1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005 Mt. Washington Cog Railway, NH.

Slides:

Advertisements

Similar presentations

3.2 Differentiability Photo by Vickie Kelly, 2003 Arches National Park Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry.

Advertisements

2.4 Rates of Change and Tangent Lines Devils Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Day 1 Lengths.

Conics. Parabolas Definition - a parabola is the set of all points equal distance from a point (called the focus) and a line (called the directrix). Parabolas.

2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

9.1 Parametric Curves 9.2 Calculus with Parametric Curves.

Greg Kelly, Hanford High School, Richland, Washington.

 We can define both elements of the ordered pair, (x, y), in terms of another variable, t, called a parameter.  Example: Given and, a) Find the points.

2012 Parametric Functions AP Calculus : BC BONUS.

10.5: Polar Coordinates Greg Kelly, Hanford High School, Richland, Washington.

1.4 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations for.

10.3 Vector Valued Functions Greg Kelly, Hanford High School, Richland, Washington.

10.3 Vector Valued Functions Greg Kelly, Hanford High School, Richland, Washington.

10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.

6.1 day 1: Antiderivatives and Slope Fields Greg Kelly, Hanford High School, Richland, Washington.

Polar Coordinates Lesson Points on a Plane Rectangular coordinate system  Represent a point by two distances from the origin  Horizontal dist,

Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

1.4 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations for.

10.2 day 2 Vector Valued Functions Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006 Everglades National Park, FL.

1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005 Mt. Washington Cog Railway, NH.

1.4 Parametric Equations. Relations Circles Ellipses Lines and Other Curves What you’ll learn about… …and why Parametric equations can be used to obtain.

10.3 day 1 Polar Coordinates Greg Kelly, Hanford High School, Richland, Washington.

Parametric Equations Greg Kelly, Hanford High School, Richland, Washington.

Solving Systems of Equations by Graphing.  I can:  Solve systems of equations by graphing  Determine whether a system of equations is consistent and.

1998 AB Exam. 1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005 Mt. Washington Cog Railway, NH.

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Greg Kelly, Hanford High School, Richland, Washington.

Prerequisites for Calculus

3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

10.1 Parametric functions Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008 Mark Twain’s Boyhood Home Hannibal, Missouri.

Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Day 1 Lengths.

Review after Christmas!. Solve the below equations for the variable..5 (6x +8) = 16 1.

3.7 Optimization Problems Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999.

6.4 day 1 Separable Differential Equations Jefferson Memorial, Washington DC Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly,

Parametric Equations Until now, we’ve been using x and y as variables. With parametric equations, they now become FUNCTIONS of a variable t.

Parametric Equations ordered pairs (x, y) are based upon a third variable, t, called the parameterordered pairs (x, y) are based upon a third variable,

Bell Ringer Solve even #’s Pg. 34.

Slope Fields Greg Kelly, Hanford High School, Richland, Washington

4.2 Implicit Differentiation

Date Calculus AB Ms. Brewer

Module 1 Review ( ) Rewrite the following equations in slope-intercept form (solve for y), then graph on the coordinate plane.

Mrs. Mongold Advanced Math Topics

10.6: The Calculus of Polar Curves

1.2 Introduction to Graphing Equations

10.3 Vector Valued Functions

Differential Equations

Polar Coordinates Lesson 10.5.

6.1: Antiderivatives and Slope Fields

3.6 Implicit Differentiation

6.1 day 1: Antiderivatives and Slope Fields

6.1 day 1: Antiderivatives and Slope Fields

6.1 day 1: Antiderivatives and Slope Fields

Parametric Equations & Plane Curves

1.4 Parametric Equations Mt. Washington Cog Railway, NH.

1.4 Parametric Equations Mt. Washington Cog Railway, NH

2.4 Rates of Change and Tangent Lines

6.1: Antiderivatives and Slope Fields

10.4 Parametric Equations Parametric Equations of a Plane Curve

10.5: Polar Coordinates Greg Kelly, Hanford High School, Richland, Washington.

5.1 day 1: Antiderivatives and Slope Fields

: Antiderivatives and Slope Fields

1.4 Parametric Equations Mt. Washington Cog Railway, NH

3.7 Implicit Differentiation

Prerequisites for Calculus

Objectives: To graph lines using the slope-intercept equation

Parametric Functions 10.1 Greg Kelly, Hanford High School, Richland, Washington.

10.3 Vector Valued Functions

1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, Washington.

Presentation transcript:

1.4 Parametric Equations Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005 Mt. Washington Cog Railway, NH

There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ). These are called parametric equations. “ t ” is the parameter. (It is also the independent variable)

Example 1: To graph on the TI-89: MODE Graph…….2 ENTER PARAMETRIC Y= 2nd T) ENTER WINDOW GRAPH

Hit zoom square to see the correct, undistorted curve. We can confirm this algebraically: parabolic function

Circle: If we let t = the angle, then: Since: We could identify the parametric equations as a circle.

Graph on your calculator: Y= WINDOW GRAPH Use a [-4,4] x [-2,2] window.

Ellipse: This is the equation of an ellipse. 

Line Segment: Plug in Endpoints: When t = 0: When t = 1: Write Parametric equation In slope intercept form: By solving first equation Now plug into second we get: