Download BC Summer Assignment 2017

AP® Calculus BC Summer Packet
Welcome to AP® Calculus BC!
While I realize that we will all want to relax throughout our summer, forgetting all about last year’s
PreCalculus or Calculus AB classes, there are certain skills that have been taught to you over the previous
years that are essential towards your success in AP Calculus BC. If you do not have these skills, you will
find that you will consistently get problems incorrect next year, even though you may understand the
calculus concepts. It is frustrating for students when they are tripped up by the algebra or trigonometry
and not the calculus. This summer packet is intended for you to retain/review/relearn these topics.
You should work these problems, in a neat, legible, organized manner. Please show all work in the space
provided on each test. NO CALCULATORS ALLOWED on any part of these packets!
Half of your
AP exam next year (which means ½ of every Calculus Quiz/Test) is taken without the calculator, and
these essential skills are typically the most costly on both in-class exams and the AP® exam.
Due to the fast pace of this AP® course, we will not re-teach prerequisite material or skills! I believe you
will benefit the most from this packet by starting it mid-June. You should try to complete a few problems
each day, as if it was a daily journal. Do not do all of it now, and do not wait and do it a week before we
start school in August. You are more likely to retain the information if you spread it out. This will be due
on the first day of class for the 2013-2014 school year.
Completion of these problems is a mandatory requirement, and the assignment will count in your first
semester grade.
Good luck and may the force be with you!
Mr. Hernandez
Below is a list of several websites that may help you when you come across a difficult problem. If you are unsure of
how to attempt these problems, please look online for help, or send me an email.
Helpful Websites
http://www.calculus-help.com/tutorials/
http://www.mathtv.com/
http://archives.math.utk.edu/visual.calculus/
Bring this completed unit circle with you to class on Day 1. You should have this memorized before Day 1.
 List the degree and radian measure for each angle, then list the correct coordinate point values.
 For the Positive/Negative Portion, list the 6 trig ratios as either +/– in that quadrant.
sin   y  coordinate
cos   x  coordinate
tan  
Reciprocal Identities
1
csc 
1
cos  
sec 
1
tan  
cot 
sin 
tan  
cos 
sin  
1
sin 
1
sec  
cos 
1
cot  
tan 
cos 
cot  
sin 
csc  
Trig Identities you MUST HAVE MEMORIZED before Day 1
Pythagorean Identities
Double Angle Identities
sin  2   2sin  cos 
sin 2   cos 2   1
tan 2   1  sec 2 
1  cot 2   csc 2 
cos  2   cos 2   sin 2 
cos  2   2 cos 2   1
cos  2   1  2sin 
y  coordinate
x  coordinate
Odd/Even Identities
sin      sin 
odd  f   x    f  x 
2
cos 2  
sin 2  
1  cos  2 
cos     cos 
2
1  cos  2 
even  f   x   f  x 
2
Bring this completed chart of parent functions with you to class on Day 1. You should have this memorized before Day 1.
f  x  x
f  x   x2
f  x   x3
Domain: ________ Range: ___________ Domain: ________ Range: ___________ Domain: ________ Range: ___________
lim f  x   _____
x 
lim f  x   _____
x  
f  x  x
f  x 
Domain: ________ Range: ___________
lim f  x   _____
x 
lim f  x   _____
x 
lim f  x   _____
x  
1
Asymptotes: ______ , ______
x
Domain: ________ Range: ___________
lim f  x   _____
x 
f  x   e x Asymptote: _________
lim f  x   _____
x  
lim f  x   _____
x  
f  x   ln x Asymptote: _________
lim f  x   _____
x 
lim f  x   _____
x  
f  x  x
Domain: ________ Range: ___________
lim f  x   _____
x 
f  x   9  x2
Domain: ________ Range: ___________ Domain: ________ Range: ___________
Domain: ________ Range: ___________
lim f  x   _____
x 
lim f  x   _____
x  
lim f  x   _____
x 
f  x   sin  x 
Domain: ________ Range: ___________
f  x   tan  x 
f  x   cos  x 
Domain: ________ Range: ___________
f  x   csc  x 
Domain: ________ Range: ___________
Domain: ________ Range: ___________
Asymptotes: _________ , ________
Asymptotes: _________ , ________
f  x   sec  x 
f  x   cot  x 
Domain: ________ Range: ___________
Domain: ________ Range: ___________
Asymptotes: _________ , ________
Asymptotes: _________ , ________
2.
Solve for x: 2  x   3 x  1   4  2 x
(a) –1
3.
(b) –3
Solve for x:
(a) 4
4.
Solve for x: 3 
(a) 
5.
4x 1 2x  3 7


