Differential Equations

Slides:

Advertisements

Similar presentations

6.1 day 1: Antiderivatives and Slope Fields Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2009 Kitt Peak National Observatory,

Advertisements

Greg Kelly, Hanford High School, Richland, Washington.

4. Slope Fields. Slope Fields We know that antidifferentiation, indefinite integration, and solving differential equations all imply the same process.

Slope Fields and Euler’s Method. When taking an antiderivative that is not dealing with a definite integral, be sure to add the constant at the end. Given:find.

AP Calculus AB/BC 6.1 Notes - Slope Fields

Integration. Indefinite Integral Suppose we know that a graph has gradient –2, what is the equation of the graph? There are many possible equations for.

6.1: Antiderivatives and Slope Fields. First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take.

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

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

First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.

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

Slope Fields and Euler’s Method Copyright © Cengage Learning. All rights reserved Day

Derivatives of Powers and Polynomials Colorado National Monument Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College.

6.1: Antiderivatives and Slope Fields. First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take.

6.1 D IFFERENTIAL E QUATIONS & S LOPE F IELDS. D IFFERENTIAL E QUATIONS Any equation involving a derivative is called a differential equation. The solution.

First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.

Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try.

–1 –2 We’ve plotted points on graphs before… Plot the following points on the empty grid at the top of your worksheet.

4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington.

Warm Up. 7.1 A – Initial Value Problems Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the.

Slide 6- 1 What you’ll learn about Differential Equations Slope Fields Euler’s Method … and why Differential equations have been a prime motivation for.

Greg Kelly, Hanford High School, Richland, Washington.

Differential equations and Slope Fields Greg Kelly, Hanford High School, Richland, Washington.

Second Fundamental Theorem of Calculus

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

1 Differential Equations 6 Copyright © Cengage Learning. All rights reserved. 6.1 DE & Slope Fields BC Day 1.

A little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.

Slope Fields Differential Equations. Slope Fields A slope field is a graphical picture of a derivative that projects the curve within the picture. Or.

Warm-Up Explain the difference between propagated error and relative error.

Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, Day 3 The Shell Method.

6.1 Differential Equations and Slope Fields. First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when.

Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College Photo by Vickie Kelly, 2001 London Bridge, Lake Havasu City,

Substitution & Separable Differential Equations

Chapter 6: Differential Equations

Antiderivatives with Slope Fields

Fluid Pressure and Forces

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

4.1 : Anti-derivatives Greg Kelly, Hanford High School, Richland, Washington.

7.1 Slope Fields and Euler’s Method

10.6: The Calculus of Polar Curves

6.2 Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA

Substitution & Separable Differential Equations

Using Derivatives for Curve Sketching

7.4 Lengths of Curves and Surface Area

and Indefinite Integration (Part I)

6.1 - Slope Fields.

Optimisation: Extreme Values of Functions

6.1: Antiderivatives and Slope Fields

Derivatives of Inverse Trig Functions

3.6 Implicit Differentiation

Quick Review 2012 Pearson Education, Inc..

Substitution & Separable Differential Equations

6.1 day 1: Antiderivatives and Slope Fields

6.1 day 1: Antiderivatives and Slope Fields

6.1 day 1: Antiderivatives and Slope Fields

The Tangent and Velocity Problems

Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA

We’ve plotted points on graphs before… • •

Derivatives of Inverse Functions

1.4 Parametric Equations Mt. Washington Cog Railway, NH

6.1: Antiderivatives and Slope Fields

6.1 day 1: Antiderivatives and Slope Fields

5.1 day 1: Antiderivatives and Slope Fields

: Antiderivatives and Slope Fields

Slope Fields Rita Korsunsky.

Substitution & Separable Differential Equations

7.4 Lengths of Curves and Surface Area

Finding constant of integration

Substitution & Separable Differential Equations

Presentation transcript:

Differential Equations Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College

First, a little review: Consider: or then: It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given: find We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

If we have some more information we can find C. Given: and when , find the equation for . This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator.

1 2 3 1 2 1 1 2 2 4 -1 -2 -2 -4 Draw a segment with slope of 2. Draw a segment with slope of 2. 1 2 3 1 2 Draw a segment with slope of 0. 1 1 2 Draw a segment with slope of 4. 2 4 -1 -2 -2 -4

If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

For more challenging differential equations, we will use the calculator to draw the slope field. On the TI-89: Push MODE and change the Graph type to DIFF EQUATIONS. MODE Go to: Y= Press and make sure FIELDS is set to SLPFLD. I Go to: and enter the equation as: Y= (Notice that we have to replace x with t , and y with y1.) (Leave yi1 blank.)

Set the viewing window: WINDOW Then draw the graph: GRAPH

Be sure to change the Graph type back to FUNCTION when you are done graphing slope fields.

Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

Integrals such as are called indefinite integrals because we can not find a definite value for the answer. When finding indefinite integrals, we always include the “plus C”.