LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Parametric Curves and Polar Coordinates Math 251, Fall 2011 J. Gerlach

We will investigate several aspects of parametric curves in the plane. The curve given by 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 and 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 will serve as an example throughout, and we begin by defining the curve once and for all. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRInRGJ0YvRjJGNS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GLDYlUSRzaW5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2I0ZVRjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSEZKRlRGNUY1LUYsNiVRInlGJ0YvRjJGNUZPRjVGVUY1RlhGNUZlbi1Gam42JC1GIzYlLUkjbW5HRiQ2JFEiMkYnRjktRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZVRjlGXm8= print(); # input placeholder JSFH

A simple modification of the plot command lets us graph a parametric curve: Enclose the two curves and the range of the parameter in a single bracket.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 %; The option scaling=constrained was added to obtain equal spacing on the coordinate axes.

JSFH

Problem: Find the slope of the curve at the point where t=\317\200/3. Solution: Finding the derivative is easily accomplished using the formula 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. 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 print(); # input placeholder 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYn print(); # input placeholder Thus the slope at 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 is -2.

JSFHJSFH

JSFH Problem: Graph the curve along with its tangent line at the point where t=\317\200/3 in a common figure. Solution: The solution requires several steps. First we denote the point which corresponds to t=\317\200/3 by (X,Y), and we compute it as 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 print(); # input placeholder In order to find the slope, we substitute t=\317\200/3 into the formula for m, and call the result M (recall that m was determined above) 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 print(); # input placeholder 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 print(); # input placeholder Thus the tangent line has slope m=-2. We have several options to obtain the curve and the tangent line in a common figure. For one we could plot the curve and the tangent line separately and then superimpose the two with the display command. This approach requires uploading the plots package, and we will not apply this method here. As an alternative we can express the tangent line in parametric form and plot the two curves simultaneously, and we will take this route. A third option will be shown below. The tangent line y -Y = M(x-X) can be expressed in parametric form by setting x = t, and we get y = Y +M(t-X) 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 This formula can be directly substituted into the plotting command. Result: 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 %; Note that in the plotting command the two parametric curves were enclosed by brackets and that the command has the structure plot( [ [first curve] , [second curve] ], options); The third option is more elegant. We compute the tangent lines for the two functions x(t) and y(t) at t=\317\200/3 as functions of t, denoted by lx(t) and ly(t), and then we plot (lx(t),ly(t)) as a parametric curve. Here are the details: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEjbHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSJ0RidGL0YyRjUtRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GNS1GLDYlUSJYRidGL0YyRjUtRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZcb0Y1LUYsNiVRIkRGJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGL0YyRjktRmNvNiQtRiM2Iy1JJm1mcmFjR0YkNigtRiw2JVElJnBpO0YnRmFvRjktRiM2Iy1JI21uR0YkNiRRIjNGJ0Y5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZfcS8lKWJldmVsbGVkR0Y9RjktRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GY282JC1GIzYlRlUtRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSEZbb0Zdb0ZecEY5LUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGSkZU print(); # input placeholder 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 print(); # input placeholder The next two pictures show that we indeed calculated ordinary tangent lines. 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 %; 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 %; Viewed as parametric curve, the pair lx(t) and ly(t) becomes the tangent line of our curve. 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 %;

JSFH

Problem: Find the point in the first quadrant where the slope equals -1. Solution: We set the slope dy/dx = m, and solve the equation m=-1 for t. 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSJtRidGL0YyLUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUY+Ni1RKiZ1bWludXMwO0YnRkFGQ0ZGRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJy9GVkZlbi1JI21uR0YkNiRRIjFGJ0ZBLUY+Ni1RIixGJ0ZBRkMvRkdGMUZIRkpGTEZORlAvRlNRJjAuMGVtRicvRlZRLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRi9GMkZBLUY+Ni1RIjtGJ0ZBRkNGXm9GSEZKRkxGTkZQRl9vRlU= print(); # input placeholder The first value appears to be the desired value for t, and we name it T (copy-and-paste) 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 print(); # input placeholder Then the point with slope m=-1 is located at x(T) and y(T). These points are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRIlRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRIlRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= print(); # input placeholder Numerically the coordinates of the point with slope -1 are 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 print(); # input placeholder Clearly, this point belongs to the first quadrant.

