A.P. Calculus AB
Summer Assignment 2015
The following packet contains topics and definitions that you will be required to know in order to
succeed in A.P. Calculus AB this year. You are advised to be familiar with each of the
concepts and to complete the included problems by September 3, 2015. All of these topics were
discussed in either Algebra II or Precalculus and will be used frequently throughout the year.
All problems are expected to be completed.
Due on September 4, 2014.
Copy each problem onto your paper and solve.
Show your procedure, not just your answer.
If you use a graph you should show a properly labeled sketch of that graph!
There will be a test covering this material.
1. Expand the following binomials
a. Expand ( x  h)3
b. Expand
x
2

3
3
2. Simplify the following, if possible… If not possible, state the reason why?
a.
c.
e.
3x  2 y
3x  y
b.
a
a

2x  h 2x
h
d.
3x  2 y
3y
 x11   1
x
x9 3
1

x
x9 3
(without using cross multiplication)
9  x 2
3  x 1
f. Show
3 5
g. Rationalize the denominator


i. Multiply 2  3 2  3


1 5

k. Multiply 3  2 5 2  4 5
j. Multiply

5 2

5 2
4  15


Add or subtract. If the operation is not possible, explain:
50  3 32  5 18
l.
m. 4 2  5 3
o. Simplify
6  15
h. Rationalize the denominator
72x3
r. Simplify 6 18  4 8  3 72
n.
74 5  23 5
p. Simplify
3
2
3x
s. Simplify
3
54 x 2 y 3  3 5 x 3 y 4
q.
Simplify
2
3
3. Write each of the following expressions in the form ca p b q where c, p and q are numbers:
a.
( 2a 2 ) 3
b
b.
3
9ab
a 1
c.
2
a 
b
3
a
e.
d.
ab  a
b2  b
f.
b 
1
 a2 / 3 
 1 / 2 
b 
a
2
 b3 / 2 
 1 / 2 
a 
4. Expand the summation notation and simplify the final answer.
4
a.

n0
3
n2
2
b.
1
n
n 1
3
5. Solve the following equations for the indicated variables:
a.
x y z
   1 , for a
a b c
b. A  P  nr P , for P
d. 2 x  2 yd  y  xd , for d
c. V  2(ab  bc  ca) , for a
e.
2x
1 x

 0 , for x
4
2
A  2r 2  2rh , for positive r
f.
6. For the following equations, complete the square and reduce to one of the standard forms
y  b  A( x  a) 2 or x  a  A( y  b) 2 .
Then sketch the graph, indicating the vertex, x and y intercepts, and axes of symmetry (if any).
a.
y  x2  4x  3
b.
3x 2  3x  2 y  0
c.
9 y2  6 y  9  x  0
7. Factor completely:
a.
x 6  16x 4
b.
4 x 3  8 x 2  25 x  50
c.
8 x 3  27
d.
x4 1
8. Find all real solutions to the following without using a calculator.
a.
x 6  16 x 4  0
b.
4 x 3  8 x 2  25 x  50  0
c.
8 x 3  27  0
c.
8 x 3  27  0
9. Find all real solutions to the following using a calculator.
a.
x 6  16 x 4  0
b.
4 x 3  8 x 2  25 x  50  0
10. Solve for x without using a calculator.:
0  x  2
a.
3 sin 2 x  cos 2 x ;
b.
cos x  sin x  sin x ;    x  
2
2
11. Solve for x; use a graphing calculator. Include a properly labeled graph on your paper and compare
your results with the problems in 10 when appropriate:
0  x  2
a.
3sin 2 x  cos2 x ;
b.
cos x  sin x  sin x ;    x  
2
2
12. Without using a calculator, evaluate the following.
(in each case, show how you used a triangle and the unit circle)
a.
cos(210)
b.
 5 
sin 

 4 
c.
tan 1 (1)
d.
sin 1 (1)
e.
 9 
cos 

 4 
f.
 3
sin 1 

 2 
g.
 7 
tan 

 6 
h.
cos 1 (1)
13. Solve the equations with a calculator.
14. Solve the equations without a calculator.
15. The equation 12 x 3  23x 2  3x  2  0 has a solution x  2 .
Find all other solutions without using a calculator.
16. Solve the inequalities with a calculator.
17. Solve the inequalities without a calculator.
18. Solve for x with a calculator.
19. Solve for x without a calculator.
20. Determine the equations of the following lines. Give your answer in point-slope form.
a. through (-1, 3) and (2, -4)
b. through (-1; 2) and perpendicular to the line 2 x  3 y  5  0
c. through (2, 3) and the midpoint of the segment from (-1, 4) to (3, 2)
21. Find the point of intersection of the lines with a calculator 3x  y  7  0 and x  5 y  3  0
22. Find the point of intersection of the lines without a calculator 3x  y  7  0 and x  5 y  3  0
23. Shade the region in the x-y plane that is described by the inequalities
3x  y  7  0
x  5y  3  0
24. Find the domain and range of each of the following and sketch a graph.
You may use the calculator where appropriate.
(a) f  x  
(c) g  x  
25. Simplify
a.
3x  1
(b) f  x   7
x2  x  2
5x  3
2x  1
(d) f  x  
f  x  h  f  x
, where
h
f ( x)  2 x  3
b.
f ( x) 
1
x 1
x
x
26. The graph of the function y  f  x  is given
Show the graphs of the functions:
27. Sketch the graphs of the functions (do not use a calculator!):
28. Find the inverses of the functions:
29. A function y  f  x  has the graph to
the right. Sketch the graph of the
inverse function
30. Express x in terms of the other variables in the picture.
(a)
(b)
Problem #31.
Find the ratio of the
area inside the square
but outside the circle
to the area of the
square.
Problem #32.
Find a formula for
the perimeter of a
window of the shape
in the picture which
is a semi-circle on
top of a rectangle.
33. A tank has the shape of a cone (like an ice cream cone without the ice cream). The tank is 10 [m]
high and has a radius of 3 [m] at the top. If water is poured into the tank until the depth is 5 [m],
what is the surface area of the top of the water? (Picture required.)
34. Two cars start moving from the same point. One travels south at 100 [km/h], the other travels west
at 50 [km/h]. How far apart are they two hours later? (Picture required.)
35. A kite is 100 [m] above the ground. If there are 200 [m] of string out, what is the angle between the
string and the horizontal. (Picture required.) (Assume that the string is perfectly straight.)
36. You should know the following trigonometric identities.
(A) sin(-x) = -sin x
(C) cos(x + y) = cos x cos y - sin x sin y
(B) cos(-x) = cos x
(D) sin(x + y) = sin x cos y + cos x sin y
(E) sin2 x + cos2 x=1
Use these to derive the following important identities, which you should also know.
a. tan( 2 x)  1  sec( 2 x)
b. sin( 2 x)  2 sin( x) cos( x)
d . cos( 2 x)  2 cos( 2 x)  1
e. cos( 2 x) 1  2 sin( 2 x)
c. cos( 2 x)  cos( 2 x)  sin( 2 x)
37. Using your graphing calculator, find all the intersections of the curves
x
3x  1
5
g  x   2e .
f  x   10  2
and
x  25
You may need more than one window (and hence more than one picture) to complete the job.