Albertson AP Calculus AB Name___________________________

AP CALCULUS AB SUMMER PACKET 2017

DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the summer. It is meant to help you practice mathematical skills necessary to be successful in Calculus AB. All of the skills covered in this packet are skills from Algebra 2 and Pre-Calculus. If you need to use reference materials please do so. While graphing calculators will be used in class the majority of this packet should be done without one. If it says to you use one then please do otherwise please refrain. As you know AP Calculus AB is a fast paced course that is taught at the college level. There is a lot of material in the curriculum that must be covered before the AP exam in May. The better you know the prerequisite skills coming into the class the better the class will go for you. Spend some time with this packet and make sure you are clear on everything covered. If you have questions please contact me via email and I will be glad to help. (If you take a picture of your work and the questions it usually makes things go faster) This assignment will be collected and graded as your first test. Be sure to show all appropriate work. In addition, there may be a quiz on this material during the first quarter. All questions must be complete with the correct work. I am very excited to see you all in August. Good Luck!!

I. Intercepts The x-intercept is where the graph crosses the x-axis. You can find the x-intercept by setting 0. The y-intercept is where the graph crosses the y-intercept. You can find the y-intercept by setting

0 Example: Find the intercepts for 3 4 Solution X-intercept Set 0.

0 3 4 Add 4 to both sides.

4 3 Take the square root of both sides

2 3 Write as two equations

2 3 2 3 Subtract 3 from both sides

5 1 Y-intercept Set 0

0 3 4 Add 0 + 3

3 4 Square 3

9 4 Add four to both sides

5

Find the intercepts for each of the following.

1. 3 2

2. 2

3.

4. 4

II. Complex Fractions When simplifying 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. Example

761

51

761

51

∙11

7 7 65

7 135

2 34

514

2 34

514

∙44

2 4 35 4 1

2 8 35 20

3 2 85 21

1.

2.

3.

III. System of Equations Use substitution or elimination method to solve the system of equations. Example:

16 39 0 9 0

Elimination Method Add the two equations and you get

2 16 30 0 8 15 0 3 5 0 3 5

Plug 3 and 5 into an original equation.

3 9 0 0

0

5 9 0 16

4 Points of intersection are 5,4 , 5, 4 , 3,0 Substitution Solve one equation for a variable

16 39 Plug into the other equation

16 39 9 0

2 16 30 0 8 15 0 3 5 0

The rest is like the previous example

Find the point(s) of intersection of the graphs for the given equations.

1. 8, 4 7

2. 6, 4

3. 4 20 64 1720, 16 4 320 641600 0

IV. Functions

Let 2 1 2 1. Find each

To evaluate a function for a given value, simply plug the value into the function for x. ° read “f of g of x”.

Means to plug the inside function (in this case in for x in the outside function (in this case, ). Example Given 2 1 and 4 find

4 2 4 12 8 16 12 16 32 1

2 16 33

1. 2

2. 3

3. 1

4. 2

5. 2

Find for the given

function f 6. 9 3

7. 5 2

V. Interval Notation Solve each equation. State your

Complete the table with the appropriate notation or graph

Solution Interval Notation

Graph

2 4

1,7

answer in BOTH interval notation and graphically.

1. 2 1 0

2. 4 2 3 4

3. 5

VI. Domain and Range The domain of a function is the set of x values for which the function is defined. The range of a function is the set of y values that a function can return. In Calculus we usually write domains and ranges in interval notation. Example: Find the domain and range for √ 3 Solution Since we can only take the square root of positive numbers 3 0 which means that 3. So we would say the domain is 3,∞ . Note that we have used a [ to indicate that 2 is included. If 3 was not to be included we would have used a (. The smallest y value that the function can return is 0 so the range is 0,∞

Find the domain and range.

1. √9

2. sin

3.

VII. Inverses To find the inverse of a function, simply switch the x and the y and solve for the new “y” value.

√ 1 Rewrite as y

√ 1 Switch x and y

1 Solve for your new y.

1 1 1

Rewrite in inverse notation

1 To prove that one function is an inverse of another function, you need to show that

Find the inverse of each function

1. 2 1

2.

Prove that f and g are inverse of each other

3.