Problem: What are the slopes as the curve passes through the origin? Solution: First we need to find the value of the parameter when the curve passes through the origin, that is, we must solve x(t)=0 and y(t)= 0 simultaneously. 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 print(); # input placeholder We find two values. Check: 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 print(); # input placeholder For the slopes we find: 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GMTYlUSJ5RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGNy1GOzYkLUYjNiMtRjE2JVElJnBpO0YnRjRGN0Y3LUYjNiVGMC1GOzYkLUYjNiMtRjE2JVEieEYnRkJGREY3RkYvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmVuLyUpYmV2ZWxsZWRHRjYtSSNtb0dGJDYtUSI7RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYn print(); # input placeholder Thus the slopes are +2 and -2.

Problem: What is the arc length of the curve? Solution: We can implement the arc length formula directly, and we obtain using the templates on the left 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 print(); # input placeholder 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 Maple couldn't find an analytic solution, and we accept a numerical approximation in its place.

Problem: Find the area enclosed by the curve. Solution: The area between a parametric curve and the x-axis can be computed as follows: 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ are to be selected to match the endpoints. In our example we use symmetry, and multiply the area in the first quadrant by 4 to get the total area. The parameter interval is [0,\317\200/2], and the area becomes 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 print(); # input placeholder JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= Here we use the example r(\316\270) = 3 + 5 sin \316\270, and we shall work several problems related to this curve. First, we define r once and for all QyQ+SSJyRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYiIiQiIiItSSRzaW5HRiU2IzkkIiImRiVGJUYlRi4= print(); # input placeholder It is easier to work with t rather than typing \316\270 all the time. However, since r is a function (change the input from t to \316\270) we have QyQtSSJyRzYiNiNJJnRoZXRhR0YlIiIi print(); # input placeholder

Plotting in polar coordinates requires the option coords=polar, everything else is pretty much the same. This is the equivalent of setting your calculator to the polar function mode. Here is the graph: QyQtSSVwbG90RzYiNiZJInJHRiU7IiIhLCRJI1BpRyUqcHJvdGVjdGVkRyIiIy9JJ2Nvb3Jkc0dGJUkmcG9sYXJHRiUvSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJSIiIg== %; The plots package offers a nice alternative. In order to use it, we need to activate it first with the command QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi Use a colon to avoid unnecessary output. Now we are ready to plot. QyQtSSpwb2xhcnBsb3RHNiI2JC1JInJHRiU2I0kidEdGJS9GKjsiIiEsJEkjUGlHJSpwcm90ZWN0ZWRHIiIjIiIi %;

Problem: What is the slope of the curve at the point (x,y)=(3,0)? Solution: First of all we realize that the point (3,0) is obtained by setting \316\270 = 0. For the slope dy/dx we follow the usual procedure for parametric curves, this time with the diff command KiYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkKiYtSSJyRzYiNiNJInRHRioiIiItSSRzaW5HNiRGJUkoX3N5c2xpYkdGKkYrRi1GLEYtLUYkNiQqJkYoRi0tSSRjb3NHRjBGK0YtRiwhIiI= print(); # input placeholder QyQtSSlzaW1wbGlmeUc2IjYjSSIlR0YlIiIi print(); # input placeholder And now we set t = 0 QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEc2IiIiIUkiJUdGKSIiIg== print(); # input placeholder QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSIlR0YoIiIi print(); # input placeholder Thus the desired slope is 3/5.

Problem: Graph the curve along with its tangent line at the point (3,0) in a common figure. Solution: We know from the last problem that the slope is 3/5, and using the point-slope form of a line, the tangent line is given by 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 For the graph we have two alternatives: Either express the line in polar coordinates and use a single ploting statement, or graph the two pieces separately and superimpose the graphs. We will demonstrate both. In polar coordinates the line becomes QyQvKiZJInJHNiIiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0kidEdGJkYnLUkiKkdGKzYkIyIiJCIiJiwmKiZGJUYnLUkkY29zR0YqRi1GJ0YnISIkRidGJw== print(); # input placeholder and solving this equation for r we find QyQtSSZzb2x2ZUc2IjYkSSIlR0YlSSJyR0YlIiIi print(); # input placeholder This is an expression for the tangent line in polar coordinates, and with a simple copy-and-paste we obtain QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNic3JC1JInJHRig2I0kidEdGKCwkKiQsJi1JJHNpbkdGJUYtISImLUkkY29zR0YlRi0iIiQhIiIiIiovRi47LCRJI1BpR0YmRjhGPS9JJ2Nvb3Jkc0dGKEkmcG9sYXJHRicvSSV2aWV3R0YoNyQ7RjQiIic7ISIjRjkvSShzY2FsaW5nR0YoSSxjb25zdHJhaW5lZEdGKCIiIg== %; The alternative requires the plotting package (which is already activated). QyQ+SSJBRzYiLUkqcG9sYXJwbG90R0YlNiRJInJHRiU7IiIhLCRJI1BpRyUqcHJvdGVjdGVkRyIiIyEiIg== QyQ+SSJCRzYiLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUkiKkdGKTYkIyIiJCIiJiwmSSJ4R0YlIiIiISIkRjQvRjM7IiIhIiInISIiJSFH %; print(); # input placeholder QyQtSShkaXNwbGF5RzYiNiRJIkFHRiVJIkJHRiUiIiI= %;

