2016 2017 summer math preparation for students who … ew a ddre ss: _ _ ci ty, s tate, z...

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2016–2017 Summer Math Preparation for Students Who Will Take AP Calculus

What is the aim of this assignment? Doing this assignment will help you review math that you will need to know for your math course next year. It will help you realize what topics you need to fix up before starting that course so that we can spend less time reviewing and more time learning new material. Printing: Please print this file out. Do your work right on the sheet. If you need additional sheets, you can work on the back or staple additional pages. Showing your work:

You must work in pencil, not pen. Edit your mistakes and be neat.

Show all work, even for any multiple-choice questions, so that your teacher can see what you were thinking, give you feedback, and help you.

Write your work clearly, so that someone else can follow your thinking. Use correct mathematics notation as well as English words.

Due date: This assignment is due the first day of school. You have all summer to do this packet, but you really should start in July and work gradually so that you have time to do a great job. Grading: This assignment will be graded as part of your Quarter 1 grade. You may also be assessed on these topics in September.

Getting Help: You are expected to work independently on this. You can ask classmates, friends, or others for suggestions, but you may NOT look at their written work. If someone asks you for help, don’t show them your work. Instead, guide them hints and ask questions that will help them.

I have read this description and agree to do this assignment independently. Name Date

2016-2017 Summer Math Preparation for Students Who Will Take AP Calculus 2

Formula Sheet


Simplifying Complex Fractions

When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction. If your simplification allows new values of the variable to be in the domain of the expression, you must prevent them explicitly, as shown below:

Simplify each of the following. Show your work carefully, and list restrictions that make your final expression equivalent to the original expression.

1.

2.

, x ≠ –1

, x ≠ 4 (Note that x = 0 is still forbidden, so it does not need to be listed separately.)


3.


Functions

Let and . Find each expression below.

1)

2)

3)

4)

5)

6)

Let . Find each expression below.

7)


Use table of the values for the function f and the graph of g shown right to evaluate each expression.

3 2

4 9

6 3

7 1

8)

9) Name a value of x for which f(g(x)) cannot be evaluated with the limited information given. Explain why not.

10)

11) Why does it make sense that your answer is 9? (Hint: think about average rate of change.)


Intercepts and Points of Intersection

Find both the x-intercepts and y-intercepts for each function or relation below. Explain your findings algebraically but also check on a graph.

1.

3.

2.

4.


5. In terms of the unspecified value a, what are the x-intercepts and y-intercepts of g(x) = x2 – a2? Justify your answer. Then sketch the graph.


Interval Notation

Inequality Interval Notation Graph

[ e

Domain and Range

Find the domain and range of each function. Write your answer in interval notation. Explain each answer algebraically.

1.

3.

2.

4.


Equations of Lines

1) Explain why the slope of a line is a constant rate of change using the example . Be sure to use the following in your explanation: steepness, ∆x, ∆y, increase, change.

2) Write the equation of the line having a slope of 3 and a y-intercept of 5.

3) Determine the equation of a line passing through the point (5, –3) with infinite slope.

4) Determine the equation of a line passing through the point (–4, 2) with a slope of 0.

5) Write the equation of the line passing through the point (0, 5) with a slope of 2/3.


6) Find the equation of a line passing through the point (2, 8) and parallel to the line

7) Find the equation of a line perpendicular to the y-axis passing through the point (4, 7).

8) Find the equation of a line passing through the points (-3, 6) and (1, 2).

9) Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3).

10) Explain why the point-slope form is correct using the definition of slope OR transformations of the parent function p(x) = x. Remember, represents any point on the line and represents a specific point.


Radian and Degree Measure

Since 180° = π radians, you can multiply by and its reciprocal to convert.

1) Covert to degrees:

a)

b)

c) 2.63 radians

2) Convert to radians:

a)

b)

c)

Angles in Standard Position

Sketch the angle in standard position. Label the (x, y) coordinates point one unit from the origin on the terminal ray (decimals are okay).

How do we use the x and y coordinates to find sine, cosine, and tangent of (or any angle)?


Trigonometry for Quadrantal Angles

a)

b)

c)

d)

e)

f)

Graphing Trig Functions

Sketch the graphs of on the same axes below, using different colors for each function. Label 5 key points on each graph.


Now, to the right of the graph, write a paragraph comparing and contrasting the graphs of sine and cosine. How are they helpful for remembering facts such as sin(0) = 0?


Inverse Trigonometric Functions

Compute each value below; express in both degrees and radians.


Solving Trigonometric Equations

Solve each of the equations for .

1)

2)

3)

4)

Solve for all x, not limited to . Your answer can use a “…” or an “n,” where n is an integer.

5) 6)


Laws of Exponents

Simplify each of the following.

1) =

2) =

3) =

4) =

If the equation is true, prove it using the definition of exponents (exponents represent repeated addition). If it’s false, give an example using numbers that shows it’s false.

5) 6)

7) Explain why x0 = 1 for any number (except x = 0, which causes all kinds of crazy issues we will study next year). Hint: make a list of positive and negative powers of x and see where x0 fits the pattern.

8) Is it true that Explain.

2016 2017 summer math preparation for students who … ew a ddre ss: _____ _____ ci ty, s tate, z...

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2016 2017 summer math preparation for students who … ew a ddre ss: _ _ ci ty, s tate, z...