√2

4. 9 , 0 √9

VIII. Symmetry x-axis substitute in –y for y into the equation. If this yields an equivalent equation then the graph has x-axis symmetry. If this is the case, this is not a function as it would fail the vertical line test. y-axis substitute in –x for x into the equation. If this yields an equivalent equation then the graph has y-axis symmetry. A function that has y-axis symmetry is called an even function. Origin Substitute in -x for x into the equation and substitute –y for y into the equation. If this yields an equivalent equation then the graph has origin symmetry. If a function has origin symmetry it is called an odd function. In order for a graph to represent a function it must be true that for every x value in the domain there is exactly one y value. To test to see if an equation is a function we can graph it and then do the vertical line test. Example 1 Is 2 a function? Solution: this is not a function because it does not pass the vertical line test.

Example 2: Test for symmetry with respect to each axis and the origin given the equation √4 0

Test for symmetry with respect to each axis and the origin.

1. √ 2

2. |6 |

3.

Solution: x-axis

4 0

√4 0 since there is no way to make this look like the original it is not symmetric to the x axis y-axis

4 0

√4 0 since there is no way to make this look like the original it is not symmetric to the y axis Origin

4 0 √4 0 since this does look like the

original it is symmetric to the origin

4. 3 1

IX. Vertical Asymptotes To find the vertical asymptotes, set the denominator equal to zero to find the x-value for which the function is undefined. That will be the vertical asymptote.

Determine the vertical asymptotes for the function (it will be a line)

1.

2.

3.

X. Holes (Points of Discontinuity) Given a rational function if a number causes the denominator and the numerator to be 0 then both the numerator and denominator can be factored and the common zero can be cancelled out. This means there is a hole in the function at this point. Example 1 Find the hole in the following function

22

Solution: When 2 is substituted into the function the denominator and numerator both are 0. Factoring and cancelling

but 2 this restriction is from the

original function before canceling. The graph of the function witll look identical to except

for the hole at 2

For each function list find the holes

1.

2.

XI. Horizontal Asymptotes Case 1: Degree of the numerator is less than the degree of the denominator. The asymptote is 0 Case 2: Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the lead coefficients. Case 3: Degree of the numerator is greater than the degree of the denominator. There is no horizontal asymptote. The function increases without bound. (If the degree of the numerator is exactly 1 more than the degree of the denominator, then there is a slant asymptote, which is determined with long division.)

Determine the horizontal asymptotes using the three cases.

1.

2.

3.

XII. Solving for indicated variables

1. 1

2. 2

3. 2 2

4. 0 for x

XIII. Absolute Value and Piecewise Functions In order to remove the absolute value sign from a function you must:

1. Find the zeros of the expression inside of the absolute value.

2. Make a sign chart of the expression inside the absolute value

3. Rewrite the equation without the absolute value as a piecewise function. For each interval where the expression is positive we can write that interval by just dropping the absolute value. For each interval that is negative we must take the opposite sign.

Example 1 Rewrite the following equation without using absolute value symbols.

|2 4| Solution:

Write the following absolute value expressions as piecewise expressions (by remove the absolute value):

1. |2 4|

Find where the expression is 0 for the part in the absolute value

2 4 0 2 4

42

2 Put in any value less than -2 into 2x+4 and you get a negative. Put in any value more than -2 and you get a positive.

Write as a piecewise function. Be sure to change the sign of each term for any part of the graph that was negative on the sign chart.

2 4 22 4 2

2. |6 2 | 1

XIV. Exponents A fractional exponent means you are taking a root.

For example is the same as √ Example 1:

Write without fractional exponent: Solution:

√ Notice that the index is the denominator and the power is the numerator. Negative exponents mean that you need to take the reciprocal. For example means and

means 2 . Example 2: Write with positive exponents:

Solution:

Example 3: Write with positive exponents and

Write without fractional exponents

1. 2

2. 16

3. 27

4. 9

5. 64

6. 8

without fractional exponents:

Solution: √ √

When factoring always factor out the lowest exponent for each term. Example 4: 3 6 33 Solution: 3 1 2 11 When dividing two terms with the same base, we subtract the exponents. If the difference is negative then the term goes in the denominator if the difference is positive then the term goes in the numerator.

Example 5: Simplify

Solution: first you must distribute the exponent.

. Then since we have two terms with x as

the base we can subtract the exponents. Thus

Example 6: Factor and simplify

4 3 3 Solution: The common terms are x and (x-3). The lowest exponent for x is 1. The lowest exponents for

(x-3) is . So factor out 3 and obtain

3 4 3 This will simplify to

3 4 12

Leaving a final solution of √

Write with positive exponents:

7. 2

8.

