10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National.

Slides:

Advertisements

Similar presentations

P ARAMETRIC AND P OLAR I NTEGRATION. A REA E NCLOSED P ARAMETRICALLY Suppose that the parametric equations x = x(t) and y = y(t) with c  t  d, describe.

Advertisements

Arc Length Cartesian, Parametric, and Polar. Arc Length x k-1 xkxk Green line = If we do this over and over from every x k—1 to any x k, we get.

PARAMETRIC EQUATIONS AND POLAR COORDINATES

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

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

Equations of Tangent Lines

Copyright © Cengage Learning. All rights reserved.

3.6 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.

Section 10.4 – Polar Coordinates and Polar Graphs.

5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

Polar Graphs and Calculus

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.

3.9: Derivatives of Exponential and Logarithmic Functions Mt. Rushmore, South Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

10.3 Polar Functions Quick Review 5.Find dy / dx. 6.Find the slope of the curve at t = 2. 7.Find the points on the curve where the slope is zero. 8.Find.

Section 12.6 – Area and Arclength in Polar Coordinates

Section 8.3 Area and Arc Length in Polar Coordinates.

Section 11.4 Areas and Lengths in Polar Coordinates.

10.3 Polar Coordinates. One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions.

10 Conics, Parametric Equations, and Polar Coordinates

10.3 Polar Coordinates. Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin.

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

Warm Up Calculator Active The curve given can be described by the equation r = θ + sin(2θ) for 0 < θ < π, where r is measured in meters and θ is measured.

TI89 and Differentiability Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park.

10.1 Parametric Equations. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function.

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

Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2.

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

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

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

Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.

5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

3.3 Rules for Differentiation Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 The “Coke Ovens”,

2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.

7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 London Bridge, Lake Havasu City, Arizona 2.3 Product and Quotient Rules.

Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 AP Calculus AB/BC 3.3 Differentiation Rules,

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park 2.1 The Derivative and the Tangent Line Problem (Part.

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

In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function. If f and g have derivatives.

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

10.6B and 10.7 Calculus of Polar Curves.

10. 4 Polar Coordinates and Polar Graphs 10

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

5.5 Bases Other than e and Applications (Part 1) Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008 Acadia National Park,

7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

Product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 London Bridge, Lake Havasu City, Arizona 3.4 Derivatives of Trig Functions.

Bell-ringer 11/2/09 Suppose that functions f and g and their derivatives with respect to x have the following values at x=0 and x=1. 1.Evaluate the derivative.

MAT 1236 Calculus III Section 10.2 Calculus with Parametric Curves

Give 4 different pairs of polar coordinates that correspond to the Cartesian Coordinates (2,2) Aim: How do we describe Curves in Polar form?

Ch. 11 – Parametric, Vector, and Polar Functions 11.3 – Polar Functions.

7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

9.3: Calculus with Parametric Equations When a curve is defined parametrically, it is still necessary to find slopes of tangents, concavity, area, and.

Parametric equations Parametric equation: x and y expressed in terms of a parameter t, for example, A curve can be described by parametric equations x=x(t),

Lecture 31 – Conic Sections

Copyright © Cengage Learning. All rights reserved.

10.6: The Calculus of Polar Curves

Try graphing this on the TI-89.

Find the area of the region enclosed by one loop of the curve. {image}

7.4 Lengths of Curves and Surface Area

Mark Twain’s Boyhood Home

Mark Twain’s Boyhood Home

3.6 The Chain Rule Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

What if we wanted the area enclosed by:

3.6 The Chain Rule Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

2.4 The Chain Rule (Part 2) Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

Copyright © Cengage Learning. All rights reserved.

7.4 Arc Length and Surface of Revolution

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

Copyright © Cengage Learning. All rights reserved.

Presentation transcript:

10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National Park, California

Try graphing this on the TI-84.

Polar-Rectangular Conversion Formulas

To find the slope of a polar curve: We use the product rule here.

To find the slope of a polar curve:

Example:

Find the slope of the curve at the given values. Find the points where the curve has horizonal or vertical tangent lines.

The length of an arc (in a circle) is given by r.  when  is given in radians. Area Inside a Polar Graph: For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula:

We can use this to find the area inside a polar graph.

Example: Find the area enclosed by:

Notes: To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

Find the area of the region that lies inside the circle r = 3 and outside the cardioid.

When finding area, negative values of r cancel out: Area of one leaf times 4:Area of four leaves:

Find the area that lies outside the four-petal rose and inside the circle.

To find the length of a curve: Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

Or… Convert to Parametric! Find the length of the cardioid

There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: 