Section 2



AP Calculus: Unit 1 (Pre-calculus) Name_________________________

Quick Review:

1. Find the value of y that corresponds to x = 3 in y = -2 + 4(x – 3)

2. Find the value of m when x = -1 and y = -3.

m = [pic]

3. Determine whether the ordered pair is a solution to the equation.

3x – 4y = 5 (a) (2, ¼) (b) (3, -1)

y = -2x + 5 (a) (-1, 7) (b) (-2, 1)

4. Find the distance between the two points.

a) (2, 1) and (1, -1/3)

b) (1, 0) and (0, 1)

5. Solve for y in terms of x.

a) 4x – 3y = 7

(b) -2x + 5y = -3

Section 1.1: Lines

Learning Targets:

• I can write an equation and sketch a graph of a line given specific information.

• I can identify the relationships between parallel/perpendicular lines and slopes.

Example 1:

If a particle moves from the point (a, b) to the point (c, d), the slope would be:

Slopes

There are many ways to denote slope. Brainstorm some with your partner:

Parallel and Perpendicular Lines

• The slopes of parallel lines are _____________________

• The slopes of perpendicular lines are _____________________ (or the product of the two slopes is _______)

Equations of Lines

• Slope-intercept form:

• Standard form (General Linear Equation):

• Point-Slope form:

Example 1:

Write an equation for the line through the point (-1, 2) that is (a) parallel, and (b) perpendicular to the line y = 3x – 4. (Leave your answers in point slope!!)

a) Parallel:

b) Perpendicular:

Section 1.2 Notes: Functions and Graphs

Learning Targets:

• I can identify the domain and range of a function using its graph or equation.

• I can recognize even and odd functions using equations and graphs.

• I can interpret and find formulas for piecewise defined functions.

• I can write and evaluate compositions of two functions.

What is a function? Brainstorm with your partner!

Viewing and Interpreting Graphs

Example 2:

Identify the domain and range, and then sketch a graph of the function. No Calculator

(A) [pic] Parent function?________ (B) [pic]

Parent function?________

Graph Viewing Skills

1. Recognize that the graph is reasonable.

2. See all important characteristics of the graph.

3. Interpret those characteristics.

4. Recognize grapher/calculator failure.

Example 3:

Use an automatic grapher (calculator) to identify the domain and range, and then draw a graph of the function.

(A) [pic] (B) [pic]

Even Functions and Odd Functions (Symmetry)

• Even functions: [pic] (Symmetric about the y-axis)

• Odd functions: [pic] (Rotation symmetric about the origin)

Example 4:

Identify the following functions as even or odd and explain why:

(A) [pic]

(B) [pic]

(C) [pic]

(D) [pic]

(E) y = cos x

(F) y = sin x

Functions Defined in Pieces

Example 5:

Graph [pic]

Absolute Value Functions

Example 6:

Draw the graph of [pic]. Then find the domain and range. (NO CALCULATORS!)

Composite Functions

Example 7:

Find a formula for [pic] if [pic] and [pic]. Then find f(g(2)) and g(f(2).

Section 1.3 Notes: Exponential Functions

Learning Targets:

• I can determine the domain, range, and graph of an exponential function.

• I can solve problems involving exponential growth and decay.

• I can use exponential regression to solve problems.

Exponential Growth

Definition: Let a be a positive real number other than 1. The function f(x) = ax is the exponential function with base a.

Example 1:

Graph the function y = 3(2x) – 4. State the domain and range.

Domain: ________________

Range: _________________

Example 2:

Find the zeros (solutions) of f(x) = (1/3)x – 4 graphically. (Sketch a picture of the solution).

Zeros: ____________

Rules for Exponents

If a > 0 and b > 0, the following hold for all real numbers x and y.

[pic] [pic]

Exponential Decay

Definition: The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a nonradioactive state by emitting energy in the form of radiation.

Example 3:

Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining?

Definition: The function f(x) = kax , k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.

The Number e

e [pic] 2.718281828

Example 4: Graph y = [pic]

Example 5: Graph y = [pic]

The number e is used in problems where interest (for example) is being compounded continuously with the formula A(t) = Pert.

