Download Summer Packet CALCULUS BC 2015 with

TO: Next Year’s AP Calculus BC Students
FROM: T. Wernau, AP Calculus BC Teacher
As you probably know, the students who take AP Calculus AB and pass the
Advanced Placement Test will place out of one semester of college Calculus;
those who take AP Calculus BC and pass the Advanced Placement Test will place
out of two semesters of college Calculus. Because Calculus BC covers two
semesters, you will be learning Calculus twice as fast as the students who take
Calculus AB. In order to have enough time to learn the material of two collegelevel courses, we will start learning Calculus as soon as possible when school begins
and will not be able to spend time reviewing the material you learned in Algebra I &
II, Geometry, and Precalculus.
Attached is a summer homework packet, which will need to be completed by first
day of Calculus class in August. The material in the packet should be material you
learned in Algebra II and Precalculus.
On the fourth day of class, you will take a test on the material in the packet.
My recommendation is that you look over the problems in the packet when you
receive it but that you wait until the week before school starts to work the
problems so that you will remember the material very well when school starts.
All of the AP Calculus AB and BC classes at Travis will be using a graphing
calculator. I feel that the TI-89 is the best calculator available. You will be
expected to have a TI-83, TI-84, or a TI-89 (recommended) the first week of
school.
I am looking forward to seeing you in Calculus in August.
Sincerely,
Mr. Wernau
Travis High School Math Department
CALCULUS BC
SUMMER NOTES
Some of the material that was briefly covered in Pre-Calculus is knowledge that I need you to
know to be able to complete the packet. Here are some notes/examples for you to use:
The Limit:
x3  1
.
x 1
What do we know about the graph of this function? __There is a hole at x=1__
Right!
Graph it.
1) Let’s take a look at the function f ( x) 
If we put the point of our pencil on f(0.2) and trace the curve towards f(1) our pencils
approaches the y-value of 3.
We denote this as : lim f ( x)  3 because as we approach f(1) from the left side (the reason for
x 1
the raise – next to the 1) the value of the graph approaches 3.
If we put the point of our pencil on f(1.2) and trace the curve towards f(1) our pencils
approaches the y-value of 3.
We denote this as: lim f ( x)  3 because as we approach f(1) from the right side (the reason for
x 1
the raised + next to the 1) the value of the graph approaches 3.
Because the limit from the left side is equal to the limit from the right side we can use the
notation: lim f ( x)  3 (notice the lack of a raised + or –). If this is not the case, we say the limit
x 1
does not exist.
______________________________________________________________
2) We can also evaluate simple limits quite easily, without even looking at the graph.
Ex 1: lim x2  3x  1
x 3
 If we first think about the function f ( x)  x 2  3x  1 we know
that it’s graph does not have any special properties
(holes/asymptotes/etc)
 substitute the x-value we are approaching with the x-values in
the function.
lim( x2  3x 1)  32  3(3)  1  17
x3
Therefore, lim( x2  3x 1) = 17.
x3
Ex 2: lim
x 2
x2  2 x  8
x2
x2  2x  8
we know
x2
there exists a hole at x = 2 and that happens to be the x-value
we want to know about, so our previous method won’t work. So
we have to be a little more creative
 If we first think about the function f ( x) 
 Think about the factors of x 2  2 x  8 . One of the factors is (x – 2).
x2  2x  8
( x  2)( x  4)
lim
 lim
 lim x  4 Now we can substitute like we did on problem #1.
x 2
x

2
x 2
x2
x2
lim( x  4)  6
x2
x2  2 x  8
6
x 2
x2
______________________________________________________________________________
x  2 1
Ex 3: lim
 Try using the conjugates
x 0
x
Therefore, lim
lim
x 0
x9 9
x
x 9 9
x 99
 lim
 lim
x 0
x  9  9 x 0 x x  9  9
x



x
x 9 9

 lim
x 0

1
x9 9


1
6
______________________________________________________________________________
x3  2 x 2  5
Ex 4: lim
 If the limit is going off to positive or negative infinity, it is
x  3 x  4 x 3  2
asking you to find the horizontal asymptote. So remember your
methods to finding the horizontal asymptote. (Ratio of the
leading coefficients)
x3  2 x 2  5
x3  2 x 2  5
1

lim

3
3
x  3 x  4 x  2
x  4 x  3 x  2
4
____________________________________________________________________________
lim
Note: To complete the optimization problems, you should have learned how to use a system of
equations, the substitution method, and your calculator to find the maximum or minimum value.
Ex 5: Find two positive numbers whose product is 115 and whose sum is a minimum.
Eqn 1: xy  115
Eqn 2: S  x  y , where x represents my first number, y represents my
second number, and S represents my sum function.
115
Combine equations by solving for x or y in eqn 1, you get: S  x 
x
Put S into Y1 and use calc to find the minimum x-value. Then use that x-value in eqn 1 to solve
for the y-value. In this case,
x  10.724
y  10.724
CALCULUS BC
SUMMER HOMEWORK
This homework packet is due the first day of school. It will be turned in the
first day of Calculus class and will count as a daily grade. You will take a test
on the material in the packet on the third day of school.
Work these problems on notebook paper. All work must be shown.
Find the x- and y-intercepts and the domain and range, and sketch the graph.
Do not use your graphing calculator on these.
1. y  2  x 2
10. y  e x
2. y   x  2 
2
11. y  ln x
1 , if x  1

