Slope Fields

Which of the following could be a specific solution for the differential equation whose slope field is shown?

• y=lnx

• y=sinx

• y=e^x

• y=x²

• y=cosx

Answer explanation

Remember that the derivative of e^x is e^x. The slope of the tangent to the graph of e^x increases without bound as x increases. Notice how the slope gets very sleep as x gets larger.

Tags

CCSS.HSA-REI.B.4B

Shown is the slope field for which of the following differential equations?

• dy/dx=x²

• dy/dx=y^3

• dy/dx=x

• dy/dx=2x+y

• dy/dx=xy

Answer explanation

Notice how all slopes at non-zero coordinates are positive. You should also notice how the slopes are zero when x=0. You might also notice how the slope field segments seem to match the shape of a cubic function. Also note how slopes in columns are the same so you know the d.e. would only contain the variable x.

Which differential equation corresponds to the slope field shown?

• dy/dx=x^2-y^2

• dy/dx=y^2-x^2

• dy/dx=x/y

• dy/dx=y/x

• dy/dx= -x/y

Answer explanation

Slopes are 0 along the y-axis (this is when x=0) and are infinite along the x-axis (this is when y=0). You can also notice the signs in the different quadrants. The slopes are positive in quadrants I and III, and in these quadrants x and y have the same sign.

The figure shows the slope field for which differential equation?

• dy/dx=2x

• dy/dx=-2x

• dy/dx=y

• dy/dx=-y

• dy/dx=x+y

Answer explanation

Notice how the slopes are the same within the same row. This means that the numerical value of dy/dx depends solely on the y-coordinate. You should also note how the tangents have negative slope when y>0 and positive slope when y<0.

Which of the following differential equations has the slope field shown?

• dy/dx=x-y

• dy/dx=√y-√x

• dy/dx=-xy

• dy/dx=2x/y

Answer explanation

Note how dy/dx=0 when the x-coordinate or the y-coordinate of the point is 0.

Shown is a slope field for which of the following differential equations?

• dy/dx=xy-x

• dy/dx=xy+x

• dy/dx=y-x²

• dy/dx=(y-1)x²

• dy/dx=(y-1)^3

Answer explanation

Note that the slopes are zero when y=1 and x=0.

If you factor the choice that says

dy/dx=xy-x, you will have

dy/dx=x(y-1). Note that this expression would be zero if you plugged in either y=1 or x=0. You also could analyze the signs of slopes to further confirm that the correct choice is valid.

Shown is a slope field for which of the following differential equations?

• dy/dx=xy

• dy/dx=xy-y

• dy/dx=xy+y

• dy/dx=xy+x

• dy/dx=(x+1)^3

Answer explanation

Notice how the slopes are zero when y=0 and when x=-1. If you factor the choice that says dy/dx=xy+y, you will have dy/dx=y(x+1). Notice how this differential equation would equal zero if you plugged in y=0 or x=-1.

Tags

CCSS.8.EE.C.8B

The slope field for a certain differential equation is shown. Which of the following could be a solution to the differential equation with the initial condition y(0)=1? (Note this is a NON-calculator example.)

• y=cosx

• y=1-x²

• y=e^x

• y=√1-x²

• y=1/(1+ x² )

Answer explanation

If you take the derivative of the choice that says y=1/(1+x^2) using the Quotient Rule, you will have

dy/dx= -2x/(1+x^2)^2. The denominator is positive for all real numbers, so you really only need analyze the sign of the numerator, -2x. Note how if x>0, the slopes would be negative, and if x<0, the slopes would be positive. This matches the slope field shown. You can also note that the graph of y=1/(1+x^2) can essentially be seen lurking in the slope field. That certainly is not the case for any of the other choices.