AP Calculus Summer Assignment
Due: The first day of school.
Directions: This assignment is to be done at your leisure during the summer. The
primary emphasis is on graphing calculator skills and reviewing your pre-calculus
knowledge. (You will need to check out a calculus book over the summer) There are
four parts to the assignment. You need to make a set of flash cards (3 x 5 or larger) of
pre-calculus formulas and definitions. These can be added to your notebook if you like.
You need to memorize the formulas before the first day of class. You will also need to
do the problems on the attached pages and review the various functions of your graphing
calculator. You will be tested on the material.
All written work should be done neatly on separate binder paper (except for the trig
chart), in order and clearly identified. All work and flashcards need to be turned in on the
first day of class in a large envelope with your name on it. The flash cards should be
secured with a rubber band.
The time spent on this assignment over the summer will help you to be successful in your
AP calculus class. You will get the most out of this assignment if you space out your
work over the summer, paying careful attention as you go along. If you have any
questions please see Mr. Little(B-6), Mr. Powers(C-3) or Ms. James(Q-206) before the
end of the school year. During the summer you can contact us by email. We look
forward to working with you next year. Have a great summer.
Remember: You will be tested on the material.
I)
Flash Cards
The following information is to be placed on index cards and mastered. Use diagrams or
sketches whenever possible.
1.
2.
3.
4.
5.
6.
7.
Definition of a function
Definition of domain and range Include different types of examples.
Symmetry, especially even and odd functions.
Definition and properties of inverse functions.
The main properties of exponential functions and graphs.
The main properties of logarithmic functions and graphs.
The rules and examples of transformations of parent graphs. (reflections,
vertical/horizontal shifts, stretching/compression, etc.)
8. Absolute value as a distance, number line solution, inequality and interval notation.
9. The unit circle with definition of sine and cosine functions, also definition of tangent,
cotangent, secant and cosecant functions.
10. The trigonometric identities, especially those involving pythagorean’s theorem, sums
and differences of angles and double and half angles. Know when and how to use
them.
11. Special right triangles and exact trigonometric values.
12. Basic properties of graphing trigonometric functions by looking at amplitude, period,
phase shift and vertical shifts.
13. The graphs of the six trig functions as well as their inverse functions. Domain, range
and any other significant details of each should be given.
14. Areas of standard geometric shapes. (square, rectangle, triangles, circle, trapezoid,
parallelogram, etc.)
15. Volumes of standard geometric shapes. (cubes, spheres, pyramids, cones, right
prisms, etc.)
16. Graph and identify key parts of conic sections. Know the general and standard
equation of each. (parabola, circle, ellipse, hyperbola)
17. Basic summation properties.
II)
Graphing Calculator Skills to Master Over the Summer.
*Find intersection of two functions.
*Finding the roots of a function.
*Finding the maximum or minimum of a function.
*Learn the zoom features of your calculator.
*Locate special features (eg. Absolute value, greatest integer function, radicals…)
*Solving an equation using solver
III) Problem Set
A) Graph the parent function of each set using your calculator. List the domain and
range of each parent graph. Describe each changed graph in a few words (eg. moved left
2 units, reflected over the x-axis, etc.). Graph all variations on graph paper. Do not
overcrowd the graphs. Put each graph on its own set of axes. Each graph should have
key units indicated on the axes. Test your own ability to graph without the calculator.
These would be considered standard functions for entering calculus students in almost
any school or university.
1) Parent Function:
y= x
2
b g
c) y = −b x − 3g + 2
b x + 4g
d) y = −2b x + 1g − 4
2) Parent Function:
y = sin(x); (set mode to radians)
b) y = sin( x ) − 2
a) y = x − 2
2
2
a) y = sin(2 x )
c) y = 3sin( x )
3) Parent Function:
a) y = cos(2 x )
c) y = 3cos( x )
b) y =
1
3
2
2
d) y = 5sin( 12 x ) + 1
y = cos(x); (set mode to radians)
b) y = cos( 14 x )
d) y = −3cos( x ) + 1
4) Parent Function:
y=
1
x
1
x+2
1
c) y =
+1
x+4
1
x
2
d) y =
5− x
a) y =
5) Parent Function:
b) y = −
y= x
3
b g
c) y = −b x + 3g + 2
d) y = −2 x + 1 − 2
6) Parent Function:
y=
a) y = x − 4
3
3
b) y = 21 x 3 + 3
b g
x
a) y = − x
b) y = x + 2
c) y = 6 − x + 2
d) y = −2 x + 4
7) Parent Function:
y = ln(x)
b) y = ln − x
b g
c) y = lnc x h
a) y = ln x + 3
b g
d) y = −3 lnb x g + 1
8) Parent Function:
y= e
a) y = e x - 3
b) y = e 2
c) y = 2 − e x
d) y = − e 3− x
x
1
9) Parent Function:
3
y= x
x
; (greatest integer function, do not connect steps)
a) y = x + 2
b) y = x − 3
c) y = 2 x
d) y = 3 − x
B) Use your graphing calculator to find the roots.
