequality: x 2  x  12  0 .
A. (, 4)  (3, )
22)
B. x  4, x  3
C. (3, 4)
D. (, 3)  (4, )
Find the perimeter of a 30 slice of cheesecake if the radius of the cheesecake is 8 inches.
Recall S = r * θ
23)
x 3  7 x 2  14 x  8
Use polynomial long or synthetic division to rewrite the expression
x4
24)
Express y = -3x2 – 24x + 11 in standard form by completing the square.
25)
Two students are 180 feet apart on opposite sides of a telephone pole. The angles of elevation from the
students to the top of the pole are 35 and 23 . Find the height of the telephone pole.
26)
Graph the piecewise function.
 x 2
 2  x 1

f ( x )   2
x 1
3 x  5 1  x  3

27)
Solve the equation 2sin 2 ( x) cos( x)  cos( x) algebraically.
28)
Find all the exact solutions to 2sin 2 ( x)  3sin( x)  2  0 on the interval 0, 2  .
29)
Give that f ( x) 
x5
. Find the asymptotes and the domain of the function.
x2
Domain: __________
Vertical Asymptote(s): _________
Horizontal Asymptote(s): ________
30)
Use a graphing calculator to approximate all of the function’s real zeros. Truncate your results to three
decimal places. f ( x)  3x6  5x5  4 x3  x 2  x  1
31)
Factor to solve the inequality. Write your answer in interval notation. 0 
x 3  64
x3
1
1

3( x  h)
3x
h
32)
Simplify the expression as much as possible
33)
1
1
4
Simplify the expression as much as possible. 10 x 2  (3x 2  4) 5 6 x  (3 x 2  4) 5  20 x
5
34)
Simplify the expression and determine where the expression is positive.
2(2 x  1)
2
x


2
x  x6 x3 2 x
35)
Use algebra to find the exact solution to x4 – 5x2 + 3 = 0. Show all work.
36.) Solve the equation
2 x  2 x  5 (Hint: Treat like a quadratic, do a temporary substitution.
Do algebraically and use calc at the end. You need to show the algebra to get credit)
Evaluate the limits that follow.
5x  3

x  12 x  6
40. lim
2 x2  6 x

x 3
x 3
41. lim
37. lim
38. lim
39. lim
x 0
x

1 1 x
5x2  3

x  12 x  6
x 2
x2  4

2 x
42) Interval Notation. Complete the table
Algebraic
Interval
Graph
−1 ≤ 𝑥
[5,3)
(
]
-3
8
Find the domain of the following functions. Make sure to use interval notation (ex: [0, 3)).
43) y  log  2 x 12
45) y 
47) y 
49) y 
x2  5x  6
x 2  3x  18
x2  4
2x  4
46) y 
22 x
x
48) y 
x3  x3
x 2  8 x  12
4
x5
44) y 
50) y 
x 2  5 x  14
2x  9
2x  9
51) y 
3x  2
4x 1
52) y 
3
x6
x  x  30
2
54) Solve the equation 2log 2 ( x  3)  log 2 ( x  1)  3
53) y 
x
cos x
(No Calc)
55) Solve the equation 2e2 x 1  6  10 (exact answer, no calculator)
56) Solve the equation for y. ln( y  1)  ln 2  x  ln x
57) Solve the following trig equations over the given domain (no calculator)
a)
1
 sin x  0 on 0, 2 
2
b) cos x 
 3
2
on 0, 2 
c) sin x  cos x on 0, 2 
58)
59) Factor and simplify the following
a) 3x  1 ( x  2) 4  4( x  1)3 ( x  2)3
2
b) 2 x  1( x  1) 2  2( x  1) 2 ( x  1) 3
2
2
3
c) 3( x  2) ( x  3)  ( x  2) (2)( x  3)
( x  3) 4
d)
3( x  2) 2  6 x( x  2)
x2
60)
Use a graphing calculator to determine the time interval where v(t)<0 (Zoom in or adjust window
to suit). Write the answer using interval notation.
61) A man 6 feet tall walks at a rate of 5 feet per second toward a light that is 20 feet above the ground. If the man’s shadow is 4 ft
long at this time, how far is he from the light?
62) Solve the equation 2log 2 ( x  3)  log 2 ( x  1)  3
(No Calc)
63) Solve the equation 2e2 x 1  6  10 (exact answer, no calculator)
64) Find the area of region A, B, and C
Find x in each right triangle shown below:
Use trig or inverse trig functions to
find sides or angles
65)
3
34°
x
66)
17
x°
8
66.
Find the x- and y-intercepts for y 
x 1
x3
67) Find the derivative of the following
a) y = x3
b)
y = x2 + 3x
c) y = -x + 4
d) y = x  5
e) y =
2
x9
Formula Sheet
Reciprocal Identities:
csc x 
1
sin x
sec x 
1
cos x
Quotient Identities:
tan x 
sin x
cos x
cot x 
cos x
sin x
Pythagorean Identities:
sin 2 x  cos 2 x  1
Double Angle Identities:
sin 2x  2sin x cos x
cot x 
tan 2 x  1  sec 2 x
1
tan x
1  cot 2 x  csc 2 x
cos 2x  cos 2 x  sin 2 x
tan 2 x 
2 tan x
1  tan 2 x
 1  2 sin 2 x
 2 cos 2 x  1
Logarithms:
y  log a x
Product property:
logb mn  logb m  logb n
Quotient property:
log b
Power property:
log b m p  p log b m
Property of equality:
If logb m  log b n , then m = n
Change of base formula:
log a n 
Derivative of a Function:
Slope of a tangent line to a curve or the derivative: lim
is equivalent to
x  ay
m
 log b m  log b n
n
log b n
log b a
h 
Slope-intercept form:
y  mx  b
Point-slope form:
y  y1  m( x  x1 )
Standard form:
Ax + By + C = 0
f ( x  h)  f ( x)
h