AP Calculus Summer Prep TLHS – 2014 1

AP Calculus Preparation Packet

Summer Review 2014 Trinity Lutheran High School

Students need a strong foundation to be ready for the work required through

the term. The problems in this packet are designed to help review topics from

previous math courses that are important to success in AP Calculus. This packet

includes:

A “Function Review”; you should be familiar with each of the graphs.

A formula and identities reference sheet. These are for your reference

and do not need to be memorized at this time. (Refer to your pink sheet

as well).

A unit circle template with which to practice your unit circle. (Degrees,

Radians, Coordinate points)

A list of skills that you will need for AP Calculus.

AP Calculus Preparation Problems, please show all work that leads you to

each solution.

o You are permitted to use notes from previous math courses to help

you, but you should do the work without any help from another

person.

o You are permitted to use a calculator to complete this packet.

o Answers are provided at the end of the packet.

The packet will be collected on the first day of class and your grade will be

determined on neatness, completeness of solutions, and accuracy. In

preparation for the AP test, you need to begin showing all work with logical

steps. DO NOT list only an answer. Work neatly and in an organized fashion.

If you need help…There will be an optional help session as indicated in the box

below. This is a help session to work on problems you are having difficulty with,

not a session in which to do your packet. Please complete as much of the

packet as possible before attending the session(s). If you have questions and

are unable to attend due to vacation, let me know (jessie.fowls@saints.org) and

we can work out another time to meet.

Optional Help Sessions

1 2 Date: Friday, August 22 Date: Tuesday, August 26 Time: 12:00-2:00pm Time: 4:00-6:00pm Location: Ms. Fowls’ classroom Location: Ms. Fowls’ classroom

AP Calculus Summer Prep TLHS – 2014 2

Function Review

Students should know the basic shape of these functions and be able to graph

their transformations without the assistance of a calculator.

Constant

f(x) = a

Identity

f(x) = x

Absolute Value

f(x) = |x|

Reciprocal

f(x) = 1/x

Quadratic

f(x) = x2

Cubic

f(x) = x3

Square Root

f(x) = √

Greatest Integer

f(x) = [x]

Exponential

f(x) = ax

Logarithmic

f(x) = ln x

AP Calculus Summer Prep TLHS – 2014 3

Trig Functions

f(x) = sin x f(x) = cos x f(x) = tan x

Polynomial Functions

A function P is called a polynomial if ( )

. Where n is a nonnegative integer and the numbers a0, a1, a2, …an are

constants.

Even Degree Odd Degree

Leading Coefficient Sign Leading Coefficient Sign

Positive Negative Positive Negative

Number of roots equals the degree of the polynomial.

Number of x intercepts is less than or equal to the degree.

Number of curves is less than or equal to (degree minus 1).

AP Calculus Summer Prep TLHS – 2014 4

Formulas & Identities

Trig Formulas

Arc Length of a circle:

Area of a sector of a circle:

Solving parts of a triangle

Law of Sines:

Law of Cosines:

Area of a Triangle

or

or

Hero’s Formula: √ ( )( )( ), where

( )

Trig Identities (See Pink Sheet)

Polar Formulas

Geometric Formulas

Area of a trapezoid:

( )

Area of a triangle:

Area of an equilateral triangle: √

Area of a circle:

Circumference of a circle: or

AP Calculus Summer Prep TLHS – 2014 5

Unit Circle – Degrees, Radians, & Coordinates

(____ , _____)

(____ , _____)

AP Calculus Summer Prep TLHS – 2014 6

Skills Needed for AP Calculus

Algebra

Exponents (operations with integer, fractional, and negative exponents)

Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes,

grouping)

Rationalizing

Simplifying rational expressions

Solving algebraic equations and inequalities (linear, quadratic, higher order using

synthetic division, rational, radical, and absolute value equations)

Simultaneous equations

Graphing and Functions

Lines (intercepts, slopes, write equations using point-slope and slope intercept,

parallel, perpendicular, distance and midpoint formulas)

Conic Sections (circle, parabola, ellipse, and hyperbola)

Functions (definitions, notation, domain, range, inverse, composition)

Basic shapes and transformations of the following functions (absolute value, rational,

root, higher order curves, log, ln, exponential, trigonometric, piece-wise, inverse

functions)

Tests for symmetry: odd, even

Geometry

Pythagorean Theorem

Area Formulas (circle, polygons, surface area of solids)

Volume formulas

Similar Triangles

Logarithmic and Exponential Functions

Simplify Expressions (use laws of logarithms and exponents)

Solve exponential and logarithmic equations (include ln as well as log)

Sketch graphs

Inverses

Trigonometry

Unit Circle (definition of functions, angles in radians and degrees)

Use of Pythagorean Identities & formulas to simplify expressions & prove identities

Solve equations

Inverse Trigonometric functions

Right triangle trigonometry

Graphs

AP Calculus Summer Prep TLHS – 2014 7

AP Calculus Preparation Problems

Algebra Review

Simplify

1. ( )

⁄ ( )

⁄ (

⁄ )

Factor Completely

2. (Hint: Grouping)

3. (Hint: Factor as difference of squares first, then cubes second)

4.

