You Will Need to Use Your Own Paper to Work the Problems in This Packet. You Will Turn

AP Calculus I

Summer Packet

You will need to use your own paper to work the problems in this packet. You will turn in ALL of your work and the attached completed answer sheet. Answers only will result in no credit.

1. Intercepts

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

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

Example:

Find the intercepts for

Solution:

x-intercept

/ set y=0
/ add 4 to both sides
/ take square root of both sides
or / / Write as 2 equations
or / / Subtract 3 from both sides

y-intercept

/ set x=0
/ Add 0+3
/ Square 3
/ subtract

Problem Set I

Find the intercepts for each of the following.

1.

2.

3.

4.

5.

1. Lines

The slope intercept form of a line is where m is the slope and b is the y-intercept. The point slope form of a line is where m is the slope and is a point on the line. In Calculus the point slope form of a line is the preferred form. If two lines are parallel then they have the same slope. If two lines are perpendicular then they have negative reciprocal slopes.

Example:

Find the slope of the lines parallel and perpendicular to

Solution: The slope of this line is

The parallel line has slope and the perpendicular line has slope .

Example: Find the equations of (a) line parallel and (b) perpendicular to that contains the point (-2,1)

Solution:

Part a (using slope from example above)

/ Using the point-slope form with and point (-2,1)

Part b (using slope from example above)

/ Using the point-slope form with and point (-2,1)

Example:

Find the slope and y-intercept of

Solution: First you must get the line in slope-intercept form.

/ Subtract 6x form both sides
/ Divide by -5
/ Simplify
The slope is m= and the y-intercept is -3

Example: Find the equation of the line that passes through (-1,3) and (4,5).

Solution: You will need to find slope using

choose one point to substitute back into either the point slope or slope-intercept form of a line.

/ Using the point-slope form with and point (4,5)

Problem Set II

Find the equation of a line:

1. contains (3,-4) and (5,2)
2. contains and
3. contains (-3,4) and m is undefined
4. contains (-2,-2) and m=2

1. x-intercept (2,0) and y-intercept (0,3)

Find the slope and y-intercept of the line:

1. x+5y = 20

Sketch a graph of the equation:

1. y=-3
2. x=4
3. y-1=3(x+4)

Write an equation of a line through the point (a). parallel to the given line and (b) perpendicular to the given line:

1. Point : (2,1) line: 4x-2y=3

1. Functions

Definition:

Let f and g be functions. The function given by (f◦g)(x)=f(g(x)) is called the composite of f with g. The domain of f◦g is the set of all x in the domain of g such that g(x) is in the domain of f.

Example: Given: f(x)=3x+5 and g(x)=2x-1

Find: f(g(2)), g(f(2)) and f(g(x))

Solution: To find f(g(2)) we must first find g(2): g(2)=2(2)-1 =4-1=3

Since g(2)=3 we can find f(g(2))=f(3)=3(3)+5=9+5=14

To find g(f(2)) we must first find f(2): f(2)=3(2)+5=6+5=11

Since f(2)=11 we can find g(f(2))=g(11)=2(11)-1=22-1=21

To find f(g(x)) we must put the function g(x) into f(x) equation in place of each x.

f(g(x))=f(2x-1)=3(2x-1)+5=6x-3+5=6x+2

Example: Given find .

Solution: To find we replace every instance of x with x+h.

Thus

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. If the domain were -1<x≤7 then in interval notation the domain would be (-1,7]. Notice that the left side has a ( because it does not include -1 but the right side includes 7 so we use a ]. When using interval notation we never use a [ or ] for infinity.

Example: Find the domain and range for

Solution: Since we can only take the square root of positive numbers x-3≥0 which means that x≥3. So we would say the domain is [3,∞). Note that we have used a [ to indicate that 3 is included. If 3 was not to be included we would have used (3,∞). The smallest y value that the function can return is 0 so the range is (0,∞).

Problem Set III

Let and find each of the following:

1.

2.

3.

Find the domain and range for each function give your answer using interval notation:

3.h(x)=

4. f(x)=

1. Asymptotes and Holes

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: Find the holes in the following function

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

Factoring and canceling:

but (x≠2) this restriction is from the original function before canceling. The graph of the function f(x) will look identical to except for the hole at x=2.

note the hole at x=2

Given a rational function if a number causes the denominator to be 0 but not the numerator to be 0 then there is a vertical asymptote at that x value.

Example: Find the vertical asymptotes for the function

Solution: When x=-1 is substituted into f(x) then the numerator is -1 and the denominator is 0 therefore there is an asymptote at x=1. See the graphs above.

Given a rational function if a number causes the numerator to be 0 but not the denominator to be 0 then the value is an x-intercept for the rational function.