4
3
12
9
(b)
2
1
4
(d) 
(c) –2
(d)
(c) 1
(d) No solution
(e) None of these
(d) 1
(e) None of these
3
2
(e) None of these
(e) None of these
4x  5 7x  9

x2
x2
(b) –1
Solve for x:
1
3
4

 2
x2 x3 x  x6
4
7
(b) 3
(a)
1
3
(c) 1
(c)
7
4
6.
The graph below is a transformation of the graph of
(a)
7.
g ( x)  x  4
(b)
g ( x)  x  4
x 1
g ( x)  x  4
(e) None of these
g ( x)   x  9 from the graph of f ( x)  x ?
(b) Reflection in the y axis
Horizontal shift 9 units to the left
(d) Reflection in the y axis
Horizontal shift 9 units to the right
(b) 2 x  2 x
2
(c)
3x 1
(d) 2 x  1
(e) None of these
(d) –14
(e) None of these
(d) 3x  x
(e) None of these
2
Given f ( x)  x  2 and g ( x)  6  2 x , find ( f  g )( 2) .
(a) 6
10.
(d)
Given f ( x )  2 x and g ( x)  x  1 , find ( fg )( x ) .
(a)
9.
g ( x)  x  4
Which sequence of transformations will yield the graph of
(a) Reflection in the x axis
Horizontal shift 9 units to the left
(c) Reflection in the x axis
Horizontal shift 9 units to the right
8.
(c)
f ( x)  x . Find an equation for the function.
Given
(b) 2
(c) –2
f ( x)  x 2  2 x and g ( x)  3x  2 x , find ( f g )( x) .
(a) 4 x  8 x  3
2
(b) 2 x  4 x  3
2
(c) 2 x  x  6 x
3
2
2
11.
Given
 x 2  1, x  4
f ( x)  
, find f (2)
6 x  7, x  4
(a) –19
12.
If
(a)
13.
9
x
(c) 4
(d) –5
(c) x – 6
(d)
(e) None of these
 f (3)  f ( x  3)
x
(b)
x2  6 x  9
18  x
x
(e) None of these
Find the domain of the equation shown at the right. Assume there are no breaks or discontinuities in the graph.
(a)
14.
f ( x)  x 2 , find
(b) 5
 ,  
(b)
 ,3
(c)
3,
(d)
3, 
(e) None of these
3, 
(d)
1,5
(e) None of these
Find the range of the function shown at the right.
(a)
 ,  
(b)
 8,1
(c)
2
15.
Evaluate:
 1 3
 
 64 
(a) 16
16.
Simplify:
(a)
3
(b)
1
16
(c)
1
512
(d) –512
(b)
8 xy 3 3xy
(c)
2 xy 3 6 xy 2
(d)
2 xy 3 3xy 2
(e) None of these
(c)
7 2
(d)
35  2
3
(e) None of these
7
4( 2  5)
(d)
1
4 2 5
(e) None of these
(e) None of these
24x 4 y5
3x 2 y 2 3 6 x 2 y 3
17.
5
7 2
35  2
47
Rationalize the denominator:
(a)
35  5 2
47
(b)
18.
Simplify by rationalizing the numerator:
(a)
1
4( 2  5)
(b)
1
4( 2  5)
2 5
12
(c)
19.
Solve xy ' y  1  y ' for y '.
20.
Solve ln y  kt for y.
21.
A seven-foot ladder, leaning against a wall, touches the wall x feet above the ground. Write an expression (in terms
of x) for the distance from the foot of the ladder to the base of the wall.
Find the exact solution of the equations for 0  x  2 – No Decimals! No Calculators!
22.
sin  x  
3
2
23.
tan 2 x  1
24.
cos
x
2

