l11 - slope fields complete 2.notebook

L11 Slope Fields COMPLETE 2.notebook

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May 02, 2018

Slope FieldsGraphical Representations of Solutions to Differential Equations

A slope field is a pictorial representation of all of the possible solutions to a given differential equation.

Remember that a differential equation is the first derivative of a function, f'(x) or . Thus, thesolution to a differential equation is the function, f(x) or y.

There is an infinite number of solutions to the differential equation = x + 3. Show your work andexplain why.

For the AP Exam, you are expected be able to do the following 4 things with slope fields:

1.

2.

3.

4.

dydx

dydx

Sketch a slope field given a differential equation, dy/dx

Given a slope field, draw a particular solution

Match a slope field to a given differential equation

Match a slope field to a solution of a differential


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May 02, 2018

#1 Sketch a slope field given a differential equation.

Given the differential equation below,compute the slope for each pointindicated on the grid to the right.

Then, make a small mark thatapproximates the slope throughthe point.

Given the differential equation below,compute the slope for each pointindicated on the grid to the right.

Then, make a small mark thatapproximates the slope throughthe point.

dydx = x ‑ 1

dydx = x ‑ y


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May 02, 2018

#2 Given a slope field, sketch a solution curve through a given point.

To the right is pictured the slope field that youdeveloped for the differential equationon the previous page.

Sketch the solution curve through the point(1, ‑1).

To do this, you find the point and then followthe slopes as you connect the lines

dydx = x ‑ 1


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May 02, 2018

#3 Match a slope field to a differential equation.

Since the slope field represents all of the particular solutions to a differential equation, and thesolution represents the ANTIDERIVATIVE of a differential equation, then the slope field shouldtake the shape of the antiderivative of dy/dx.

Match the slope fields to the differential equations on the next page.


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May 02, 2018

Separate the variables and find the general solution to each differential equation below to determinewhat the slope field should look like fo reach. Then, match to the graphs of slope fields on theprevious page.


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May 02, 2018

#4 Match a slope field to a solution to a differential equation.

When given a slope field and a solution to a differential equation, then the slope field should look likethe solution, or y.

Match the slope fields to the solutions on the next page.


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May 02, 2018


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May 02, 2018

Shown below is a slope field for which of the following differential equations? Explain your reasoningfor each of the choices below.

Consider the differential equation = to answer the following questions.

a. On the axes below, sketch a slope field for the equation.

b. Sketch a solution curve that passes through the point (0, ‑1) on your slope field.

c. Find the particular solution y = f(x) to the differential equation with the initial condition f(0) = ‑1.

y x dy

dx

l11 - slope fields complete 2.notebook

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