4
3
2
10)
Given:
f ( x) = x − 3x + 2 x − 7 x − 11
Find all roots to the nearest 0.001 using your calculator
11)
Given:
f (x) = 3sin x − 4x + 1
Find all roots to the nearest 0.001 using your calculator
C) Solve the following inequalities. No calculator.
2
12) x − x − 6 > 0
13) x 3 − 4 x − 5 ≤ 0
D) For problem 14, sketch the graph of y as described in parts a-e and write a brief
description of the changes. No calculator.
a) graph y = f x
14) f x = x 2 − 5x − 3
b) graph y = − f x
c) graph y = f − x
d) graph y = 2 f x
bg
bg
b g
bg
e) graph y = f b2 x g
bg
E)
15) Solve each equation for x unless indicated differently. (no calculator)
a) lnx – ln5 = ln7
b) lnx + ln5 = 7
c) sin2x – cosx = 0 (exact solutions)
3x 2
d)
= 2y + b
y
3x 2
e)
= 2y + b
solve for b
y
3x 2
= 2y + b
solve for y
f)
y
16) Change each radical expression into an exponent expression.
a) x
b) 7 b3 y
c) 3 m 5
d)
4
3x 2 − 7x
17) Simplify each expression.
c hb xg
a) 7 x
3
1
2
b5y − 2g b5y − 2g
c)
b5y − 2g
2
2
3
4
b) m m
1
3
−7
3
18) If sec θ = 3 in quadrant IV, find each of the following:
a) cosθ
b) tanθ
c)sin θ
d)cos2θ
F) Solve the trigonometric equations for all solutions in the domain. Use exact values
and be sure to put angle measures in radians.
19) 2 cos 3x − 1 = 0
(all reals)
b g
2
20) sin x − 3 sin x − 2 = 0
[0,2π )
G) Find the domain and range of each. Show work. No calculator.
x 2 − 3x
21a) y = sin x
b) y = 2
c) ln x − 2
x −9
b g
H) Solve for x. Show all work.
22) 80e 0.045 x = 240
b g
24) log 4 1 − x = 1
23) 2 x
2
+1
=8
25) 100sin x = 10 for x in 0, π2
I) Functions and modeling. Function notation can be written as y=f(x), where f is the
name for the function. Example: height and age can be written as h=f(a) since height is a
function of age. Here are some pairs of related variables. In each case, a function can be
used to model the relationship between them. For each pair:
-Decide which to take as independent and dependent variable.
-Express the relationship in function notation.
-Write down a possible domain for the function.
-Sketch a possible graph of the function.
26) the number of people in a room who are smoking and the concentration of smoke in
the air.
27) the number of chocolate bars someone buys and the amount paid for them.
28) the number of people sharing a money prize equally and the amount each person gets.
29) the amount of cans of beer a person and his/her blood alcohol level.
J) More Modeling. Read the examples carefully. Then write the specified quantity as a
function of the specified variable. It will help to draw a picture.
30) A rectangular field of area 20,000 sq. ft. is to be fenced off next to a river, with
fencing on three sides and the river making the fourth side. Express the length of fencing
necessary as a function of the width of the field.
31) The shape of a window is given by two squares, one on top of the other, with a
semicircle on top of that. Find the perimeter and area of the window as a function of the
width of the window.
32) A cylindrical can is to be made up from sheet steel so that its surface area is 100sq.
in. Express the height as a function of the radius. Hint: Imagine removing the top and
bottom with a can opener, and splitting the rest down the side and unrolling flat.
33) An open cardboard box is to be constructed from a rectangular 8-inch by 10-inch
sheet by cutting identical squares of side x out of each corner, and folding up the resulting
edges. Determine the volume of the box as a function of x.
IV) Book Problems.
K) Do the following problems from the calculus book.
34) Pg. 8 (44, 47)
35) Pg. 53 (65, 66)
36) Pg. 62 (32-34, 40-42)