Example: Discuss the zeroes in the numerator and denominator

Solution: When x=-3 is substituted into the function the numerator is 0 and the denominator is -6 so the value of the function is f(-3)=0 and the graph crosses the x-axis at x=-3. Also note that for x=0 the numerator is 3 and the denominator is 0 so there is a vertical asymptote at x=0.The graph is below.

Example: Find the holes, vertical asymptotes and x-intercepts for the given function:

Solution: First we must factor to find all the zeroes for both the numerator and denominator:

Numerator has zeroes x=0 and x=3

Denominator has zeroes x=0 and x=-2.

x=0 is a hole

x=-2 is a vertical asymptote

x=3 is a x-intercept

Also, recall horizontal asymptotes can be found by comparing the degree of the numerator and denominator of a rational function. If the denominator is of higher degree, the horizontal asymptote is . If the numerator is of higher degree, there is no horizontal asymptote. If the numerator and denominator are the same degree, the horizontal asymptote is where k is the ratio of the lead coefficients.

Example: Find the horizontal asymptote of

Solution: Since the degrees of both polynomials is 2, we use the ratio of the lead coefficients to get an asymptote of .

Problem Set IV

For each function below list all holes, vertical asymptotes and x-intercepts, and horizontal asymptotes.

1.

2.

3.

4.

1. Trig. Equations and Special Values

You are expected to know the special values for trigonometric functions. Fill in the table below and study it.

(degrees) / (radians) / cos / sin / Quadrant
0°
30°
45°
60°
90°
120°
135°
150°
180°
210°
225°
240°
270°
300°
315°
330°
360°

You should study the following trig identities and memorize them before school starts:

Reciprocal identities

Tangent Identities

Pythagorean Identities

Reduction Identities

We use these special values and identities to solve equations involving trig functions.

Example: Find all solutions to

Solution:

/ Original Problem
/ Get one side equal to 0.
/ Factor
and / Set each factor equal to 0
and / Get the trig function by itself
and / Solve for x (these are special values)

Problem Set V

Find all solutions to the equations. You should not need a calculator.

1.

2.

3.

4.

VI. Exponents

A fractional exponent means you are taking a root. For example is the same as .

Example: Write without fractional exponent:

Solution: Notice that the root is the bottom number in the fraction and the power is the top number in the fraction.

Negative exponents mean that you need to take the reciprocal. For example means and means .

Example: Write with positive exponents:

Solution:

Example: Write with positive exponents and without fractional exponents:

Solution:

When factoring, always factor out the lowest exponent for each term.

Example:

Solution: The lowest exponent for x is -2 so can be factored from each term. Leaving. Notice that for the exponent for the 6x term we take 1- (-2) and get 3. For the term we take -1-(-2) and get 1 as our new exponent.

When dividing two terms with the same base, we subtract the exponents (numerator exponent- denominator exponent). 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: Simplify

Solution: First you must distribute the exponent. . Then since we have two terms with x as the base we can subtract the exponents. Since 3-8 results in -5 we know that we will have

in the denominator. .

Example: Simplify

Solution: First we must factor both the numerator and denominator. . Then we can see that we have the term (x-1) in both the numerator and denominator. Subtracting exponents we get 2-1=1 so the term will go in the numerator with 1 as it’s exponent. .

Example: Factor and simplify

Solution: The common terms are x and (x-3). The lowest exponent for x is 1. The lowest exponent for (x-3) is -1/2. So factor out and obtain . This will simplify to . Leaving a final solution of .

Problem Set VI

Write without fractional exponents.

1.

2.

Write with positive exponents:

3.

4.

5.

Factor then simplify:

6.

7.

8.

Simplify:

9.

10.

11.

You will need to use your own paper to work the problems in this packet. You will turn in ALL of your work and the attached completed answer sheet. Answers only will result in no credit.

AP CALC 1 Summer PacketName ______

Problem Set I

1) x-int ______y-int ______

2) x-int ______y-int ______

3) x-int ______y-int ______

4) x-int ______y-int ______

5) x-int ______y-int ______

Problem Set II

1) y = ______

2) y = ______

3) y = ______

4) y = ______

5) y = ______

6) m = _____b = ______

7-9)(put on the same graph)

10) parallel, y = ______and perpendicular,y = ______

Problem Set III

1) f(g(x))= ______

2)

3)

4) domain ______range ______

5) domain ______range ______

Problem Set IV

1) holes ______vertical asymptotes ______x-int ______horizontal asymptote ______

2) holes ______vertical asymptotes ______x-int ______horizontal asymptote ______

3) holes ______vertical asymptotes ______x-int ______horizontal asymptote ______

4) holes ______vertical asymptotes ______x-int ______horizontal asymptote ______

Problem Set V

1) x = ______

2) x = ______

3) x = ______

4) x = ______

Problem Set VI

1) y =2) f (x) =

______

3) f (x) = 4) y =

______

5) f (x) =6) f (x) =

______

7) f (x) =8) f (x) =

______

9) f (x) =10) y =

______

11) y =

______