7.3.5What is the slope field pattern?

Plotting Slope Fields Efficiently

7-150.

Theresa enjoys tedious tasks, so slope fields are among her favorite graphs to sketch. However, her teammates desire a more efficient method to plot slope fields. Sketch each of the following slope fields as efficiently as possible.

$\frac { d y } { d x }=\frac { x } { y }$

$\frac { d y } { d x }=\frac { y } { x }$

$\frac { d y } { d x }=\frac { - x } { y }$

$\frac { d y } { d x }=\frac { - y } { x }$

Plot each slope field and trace a few particular solutions. Describe the shape of the particular solution—is it familiar?
Describe any patterns you used. For example, where is the slope horizontal? Where is it undefined (or vertical)?
Use implicit integration to solve the differential equation. Compare the solution to the slope field. Explain any inconsistencies. Hint: Consider looking at the standard form of the solution.

7-151.

Discuss with your team how to plot the following slope fields. Will the tangent lines be parallel? If so, in which direction? Then sketch each slope field on your own paper.

$\frac { d y } { d x }= y + 1$

$\frac { d y } { d x }= x + 1$

Write an equation for the general solution to each differential equation using implicit integration. Discuss any similarities and differences.

7-152.

Your teacher will provide you with a model. DO ALL SOLUTIONS LOOK THE SAME?

The slope field at right represents the differential equation

$\frac { d y } { d x }= −0.1x + 0.2y$ .

On your paper, plot a particular solution containing the point $(0, 1)$ . Plot another solution containing the point $(0, 4)$ .
For this slope field, the different solutions do not look the same. Why not?

7-153.

Compute without a calculator SLOPE FIELD SORT

Your teacher will give you a slope field resource page and two sets of cards. Your task is to match each graph with its differential equation and its general solution. Be sure to discuss your choices with your team and explain your reasoning. Do not use a calculator!

Solving Differential Equations with Slope Fields

A differential equation is an equation that relates a function, $y$ , and its derivative.

Solutions to differential equations are functions or families of functions (or relations). The equation of the solution function and its derivatives can be substituted into the differential equation to verify that the solution function is correct. This equation can be in the form of a particular solution (when a coordinate point on the graph of the function is known) or a general solution (which includes a generic constant).

Your teacher will provide you with a model. Integration is a useful tool to solve for a differential equation. There are many integration techniques, but not every solution can be found analytically. When analytic methods fail (or become too time consuming), the graph of the solution can be approximated with a slope field, or direction field.

A slope field is a sketch of a collection of small tangent lines whose slopes are equal to $\frac { d y } { d x }$ at select values of $(x, y)$ on a coordinate plane. The collection of tangent lines reveals the shape of the curves of many general solutions to the differential equation. If a single coordinate point is known on the graph of the solution equation, then a particular solution can be sketched by connecting tangent lines, starting at the known point, and extending throughout the region in which the solution curve is continuous.

For example, the slope field of the differential equation $\frac { d y } { d x }=-\frac { y } { x }$ is shown at right. Using implicit integration, the solution to this differential equation can be shown to be $y = \frac { C } { x }$ , a hyperbola. The particular solution through $(3, 3)$ is shown on the graph. Notice that only one arm of the hyperbola is revealed on the slope field.

It is important to remember that a slope field cannot cross a vertical asymptote, locate a point of discontinuity, or have a sudden jump.

7-154.

For each of the derivative functions below, write the equations of two different possible functions for $f(x)$ . Homework Help ✎

$f^\prime(x) = 1$

$f^\prime(x) = x^2$

7-155.

Use the identity $\cos(2x) = 2 \cos^2(x) - 1$ to evaluate $\int 4 \operatorname { cos } ^ { 2 } ( x ) d x$ . Homework Help ✎

7-156.

In order to calculate the average value of a function, sometimes it makes sense to integrate while other times the slope of a secant line is determined. When do you need to use each strategy? Homework Help ✎

7-157.

Solve for $y$ if $\frac { d y } { d x }= 0.5x^2$ . Homework Help ✎

Then use the slope field for $\frac { d y } { d x }$ at right to help graph a family of solution functions (place your paper over the slope field and use the tangents as guides).

7-158.

Let $y(t)$ denote the temperature (in $^\circ\text{F}$ ) of a cup of tea at time $t$ (in minutes). The temperature of the tea starts at $190^\circ$ , while the room temperature is $70^\circ$ . The tea’s change in temperature is described by the equation: Homework Help ✎

$\frac{dy}{dt}=−0.1(y−70)$

Describe the change in temperature of the tea in relation to the room temperature.
What is the temperature of the tea at any time $t$ ?
What is the temperature of the tea after $10$ minutes?

7-159.

Use Newton’s Method to calculate the $x$ -intercept of the function $y = x - 2 + \sin(x)$ , correct to five decimal places. Homework Help ✎

7-160.

Multiple Choice: Two particles start at the origin and move along the $x$ -axis. For $0 ≤ t ≤ 2π$ , the position functions are given by $x_1(t) = -\cos(2t)$ and $x_2(t) = e^{(t–1)/4} - 0.5$ . For how many values of t in the given interval do the particles have the same velocity? Homework Help ✎

Calculus Third Edition

Solving Differential Equations with Slope Fields