# 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.