# Worksheet: Integral Curves of Vector Fields

In this worksheet, we will practice finding the integral curve of a vector field.

Q1:

The figures show the vector field , together with several of its flows.

Suppose we know that for some numbers the integral curves are such that and are linear combinations of some . What are the values of ?

• A and 3
• B and 2
• C and 3
• D and 2
• E and 2

What are the parametric equations of the integral curve that is at when ?

• A
• B
• C
• D
• E

What are the parametric equations of the integral curve that is at when ?

• A
• B
• C
• D
• E

What are the parametric equations of the integral curve that is at when ?

• A
• B
• C
• D
• E

As and as along an integral curve, the secant between and approaches one of the lines and shown. What are the slopes of these two lines?

• Aslope of , slope of
• Bslope of , slope of
• Cslope of , slope of
• Dslope of , slope of
• Eslope of , slope of

Q2:

The figures show the vector field , together with several of its flows.

Suppose we know that, for some numbers , the integral curves and are such that and are linear combinations of some . What are the values of ?

• A and 2
• B and
• C and 2
• D and
• E and

What are the parametric equations of the integral curve that is at when ?

• A ,
• B ,
• C ,
• D ,
• E ,

What are the parametric equations of the integral curve that is at when ?

• A ,
• B ,
• C ,
• D ,
• E ,

What are the parametric equations of the integral curve that is at when ?

• A ,
• B ,
• C ,
• D ,
• E ,

Using the fact that , find a Cartesian equation satisfied by the points of the integral curve that is at when . You need not simplify your expression.

• A
• B
• C
• D
• E

As and along an integral curve, the secant between and approaches one of the lines and shown. What are the slopes of these two lines?

• Aslope of , slope of
• Bslope of , slope of
• Cslope of , slope of
• Dslope of , slope of
• Eslope of , slope of

Q3:

Consider the parametric curve , with constants and . The figure shows the case and for .

Find a vector field such that the curve and is its integral curve.

• A
• B
• C
• D
• E

Find a linear second-order differential equation satisfied by .

• A
• B
• C
• D
• E

You can check that is also a solution to this differential equation and therefore any for constants and . Using the vector field, determine the corresponding function so that and is an integral curve.

• A
• B
• C
• D
• E

For the case and , find parametric equations for the integral curve that starts at the point when .

• A ,
• B ,
• C ,
• D ,
• E ,

Q4:

The figures show the vector field , together with several of its flows.