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Welcome Back!. WO graduate, Class of ’98 GVSU graduate, Class of ‘02 Cornerstone University , masters 2008 MATH NERD!. Running in the Hot Chocolate 15K in November Love to travel (but not rustic camping!!) I love to read!. Who is Mrs. Meyer?. WO Powder Puff, Class of 1998.
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WO graduate, Class of ’98 GVSU graduate, Class of ‘02 Cornerstone University , masters 2008 MATH NERD! • Running in the Hot Chocolate 15K in November • Love to travel (but not rustic camping!!) • I love to read! Who is Mrs. Meyer?
Chapter 1 Prerequisites for Calculus WE NEED TEXTBOOKS
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 • With your partner, come up with as many ways to name slope as you can. (4 min)
Parallel and Perpendicular Lines equivalent • The slopes of parallel lines are ______________ • The slopes of perpendicular lines are _____________________ (or the product of the two slopes is _______) Opposite reciprocals -1
Equations of Lines • Slope-intercept form: • Standard form (General Linear Equation): • Point-Slope form: y = mx + b Ax + By = C y – y1 = m(x – x1)
Example 1: • 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!!) Parallel: 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 w/ partners • Dependent variables: • Independent variables: • Domain: • Range: y x {x: } input [ ] output {y: } [ ]
Viewing and Interpreting Graphs • Identify the domain and range, and then sketch a graph of the function. No Calculator
Graph Viewing Skills • Recognize that the graph is reasonable. • See all important characteristics of the graph. • Interpret those characteristics. • Recognize grapher/calculator failure.
Example 3: • Use calculator to identify the domain and range, and then draw a graph of the function.
Even Functions and Odd Functions • Even functions: (Symmetric about the y-axis) • Odd functions: (Rotation symmetric about the origin) f(-x) = f(x) f(-x) = -f(x)
Example 6: Absolute Values • Draw the graph of . Then find the domain and range.
Example 7: Composites • Find a formula for f(g(2)), and g(f(2)) f(g(2)) g(f(2))
Warm Up Solve the equations: 1. x3 = 17 2. x10 = 1.4567 Simplify: 3.
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.
Example 2: • Find the zeros (solutions) of f(x) = (1/3)x - 4 graphically. (Sketch a picture of the solution).
Rules for Exponents • If a > 0 and b > 0, the following hold for all real numbers x and y.
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?
Growth and Decay • 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 2.718281828 • 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 4 • Graph y =
Example 5 • Graph y =
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?
Study note! • 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 Equations • Please complete the activity on pages 10 – 12 in your packets. Work with your table partner. Your calculator should be in radian mode and parametric mode.
Section 1.4 Notes: Parametric Functions • Objectives: Relations, circles, ellipses, lines and other curves. Parametric equations can be used to obtain graphs of relations and functions.
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 when . Indicate the direction in which the curve is being traced. Find a Cartesian equation for a curve that contains the parametrized curve.
Definitions • Definition: If x and y are given as functions over an interval of t-values, then the set of points 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.
Circles • Open your book to page 31. With your partner, complete the Exploration 1: Parametrizing Circles. Record your answers/responses below so we can discuss as a group: • 1. • 2. • 3. • 4.
Example 2: • Describe the graph of the relation determined by x = 2 cos t, y = 2 sin t, when . 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.
EllipsesExample 3: • Graph the parametrized curve x = 3 cos t, y = 4 sin t, . 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 CurvesExample 4: • Draw and identify the graph of the parametric curve determined by x = 3t, y = 2 – 2t, .
Example 5: • Find a parametrization for the line segment with endpoints (-2, 1) and (3, 5).
Section 1.5 Notes: Functions and Log 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.
Definition: A function f(x) is one-to-one on a domain D if f(a) f(b) whenever ab. NOTE: A one-to-one function passes the vertical line test AND the horizontal line test! One-to-one Functions
Example 1: • Determine if the following functions are one-to-one: • f(x) = |x| • g(x) = No, although this is a function it does not pass horizontal line test Yes, passes both! Also, note: we know this does not include the bottom half of the square root. If it included a + / - in front it would!