5. ⁄

⁄

⁄ (Hint: Factor GCF

⁄ first)

Rationalize

6.

√

Simplify

7. ( ) ( ) ( )

( )

Use synthetic division to help factor, state all factors and roots.

8. ( )

Solve (You may use your graphing calculator to check solutions)

9. (Factor first)

10.

AP Calculus Summer Prep TLHS – 2014 8

Graphing and Functions

Write the equation of the line described below.

11. Passes through the point (4, -3) and is perpendicular to .

12. Passes through the point (-1, -2) and is parallel to

.

Find the domain and range of the following.

13. y = log(x-3)

14. √

15. y = |x – 5|

Find the composition or inverses as indicated.

Let ( ) , ( ) , ( )

16. ( )

17. ( ( ))

18. ( ( ( )))

Let ( ) , ( ) , ( )

19. ( )( )

Sketch the graphs. You may use your graphing calculator to verify your graph,

but you should be able to graph the general shape of the curve without the use

of the calculator, by plotting a few points, and by your knowledge of

transformations.

20.

21. (

)

AP Calculus Summer Prep TLHS – 2014 9

Identify as odd, even, or neither and justify your answer. To justify your answer

you must show substitution using –x! It is not enough to simply check one

number.

Even: f(x) = f(-x) Odd: f(-x) = -f(x)

22. ( )

23. ( ) | |

Logarithmic and Exponential Functions

Simplify

24.

(

)

25.

26.

27.

Solve

28. ( ) ( )

29.

Trigonometry

Unit Circle: Know the unit circle – both radian & degree measures, coordinates

and trig functions.

Verify

30. ( )( )

Solve the equation

31.

AP Calculus Summer Prep TLHS – 2014 10

Solve the inverse trig functions

32. (√

)

33. ( (√

))

34. Find the value of x.

35. Find the value of x.

Be able to do the following on the graphing calculator.

Be familiar with the CALC commands; value, root, minimum, maximum,

intersect. You may need to zoom in on areas on your graph to find the

information. Sketch the graph.

#36 - #39 Given the following function ( ) .

(Window: x min: -10, x max: 10, scale 1, y min: -100, y max: 60, scale 10)

36. Find all roots.

37. Find all local maxima.

38. Find all local minima.

39. Find the following values: f(-1), f(2), f(0), f(.125)

10 X

50°

70° 70°

X

10

AP Calculus Summer Prep TLHS – 2014 11

40. A rectangle has perimeter 20 m. Express the area of the rectangle as a

function of the length of one of its sides.

41. Biologists have noticed that the chirping rate of crickets of a certain

species is related to temperature, and the relationship appears to be very

nearly linear. A cricket produces 113 chirps per minute at 70° and 173

chirps per minute at 80° F.

a. Find a linear equation that models the temperature T as a function

of the number of chirps per minute N.

b. What is the slope of the graph? What does it represent?

c. If the crickets are chirping at 150 chirps per minute, estimate the

temperature.

AP Calculus Summer Prep TLHS – 2014 12

Answers

1.

⁄ ⁄

2. ( )( ) 3. ( )( )( )( ) 4. ( )( )

5. ⁄ ( )( )

6. √

7.

( )

8. ( )( )( )

9.

10. [ ) [ )

11.

12.

13.

14. ⁄

15.

16. ( )

17. ( ( )) 18.

19. -8

20.

21.

AP Calculus Summer Prep TLHS – 2014 13

22. Even

23. Even

24. -1

25. 3/2

26. 45

27. 2

28. -1

29. (ln 15/ln 3) - 1

30. Yes – show work

31. ⁄

32. ⁄

33. ½

34. 10 sin 50 = 7.66

35. Draw altitude. x = 14.619

36. -1.5, 0, 2, 5

37. Rel. Max (1.07, 20.1)

38. Rel. Min (-.89, -18.48); (3.94, -88.16)

39. f(-1) = -18; f(2) = 0, f(0) = 0, f(.125) = 3.713

40. A(L) = 10L – L2 Domain: 5 < L < 10

41. a) ⁄ ⁄ b) 1/6 means that for each increase of 6 cricket

chirps per minute corresponds to an increase of 1 degree F. c) 76

degrees F