What is the formula that we use if we are not compounding continuously?

Example 6:

Chenelle opened a bank account at a 1.25% interest rate compounded quarterly. She put $500 in the account 10 years ago and has not touched the account since then. How much should be in her account today?

Example 7:

How long would it take Chenelle’s investment to double if the account was compounded continuously?

To help you study for your quiz over 1.1-1.3, you may want to practice the quiz using the “Quick Quiz” on page 29 of your textbook. These serve as a good review, but also great AP testing practice!

Parametric Relations: Activity

Objectives:

• Given parametric equations, plot relations by hand or calculator.

• Control the speed and direction of the plot by varying t and its increments or by varying the equations.

• Produce parametric equations for Cartesian equations.

• Convert parametric equations to Cartesian equations (by eliminating the parameter)

• Model motion problems.

Big Picture:

Parametrics offer a powerful method to plot many relations whether or not they are functions. They also allow us to model motion, since we have more control over how points are plotted. Beyond this introductory section, you will work with the calculus of parametrics, so gaining a high level of comfort with them now will assure future success.

Activity:

When you first learned to plot lines, you probably used a chart where you chose x-values and plugged them into an equation to produce y-values. With parametrics, the x- and y-values are produced independently by substituting for a third variable, t, called a parameter. In modeling motion, t usually represents time.

1. Given the following parametric equations, produce a table of values and plot the relation. The table has been started for you.

|t |x |y |

|-2 |6 |-4 |

|-1 | | |

|0 | | |

|1 | | |

|2 | | |

|3 | | |

2. Using substitution, convert the parametric equations in #1 to Cartesian form. This is called eliminating the parameter. [Hint: Solve for x as a function of y.]

Note: When parametric equations contain trig functions, we often rely on a trig identity rather than substitution to eliminate the parameter. Consider the parametric equations:

[pic]. If we use the trig identity [pic], the equation becomes 1 + 2y = x2.

3. Convert to Cartesian coordinates: x = 3 sin(t), y = 4 cos(t). [Hint: Divide each by the constant first. Then refer to the note above!]

Note: Any function can be converted to parametric form simply by letting the independent variable be t. So, for instance, [pic] can be converted to x = t, [pic]. We must realize, though, that due to limitations on the values of t we will not always produce a complete graph. For instance, y = 2x – 1 can be defined parametrically as x = t, y = 2t – 1, but if t goes from [-10, 10], we would only see a plot of a segment from (-10, -21) to (10, 19).

4. Determine parametric equations to plot the right half of the parabola y = (x – 2)2. Graph it on your calculator to see if you have achieved your goal.

[pic] [pic]

5. What is the effect of changing the increments of t or [pic] on the calculator? Find out by exploring. Try the following examples, comparing A to B and A to C.

(A) [pic] [pic] (B) [pic] [pic] (C) [pic] [pic]

a. A compared to B: What was the effect of making the [pic]smaller? Explain why it caused that effect.

b. A compared to C: What was the effect of a negative [pic]? Explain why it caused that effect.

6. Compare the next two plots, where just the functions were changed slightly. (Make sure you plot in radians.) Explain the similarities and differences in the plots.

(D) [pic] [pic] (E) [pic] [pic]

Section 1.4 Notes: Parametric Functions

Learning Targets:

• I can graph curves that are described using parametric equations.

• I can find parameterizations of circles, ellipses, line segments, and other curves.

Relations

Definition: A relation is a set of ordered pairs (x, y) of real numbers. The graph of a relation is the set of points in the plane that correspond to the ordered pairs of the relation. If x and y are functions of a third variable t, called a parameter, then we use the parametric mode of our calculator.

Example 1:

Describe the graph of the relation determined by [pic] when[pic]. Indicate the direction in which the curve is being traced. Find a Cartesian equation for a curve that contains the parametrized curve.

Definition: If x and y are given as functions [pic] over an interval of t-values, then the set of points [pic]defined by these equations is a parametric curve.