2
2
25.
2sin 2 x  sin x  1  0
26.
sin  2 x   sin  x   0
27.
2 tan x  sec2 x  0
Find the exact solution to each exponential or logarithmic equation – No Decimals! No Calculators!
log 1 x  4
28. log 2 x  3
29.
30. log3 81  x
2
31.
log x 16  4
32.
2x  3
33.
ln  x  5  ln  x  1  ln  x  1
34.
5x  125
35.
8x  16 x
36.
log3  x  7   log3  2 x  1
2.
The graph at the right is a transformation of the graph of
(a) g ( x)   x  3
3.
y  3  2  x  4
2
11
x
3
3
(c) g ( x)   x  3
2
(d)
g ( x)  x 2  3
(e) None of these
(b)
y  2 x  1
(c) y 
3
3
x
2
2
(d) y  2 x  5
(e) None of these
(b)
y
4
x4
5
(c) y  
2
x4
3
(d) y  
4
7
x
5
2
What is the slope of the line perpendicular to the line 3 x  4 y  12?
(a) Undefined
6.
g ( x)  x 2  3
Find an equation of the line shown at the right.
(a) y 
5.
(b)
Find an equation of the line that passes through the points (1,–3) and (4, 3).
(a)
4.
2
f ( x)  x 2 . Find an equation for the function.
(b) 0
(c) 
4
3
(d) 
3
4
(e) None of these
Find an equation of the line that passes through (3, 10) parallel to the line x  3 y  1 .
(a) y 
1
x9
3
(b)
y  3x  1
(c) y  3 x  19
(d) y  
1
x  11
3
(e) None of these
7.
Given
f ( x)  6  2 x 2 , find f ( 3).
(a) 12
8.
(b) 24
(b) II
( x  3)  ( y  2)  49
2
(d) IV
(e) None of these
(b)
2 x  4 y  31  0
(c) x  2 y  7  0
(d) 2 x  8 y  45  0
2
(e) None of these
7.
(b)
(c)
(d)
( x  3)2  ( y  2)2  7
( x  3)2  ( y  2) 2  49
( x  3)2  ( y  2)2  7
(e)
None of
these
Determine the slope of the line that passes through the points (1, 3) and (–2, –2)
(a)
12.
(c) III
Find the standard equation of the circle with center (–3, –2) and radius of
(a)
11.
(e) None of these
Find a relationship between x and y so that (x, y) is equidistant from the two points (4, -1) and (6, 3).
(a) x  4 y  3  0
10.
(d) –24
Determine the quadrant in which the point (x, y) must be located if x>0 and y<0.
(a) I
9.
(c) –12
3
5
(b)
1
3
(c)
5
3
(d) 1
(e) None of these
Find an equation of the line that passes through the point (1, –1) and has a slope of–3.
(a) y  3 x  2
(b)
y  3x  2
(c) y  3x  1
(d) y  3 x  4
(e) None of these
13.
Add, then simplify:
x 1
x3
 2
x  x  2 x  4x  3
(a)
2
(b)
2x  3
( x  1)( x  3)( x  2)
14.
15.
3
2
x

Simplify: x  1 x  2
1
x 1
2
x  3x  4
 x2  x  4
(a)
(b)
( x  2)
( x  2)
2x  4
3 x  1
2
(c)
2 x
x2
(d)
2 x 2  3x  3
( x  1)( x  3)( x  2)
(e)
None
these
(d)
 x 2  3x  3
( x  1)( x  2)
(e) None of these
(d)
2 65
(e) None of these
(d)
120x4
(e) None of these
Find the distance between the points (–6, 10) and (12, 2)
(a)
16.
(c)
2x  x  3
2
( x  x  2)( x 2  4 x  3)
2
2 7
Simplify:
(a)
30x 6
(b)
2 97
(c) 10
6 6
x
5
(c)
3 x 2 (2 x)3 (5 x 1 )
(b)
24 4
x
5
of
Expand each completely of the following using Log Laws:
17.
log3 5x2
18.
Find the exact answer to each, using the Unit Circle.
 2 
 
19. sin  
20. cos 

6
 3 
 
ln
5x
y2
21.
 
tan  
4
22.
  
sin 

 6 
23.
sin  
24.
 5 
csc  
 6 
25.
 
sin  
2
26.
 
cos  
4
27.
 
cos  
2
28.
 3 
cos  
 4 
29.
 
tan  
6
30.
sin  0 
31.
cos  0
32.
 
tan  
2
33.
 3
arccos 

 2 
34.
1
cos1  
2
35.
 2
sin 1 

 2 
36.
tan 1 1
Find each limit algebraically.
17.

lim x 2  2
x 3

20.
x4
x 2 x  2x  8
22.
lim
25.
lim
x 5
18.
 x  3 x  4
x   3  x  3 x  1
lim
21.
2
x5
x5
2

 x  1;
f  x  

4  x;
a) lim f  x 
x 1
b) lim f  x 
19.
23.
1
x 8 x  8
lim
3x  1;
f  x  
3  x;
x 1
x 1
26.
x 1
a) lim f  x 
x 1
b) lim f  x 
x 1
x 5
x  25
x2  2 x  3
x   3 x 2  7 x  12
lim
24.
x3  8
x 2 x  2
lim
 x 2 ;

f  x   2;
 x  2;

x 1
x 1
27.
a) lim f  x 
x 1
x 1
b) lim f  x 
x 1
c) lim f  x 
c) lim f  x 
c) lim f  x 
lim
x  25
x 1
x 1
Determine the following limits graphically:
28.
29.
30.
a) lim f  x 
a) lim f  x 
a) lim f  x 
b) lim f  x 
b) lim f  x 
b) lim f  x 
c) lim f  x 
c) lim f  x 
c) lim f  x 
x 1
x 1
x 1
x 1
x 1
x 1
x 1
x 1
x 1
x 1
x 1
x 1