12. y  3x  2, if x  1
7  2 x, if x  1

3. y  x  1
4. y  9  x 2
 x 2  1, if x  0
13. y  
2 x  2, if x  0
5. y  x  2
3
6. y 
x
x
7. y  sin x,  2  x  2
8. y  cos x,  2  x  2
9. y  tan x,  2  x  2
______________________________________________________________________________
Find the asymptotes (horizontal, vertical, and slant), symmetry, and intercepts, and sketch the
graph. Do not use your graphing calculator on these.
2 x2  9
1
14. y 
16. y 
x2  4
x 1
1
x2  2 x  4
15. y 
17.
y

2
x 1
 x  2


_____________________________________________________________________________
Solve.
3x  2
0
18. x 2  x  12  0
20.
x4
19.
 x  2   x  1  x  5  0
2
3
21.
 2 x  5 x  1
3
 x  2
2
0
Find the limit.

x 2
22. lim 4 x 2  5 x  3

23. lim
x 0
24. lim x  4
2
25.
x0
x 1 1
x
 x  5
lim
2
 25
x 0
x
3x  5 x 2
27. lim
x  4 x 2  1
2  3 x3
29. lim 2
x  x  4
x  x6
x  3
x3
x6
28. lim 2
x  6 x  3 x  18
x3  8
30. lim
x  2 x  2
______________________________________________________________________________
Use the figure to find the limit.
y
31. lim f  x 
32. lim f  x 
y
2
26. lim
x 3
33. lim f  x 
x 
35.
lim f  x 
x 2
x0
34. lim f  x 
x  
36.
lim f  x 
x
x  5
Graph of f
______________________________________________________________________________
Write a function for each problem, and use your graphing calculator to solve. Give decimal
answers correct to three decimal places. Be sure to sketch the graph you used, and label it with
your answer.
37. Find two positive numbers such that their product is 192 and their sum is a minimum.
38. Find two positive numbers such that their product is 192 and the sum of the first plus
three times the second is a minimum.
39. A rancher has 200 feet of fencing with which to enclose two adjacent
rectangular corrals, as shown. What dimensions should be used so
that the enclosed area will be a maximum?
40. A rectangle is bounded by the x-axis and the semicircle y  25  x 2
as shown. What length and width should the rectangle have so that
its area is a maximum?
41. A rectangular pace is to contain 24 square inches of print. The margins at the top and
1
bottom of the page are to be 1 inches, and the margins on the left and right are to
2
be 1 inch. What should the dimensions of the page be so that the least amount of paper
is used?
42. Four feet of wire is to be used to form a square and a circle. How much of the wire
should be used for the square and how much should be used for the circle to enclose
the maximum total area?
43. The sum of the perimeters of an equilateral triangle and a square is 10. Find the
dimensions of the triangle and the square that produce a minimum total area.
44. Two posts, one 12 feet high and the other 28 feet high,
stand 30 feet apart. They are to be stayed by two wires,
attached to a single stake, running from ground level to
the top of each post. How far from the left post should
the stake be placed to use the least wire?
Know your Trig. Values: here is a worksheet to help you.
To use the hand trick:
1. Always use your left hand, palm facing you.
2. Your hand represents the first quadrant only in radians-must use reference angles for any
angle out of the first quadrant.
1
3. Pinky finger is x-axis (0 radians) and thumb is the y-axis (  radians).
2
4. Whatever radian measure, pull that finger-that represents the comma in the ordered pair
of trig. values (cos, sin).
5. All the fingers on top of the pulled down finger are the cosine, on bottom the sine.
fingers
6. Formula: for sin, cos, csc, and sec =
2
fingers
sin
7. Formula for Tangent =
. However recall that Tangent is
. You may need
cos
fingers
to flip hand to palm facing away from you.
Let’s try a few: Find all six Trig values for the following.
Angle(radians)
0

6

4

3

2
Cos
Sin
Sec
Csc
tan
Cot
Solve. Give exact answers in radians, 0  x  2 .
45. 2cos x  3  0
46. 2 cos 2 x  3cos x  2  0
47. sin 2x  1
48. 2 cos 2 x  sin x  1  0
49. sin 2x  cos x
50. tan 2 x  sec x  1
Know the following Trigonometric Identities:
Pythagorean: sin 2 x  cos 2 x  1
Division: sec x 
1
cos x
csc x 
1  tan 2 x  sec 2 x
1
sin x
cot x 
1
tan x
tan x 
1  cot 2 x  csc 2 x
sin x
cos x
cot x 
cos x
sin x
Double Angle: sin( 2 x)  2 sin x cos x
Product to Sum:
1
sin x  sin y  [cos( x  y )  cos( x  y )
2
1
cos x  cos y  [cos( x  y )  cos( x  y )
2
1
sin x  cos y  [sin( x  y )  sin( x  y )
2
Logarithms:
Inverse Property: log b y  x iff
Recall: ln x  log e x
bx  y
Product Property: log b (mn)  log b m  log b n
m
Quotient Property: log b ( )  log b m  log b n
n
n
Power Property: log b (m )  n  log b m
Expand the following:
 x 4
53. ln 

 3x 
52. log 9 (27m 2 )
51. log 4 (3m)
Solve the following log and exponential equations for x:
x
54. 3  3
x4
x 1
55. 4  16
56. 3  9
58. log 4 x  3
59. log 2 8  x
60. log 3 9  x  1
61. 2 log 3 x  8
62. log 4 ( x  3)  2
x
x
x
1
57.    4
2