NOTE: If we are graphing a parametric curve on a closed interval [a, b], we consider the point (f(a), g(a)) the initial point and (f(b), g(b)) the terminal point.

Example 2:

Describe the graph of the relation determined by x = 2 cos t,

y = 2 sin t, when [pic]. Find the initial and terminal

points, if any, and indicate the direction in which the curve

is traced. Find a Cartesian equation for a curve that contains

the parametrized curve.

Ellipses

Example 3:

Graph the parametrized curve x = 3 cos t, y = 4 sin t,

[pic]. Find the Cartesian equation for a curve that

contains the parametric curve. Find the initial and terminal

points, if any, and indicate the direction in which the curve

is traced.

Lines and Other Curves

Example 4:

Draw and identify the graph of the parametric curve

determined by x = 3t, y = 2 – 2t, [pic].

Example 5:

Find a parametrization for the line segment with endpoints (-2, 1) and (3, 5).

Section 1.5 Notes: Functions and Logarithms

Learning Targets:

• I can identify one to one functions.

• I can determine the algebraic representation and the graphical representation of a function and its inverse.

• I can use parametric equations to graph inverse functions.

• I can apply the properties of logarithms.

• I can use logarithmic regression equations to solve problems.

One-to-one Functions

Definition: A function f(x) is one-to-one on a domain D if f(a) [pic] f(b) whenever a [pic] b.

NOTE: A one-to-one function passes the vertical line test AND the horizontal line test!

Example 1:

Determine if the following functions are one-to-one:

(A) f(x) = |x| (B) g(x) = [pic]

Finding Inverses

Definition: The function defined by reversing a one-to-one function f is the inverse of f.

NOTE: If [pic], then f and g are inverses.

Writing f-1 as a function of x:

1. Solve the equation y = f(x) for x in terms of y.

2. Interchange x and y. The resulting formula will be y = f-1(x).

Example 2:

Show that the function y = f(x) = -2x + 4 is one-to-one and find its inverse function.

Example 3:

a) Graph the one-to-one function f(x) = x2, [pic],

together with its inverse and the line y = x, [pic].

b) Express the inverse of f as a function of x.

Logarithmic Functions

Definition: The base a logarithm function [pic] is the inverse of the base a exponential function [pic] ([pic]).

Properties of Logarithms

Example 4: Solve for x.

(A) ln x = 3t + 5 (B) e2x = 10

Properties of Logarithms:

For any real numbers x > 0 and y > 0,

1. Product Rule:

2. Quotient Rule:

3. Power Rule:

Example 5: Solve

a) ln(2x – 1) = ln 16 b) ln 56 – ln x = 4 c) ln (x + 4) + ln x = ln 12

Definition: Change of Base Formula:

[pic]

Example 5:

Graph [pic].

Example 6:

Sarah invests $1000 in an account that earns 5.25% interest compounded annually. How long will it take the account to reach $2500? (Solve algebraically and confirm graphically!)

Section 1.6 Notes: Trigonometric Functions

Learning Targets:

• I can convert between radians and degrees, and find arc length.

• I can identify the periodicity and even-odd properties of the trigonometric functions.

• I can find values of trigonometric functions.

• I can generate the graphs of the trigonometric functions and explore different transformations on these graphs.

• I can use the inverse trigonometric functions to solve problems.

Example 1:

Find all the trigonometric values of x if sin x = -3/5 and tan x < 0.

Transformations of Trigonometric Graphs

[pic]

Example 2:

Determine the (a) period, (b) domain, (c) range,

and (d) draw the graph of the function

[pic]

[pic]

Inverse Trig Functions:

|Function |Domain |Range |

|[pic] | | |

|[pic] | | |

|[pic] | | |

|[pic] | | |

|[pic] | | |

|[pic] | | |

Example 3:

Find the measure of cos-1 (-0.5) in degrees and radians (NO CALCULATOR!). How many solutions should you expect to have?

Example 4:

Solve: (A) sin x = 0.7, [pic] (B) tan x = -2, [pic]

How many solutions should you How many solutions should you

expect to have? expect to have?

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[pic]

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