- Background
- Getting Started
- Vector functions/Parametric curves
- Derivatives and the slope
- The parametric form of a line
- Plotting Polar Curves

- Area in Polar Coordinates
- Exercises

\\storage\academics\math\calclab\MA1023\Parametric_polar_start_A14.mwYou can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read through the exercises, start up Maple, load the worksheet, and go through it carefully. Then you can start working on the exercises.

>r:=t->[3*sin(t),3*cos(t)];You can evaluate this function at any value of t in the usual way.

>r(0);This is how to access a single component.

>r(t)[1]; >r(t)[2];

In Maple, the slope of a parametric curve can be calculated using the command as in the example below.

> r := -> [3*sin(t),3*cos(t)]; > slope := diff(r(t)[2],t)/diff(r(t)[1],t);

Below is an example in Maple using this parametric form of a line that is tangent to the curve defined above at . The plot of the curve and the line on the same graph verifies that the line is tangent at the given point.

>eval(slope,t=Pi/4);Since the slope at is -1, we want the line through the point , parallel to the vector .

>line:=t->r(Pi/4)+[t,-t]; >with(plots): >a:=VPlot(r(t),t=-Pi..Pi): >b:=VPlot(line(t),t=-Pi..Pi): >display(a,b);

Name | Equation |

cardioid | or |

limaçon | or |

rose | or |

Below is an example of a cardiod.

>plot(1-cos(theta),theta=0..2*Pi,coords=polar);

>Area1:=1/2*int((1-cos(theta))^2, theta=0..2*Pi); >evalf(Area1);

- Consider the vector function
for
.
- Calculate the coordinate point on the curve at and the slope of the curve at .
- Define the vector equation of the line through the point above tangent to the curve at that point.
- Plot the graph of and this tangent line on the same graphover the interval . Identify the graph as a cardioid, limaçon, or rose.

- Find all points of intersection for each pair of curves in polar
coordinates.
- and for .
- and for .

- Find the angles that create only one petal of the five petal rose given by the equation . Plot only one petal and find the area of that petal.

2014-10-06