Factor then simplify

9. 4 2 18

10. 5 2 2 3

11. 6 2 1 4 21

XV. Natural Logarithms Recall that ln and are inverse to each other. Properties of Natural Log:

ln ln ln Example 1: ln 2 ln 5 ln 10

ln ln ln

Example 2: ln 6 ln 2 ln 3

ln ln Example 3: ln 4 ln

3ln 2 ln 2 ln 8

ln , ln 1, ln 1 0, 1 Example 4: Use the properties of natural logs to solve for x.

2 ∙ 5 11 ∙ 7 57

112

ln57

ln112

ln 5 ln 7 ln 11 ln 2 5 ln 7 ln 11 ln 2 ln 5 ln 7 ln 11 ln 2

ln 11 ln 2ln 5 ln 7

Express as a single logarithm:

1. 3 ln 2 ln 4 ln Solve for x

2. 3 ln 1

3. 7

4. 3 5 ∙ 2

XVI. Radians and Degree Measure

Use °

to get rid of radians and convert to

degrees

Use °

to get rid of degrees and convert to

radians

Convert to degrees

1.

2. 2.63 radians

Convert to radians

3. 45°

4. -17°

5. 237° XVII. Trig. Equations and special values

You are expected to know the special values for trigonometric functions. Fill in the table to the right and study it. (Please) You can determine sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate is the cosine and the y-coordinate is the sine of the angle.

Example:

sin 90° 1

cos2

0

1. sin 180°

2. cos 270°

3. sin 90°

4. cos

XVIII. Trig. Identities You should study the following trig identities and memorize them before school starts (we use them a lot)

Find all the solutions to the equations. You should not need a caluclator. (hint one of these has NO solution)

1. sin

2. 2 cos √3

3. 4 cos 4 cos 1

4. 2 sin 3 sin 1 0

5. 2 cos 1 cos 0

XIX. Graphing Trig Functions

sin and cos have a period of 2 and an

amplitude of 1. Use the parent graphs above to help you sketch a graph of the function below. For sin , A = amplitude, = period,

phase shift (positive C/B shift left, negative C/B shift right) and K = vertical shift

Graph two complete periods of the function.

XX. Inverse Trig Functions Inverse Trig Functions can be written in one of two ways:

arcsin sin Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains.

Example 1: Express the value of “y” in radians

arctan1

√3

Solution: Draw a reference triangle This means the reverence angle is 30° or . So

so it falls in the

interval from

Thus

Example 2: Find the value without a calculator

cos56

Solution Draw the reference triangle in the correct quadrant fits. Find the missing side using the Pythagorean

For each of the following, express the value for “y” in radians

1. arcsin √

2. arccos 1

3. tan 1 For each of the following give the value without a calculator.

4. tan arccos

Theorem. Find the ratio of the cosine of the reference triangle.

cos6

√61

5. sec sin

6. sin arctan

7. sin sin

XXI. Transformations of a Graph Graph the parent function of each set using your calculator. Draw a quick sketch on your paper of each additional equation in the family. Check your sketch with the graphing calculator.

1. Parent Function a. 5 b. 3 c. 10 d. 8 e. 4 f. 0.25 g. h. 3 6 i. 4 8 j. 2 1 4

2. Parent Function sin (set mode to radians)

a. sin 2 b. sin 2 c. 2sin d. 2 sin 2 2

3. Parent Function cos

a. cos 3

b. cos

c. 2cos 2 d. 2cos 1

4. Parent Function a. 2 b. c. 5 d. 3 e. 4 f. 1 4

5. Parent Function √

a. √ 2 b. √ c. √6 d. √ e. √ f. √ 2 g. √4

6. Parent Function ln a. ln 3 b. ln 3 c. ln 2 d. ln e. ln f. ln 2 4

7. Parent Function

a. b. c. 3 d. e.

8. Parent Function

a. 5 b. 2 c. 3

d.

e. 4

9. Resize your window to 0,1 0,1 Graph all of the following functions in the same window. List the functions from the highest graph to the lowest graph. How do they compare for values of 1? a. b. c. √

d. e. | | f.

10. Given 3 2 7 11 Use your calculator to find all roots to the nearest 0.0001

11. Given | 3| | | 6 Use your calculator to find all the roots to the nearest 0.001

12. Find the points of intersection. a. 3 2, 4 2 b. 5 2, 3 2

(If you found any errors in the packet please let me know so I can correct it. Thanks!)