Slope FieldsSlope Fields

What are slope fields?What are slope fields?

Graphical representation of the Graphical representation of the slope of a function at different slope of a function at different points in the plane.points in the plane.

Since we are given the differential Since we are given the differential equation, the graph that’s equation, the graph that’s sketched through the points is sketched through the points is called the solution curve (an called the solution curve (an approximation of the real solution)approximation of the real solution)

Differential EquationDifferential EquationA differential A differential equation tells us that equation tells us that the slope of the the slope of the curve at any point x curve at any point x is the x-value at that is the x-value at that point.point.

In slope field In slope field problems, they problems, they always give us the always give us the differential equation differential equation or it’s graph.or it’s graph.

1. Let’s say: 1. Let’s say: f’(x)=2x/yf’(x)=2x/y

So at point (6,4), f’(x) So at point (6,4), f’(x) or the slope = 2(6)/4 or the slope = 2(6)/4 = 3.= 3.

2. Let’s say: f’(x)=3xy2. Let’s say: f’(x)=3xy

So at point (-2, -1), So at point (-2, -1), f’(x) or the slope = 3(-f’(x) or the slope = 3(-2)(-1) =62)(-1) =6

We have differential We have differential equation… now what?equation… now what?

1.1. Select a coordinate (x,y) and plug it into Select a coordinate (x,y) and plug it into the diff. equation.the diff. equation.

2.2. Evaluate at that point (x,y) and you will Evaluate at that point (x,y) and you will get the slope at (x,y).get the slope at (x,y).

3.3. Draw a short line segment at point (x,y) Draw a short line segment at point (x,y) on the coordinate plane with that slope.on the coordinate plane with that slope.

4.4. Do this until you have enough points to Do this until you have enough points to draw a solution curve through all the draw a solution curve through all the points.points.

5.5. If given an initial point, then draw graph If given an initial point, then draw graph with the initial point on the graph.with the initial point on the graph.

Some helpful tips before we Some helpful tips before we startstart

Make a chart of points (x,y) and their Make a chart of points (x,y) and their respective slopes to make it easier respective slopes to make it easier for you to graph.for you to graph.

Draw line segments in all 4 Draw line segments in all 4 quadrants.quadrants.

Draw enough line segments so you Draw enough line segments so you can predict the slopes of other points can predict the slopes of other points without evaluating.without evaluating.

Let’s draw a slope field Let’s draw a slope field 1.1. Select a coordinate Select a coordinate

(x,y) and plug it into (x,y) and plug it into the diff. equation.the diff. equation.

2.2. Evaluate at that point Evaluate at that point (x,y) and you will get (x,y) and you will get the slope at (x,y).the slope at (x,y).

3.3. Draw a short line Draw a short line segment at point (x,y) segment at point (x,y) on the coordinate on the coordinate plane with that slope.plane with that slope.

4.4. Do this until you have Do this until you have enough points to enough points to draw a solution curve draw a solution curve through all the points.through all the points.

Ex. 1: f’(x)=2x. Sketch a slope field for the Ex. 1: f’(x)=2x. Sketch a slope field for the given differential equation at the 9 given differential equation at the 9 points indicated. Initial point f(2)=0points indicated. Initial point f(2)=0

http://www.math.lsa.umich.edu/courses/116/slopefields.html

Let’s draw another slope Let’s draw another slope field ;)field ;)

Ex. 2: f’(x)=1/x. Sketch a slope field Ex. 2: f’(x)=1/x. Sketch a slope field for the given differential equation at for the given differential equation at 9 the points indicated. Initial point: 9 the points indicated. Initial point: f(1)=1f(1)=1

1.1. Select a coordinate Select a coordinate (x,y) and plug it into (x,y) and plug it into the diff. equation.the diff. equation.

2.2. Evaluate at that point Evaluate at that point (x,y) and you will get (x,y) and you will get the slope at (x,y).the slope at (x,y).

3.3. Draw a short line Draw a short line segment at point (x,y) segment at point (x,y) on the coordinate on the coordinate plane with that slope.plane with that slope.

4.4. Do this until you have Do this until you have enough points to enough points to draw a solution curve draw a solution curve through all the points.through all the points.

http://www.ies.co.jp/math/java/calc/DiffEqu/DiffEqu.html

Quiz time!Quiz time!Multiple Choice:Multiple Choice:

Choose the slope field of the Choose the slope field of the following graph with following graph with particular solution (0,0) particular solution (0,0) to the correct differential to the correct differential

equationequation..

a)a) y’=y+xy’=y+x

b)b) y’=y-xy’=y-x

c)c) y’=x2y’=x2

d)d) y’=7-xy’=7-x

Graph of differential Graph of differential equation:equation:

And the answer is…And the answer is…

A) is the correct answer! Congrats on A) is the correct answer! Congrats on those who figured it out.those who figured it out.

Quiz timeQuiz time

a)a) f’(x)=yf’(x)=y

b)b) f’(x)=y+xf’(x)=y+x

c)c) f’(x)=sin(x+y)f’(x)=sin(x+y)

d)d) f’(x)=-x/yf’(x)=-x/y

And the answer is…And the answer is…1. b)1. b)2. a)2. a)3. d)3. d)4. c)4. c)

Yeah!Yeah!

More Quiz!More Quiz!

FREE RESPOSE FREE RESPOSE QUESTION:QUESTION:

a)a) Sketch the slope field for Sketch the slope field for the differential equation the differential equation y’=6x2y’=6x2

b)b) Explain why a solution Explain why a solution could not have the graph could not have the graph shown to the right.shown to the right.

c)c) Draw the exact solution Draw the exact solution curve through the points curve through the points with initial value (0,1) with initial value (0,1) using antidifferentiation.using antidifferentiation.

And the answer is…And the answer is…

A) I will show on the board.A) I will show on the board.

B) I will show on the board.B) I will show on the board.

C) I will show on the board.C) I will show on the board.

Slope field websitesSlope field websiteshttp://www.sosmath.com/diffeq/slopehttp://www.sosmath.com/diffeq/slope/slope1.html/slope1.htmlhttp://archives.math.utk.edu/visual.chttp://archives.math.utk.edu/visual.calculus/4/slope_fields.3/alculus/4/slope_fields.3/http://www.math.montana.edu/frankhttp://www.math.montana.edu/frankw/ccp/calculus/des/slope-fields/learn.w/ccp/calculus/des/slope-fields/learn.htmhtmhttp://www.calculusapplets.com/slophttp://www.calculusapplets.com/slopefields.htmlefields.html

A cool A cool ThingThing!!