Download calculus bc - Fort Bend ISD

TO: Next Year’s AP Calculus AB Students
FROM: Mr. Wernau, AP Calculus AB 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. Calculus AB will basically start where
Pre-Calculus left off and move on from there.
Attached is a summer homework packet, which will be due the first day of Calculus
class in August. The material in the packet should be material you learned in
Algebra II and Precalculus.
You will turn in the packet the first week of Calculus class, and it will count as a
daily grade. On the fifth 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. You will be expected to have a TI-83, TI-84, or a TI-89 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 AB
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.
Parent Graphs: need the know the following:
1. y  x
2. y  x 2
3. y  x 3
x
5. y  e x
6. y  ln x
7. y 
x
8. y  sin x,  2  x  2
9. y  cos x,  2  x  2
4. y  x
Find the x- and y-intercepts and the domain and range, and sketch the graph.
Do not use your graphing calculator on these.
2
10. y  2  x 2
11. y   x  2 
12. y  x  1
13. y  9  x 2 (semi-circle)
14. y  x3  2
1 , if x  1

12. y  3x  2, if x  1
7  2 x, if x  1

______________________________________________________________________________
Find the asymptotes (horizontal, vertical, and slant), symmetry, and intercepts, and sketch the
graph. Do not use your graphing calculator on these.
16. y 
1
x 1
17. y 
x3
x2  9
18. y 
1
 x  2
2
3 x( x  2)
x2  2 x  4
20. y  2
x 1
x  5x  6
_____________________________________________________________________________
Solve the following polynomial inequalities.
3x  2
0
21. x 2  x  12  0
22.
x4
19. y 
23.
 x  2   x  1  x  5  0
2
3
24.
 2 x  5 x  1
3
 x  2
2
0
Optimization Problems (quadratic word problems)
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.
These are sometimes called quadratic word problems.
Ex 1: 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 the
sum of the two numbers.
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.
The minimum value is (10.724, 21.448)
Use the x-value of the minimum and substitute back into Eqn 1: (10.724)y=115
Solve for y.
Solution: x = 10.724
y = 10.724
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.
25. Find two positive numbers such that their product is 192 and their sum is a minimum.
26. Find two positive numbers such that their product is 192 and the sum of the first plus
three times the second is a minimum.
27. 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?
KNOW THE UNIT CIRCLE: attached is a handout that should help. We will go over in class
during the first week.
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)
Cos
Sin
Sec
Csc
tan
Cot
0

6

4

3

2
Simplify the following Trigonometric Values:

3
28. sin
29. cos
30. tan 
3
4

5

33. csc
35. sec
37. tan
6
4
2

41. tan
42. sec 0
6
3
2
7
39. sec
6
31. sin
Solve. Give exact answers in radians, 0  x  2 .
3
1
1
43. sin x 
45. cos x  
46. sin x  
2
2
2
47. tan x  1
48. 2cos x  3  0 49. sin 2x  1
32. cot
2
3
40. sin 2
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
Simplify:
50. (1  cos  )(1  cos  )
53.
sin x cos x
1  cos 2 x
51. tan x  sec x
2
54. csc x tan x
2
sin 2 A
52. 1 
tan 2 A
56. 4 sin
55. tan x cos x
7
7
cos
12
12
Logarithms:
Inverse Property: log b y  x iff b x  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:
57. log 4 (3m)
60. ln 3 3 x
Recall: ln x  log e x
58. log 9 (27m 2 )
59. log 5 3x  7
 x 4
61. ln 

 3x 
Solve the following log and exponential equations for x:
62. 3 x  3 x  4
63. 4 x  16
64. 2 3 x 1  2 x 1
65. 3 x  9 x 1
67. log 4 x  3
68. log 2 8  x
69. log 3 9  x  1
x
1
66.    4
2
70. 2 log 3 x  8
71. log 4 ( x  3)  2