Problem: What are the slopes of the curve as it passes though the origin? Solution: First we set r=0 to find the points where the curve passes through the origin. QyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLy1JInJHRig2I0kidEdGKCIiIUYuIiIi print(); # input placeholder This result is of limited use. But applying our knowledge of trigonometry we conclude that r(t)=0 for t = 2\317\200 - arcsin(3/5) and t = \317\200 + arcsin(3/5) Let's confirm this: QyY+SSNUMUc2IiwmSSNQaUclKnByb3RlY3RlZEciIiItSSdhcmNzaW5HNiRGKEkoX3N5c2xpYkdGJTYjIyIiJCIiJkYpRiktSSJyR0YlNiNGJEYp print(); # input placeholder QyY+SSNUMkc2IiwmSSNQaUclKnByb3RlY3RlZEciIiMtSSdhcmNzaW5HNiRGKEkoX3N5c2xpYkdGJTYjIyIiJCIiJiEiIiIiIi1JInJHRiU2I0YkRjM= print(); # input placeholder Now we calculate the slope at these points. QyQqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQqJi1JInJHNiI2I0kidEdGKyIiIi1JJHNpbkc2JEYmSShfc3lzbGliR0YrRixGLkYtRi4tRiU2JComRilGLi1JJGNvc0dGMUYsRi5GLSEiIkYu print(); # input placeholder QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kidEc2IkkjVDFHRilJIiVHRikiIiI= print(); # input placeholder QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSIlR0YoIiIi print(); # input placeholder QyQqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQqJi1JInJHNiI2I0kidEdGKyIiIi1JJHNpbkc2JEYmSShfc3lzbGliR0YrRixGLkYtRi4tRiU2JComRilGLi1JJGNvc0dGMUYsRi5GLSEiIkYu print(); # input placeholder QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUklc3Vic0dGJjYkL0kidEdGKEkjVDJHRihJIiVHRigiIiI= print(); # input placeholder There is a Much easier way to determine the slope. We know from class that the slope is the tangent of the respective angles. QyYtSSR0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kjVDFHRigiIiItRiQ2I0kjVDJHRihGKw== print(); # input placeholder JSFH

Problem: Find the arc length of the curve. Solution: Here we implement the familiar formula for the arc length of a polar curve directly, and let maple do the work. LUkkaW50RzYiNiQtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJiokLUkickdGJDYjSSJ0R0YkIiIjIiIiKiQtLUkiREdGJDYjRi9GMEYyRjMvRjE7IiIhLCRJI1BpR0YpRjI= print(); # input placeholder 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 LSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiM1NiMsKC1JKkVsbGlwdGljRUc2JEYlSShfc3lzbGliRzYiNiMsJCokIiM6IyIiIiIiIyNGNCIiJSIjWy1JK0VsbGlwdGljUGlHRiw2JSwkKiQiIzxGMyNGN0Y+I0YyIiM7RjAhIiItRis2JCwkKiRGNUYzRjNGMCEjOw== JSFH 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 The result involves special functions, but numerical evaluation leads to an acceptable result.

Problem: Compute the area enclosed by the inner loop of the curve. Solution: The inner part of the curve is traced when T1 < t < T2 (the numbers T1 and T2 were found above). Using the area formula for polar curves we find that LCQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCokLUkickdGKDYjSSJ0R0YoIiIjL0YuO0kjVDFHRihJI1QyR0YoIyIiIkYv print(); # input placeholder QyQtSSZldmFsZkclKnByb3RlY3RlZEc2I0kiJUc2IiIiIg== print(); # input placeholder The area is roughly two square units.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=