9.2 Calculus with Parametric Curves

1-2. Find .

1. .

.

2. .

.

3-6. Find an equation of the tangent to the curve at the point.

3. .

At the point with parameter value , the slope is .

Slope of the tangent at is .

Hence, the equation of the tangent line is => .

4. .

At the point with parameter value , the slope is .

Slope of the tangent at is .

Hence, the equation of the tangent line is .

5. .

At the point with parameter value , the slope is .

The point (1,1) corresponds to the parameter value ,

so the slope of the tangent at the point is .

Hence, the equation of the tangent line is .

6. .

7-10. Find . For what values of is the curve concave upward or downward?

7. .

,

.

¢¡ Since , for all .

Hence, the curve is downward everywhere.

8. .

,

.

¢¡ If , . Thus if , .

Hence, the curve is concave upward on ,

and is concave downward on .

9. , , .

,

.

¢¡ If , .

Thus if , .

Hence, the curve is upward on , and the curve is downward on .

10. .

11-12. Find the point on the curve where the tangent is horizontal or vertical.

11. .

12. .

, so for . Points at which the tangent is horizontal and , so for .

The curve has horizontal tangents at , and vertical tangents at .

13.Show that the curve , has two tangents at (0, 0) and find their equations.

.

Now is 0 when , so there are two tangents at the points (0,0) since both correspond to the origin. The tangent corresponding to has slope 2, and its equation is . The tangent corresponding to

has slope -2, and its equation is .

14.At what point does the curve , cross itself? Find the equations of both tangents at that point.

http://matrix.skku.ac.kr/cal-lab/cal-9-2-14.html

t=var('t'); parametric_plot(((1-2*cos(t)^2),tan(t)*(1-2*cos(t)^2)),(t,-2,2)). show(aspect_ratio=1,xmin=-5, xmax=5, ymin=-7, ymax=7) |

At the curve does cross itself. Moreover, is 0 when , , , .

And the tangent corresponding to t has slope 1 or -1. Then the equations of tangents at (0,0) are .

15.At what points on the curve is the tangent parallel to the line with equations ?

16.Find the area bounded by the curve , and the line .

17. Find the area bounded by the curve , , and the lines and .

The curve intersects the -axis when .

Then, the corresponding values of are .

http://matrix.skku.ac.kr/cal-lab/cal-9-2-17.html

t=var('t'); parametric_plot((cos(t),e^t),(t,0,pi)).show(aspect_ratio=1, xmin=-5, xmax=10, ymin=-5, ymax=15) |

x,y,t=var('x,y,t') x=cos(t) y=e^t integral((9-y)*diff(x,t),t,0,pi) |

1/2*e^pi - 35/2

18. Find the area of a region enclosed by the astroid .

http://matrix.skku.ac.kr/cal-lab/cal-9-2-18.html

t=var('t'); parametric_plot((cos(t)^3,sin(t)^3),(t,0,2*pi)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2) |

x,y,t=var('x,y,t') x=cos(t)^3 y=sin(t)^3 4*integral(y*diff(x,t),t,pi/2,0) |

3/8*pi

19. .

. Then, .

Hence, .

20. .

21. , ;

http://matrix.skku.ac.kr/cal-lab/cal-9-2-21.html

x,y,t=var('x,y,t'); x=2*ln(t); y=t+(1/t); parametric_plot((x,y),(t,1,4)).show(aspect_ratio=1, xmin=-1, xmax=3, ymin=-2, ymax=5) |

dxdt=diff(x,t) dxdt |

2/t

dydt=diff(y,t) dydt |

-1/t^2 + 1

L=integral(sqrt(dxdt^2+dydt^2),t,1,4) L |

15/4

22. , ; .

http://matrix.skku.ac.kr/cal-lab/cal-9-2-22.html

x,y,t=var('x,y,t'); x=t^2*cos(t); y=t^2*sin(t); parametric_plot((x,y),(t,0,2*pi)).show(aspect_ratio=1, xmin=-20, xmax=40, ymin=-30, ymax=10) |

dxdt=diff(x,t) dxdt |

-t^2*sin(t) + 2*t*cos(t)-t^2*sin(t) + 2*t*cos(t)

dydt=diff(y,t) dydt |

t^2*cos(t) + 2*t*sin(t)

L=integral(sqrt(dxdt^2+dydt^2),t,0,2*pi) L |

1/3*(4*pi^2 + 4)^(3/2) - 8/3

23. Find the length of one arch of the cycloid , .

. Then, .

Hence, .

24-26. Find the area of the surface obtained by rotating the given curve about the -axis.

24. , : .

. Then, .

Hence, .

25. , : .

. Then, .

Hence,

.

26. , ; .

27-29. Find the area of the surface generated by rotating the given curve about -axis.

27. , ; .

28. , ; .

29. , ; .

30. The curvature at a point of a curve is defined as , when is the angle of inclination of the tangent line at , as shown in the figure. Thus, the curvature is the absolute value of the rate of change of with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve ar and will be studied in greater detail in Chapter 12.

(a)For a parametric curve , derive the formula

.

where the dots indicate derivatives with respect to , that is, . [Hint: Use and Equation to find . Then use the Chain Rule to find .]

(b)For a curve as the parametric curve , , show that the formula in part (a) becomes

.

(a) Here ,

, and

. Thus .

(b) Here .

By applying the result of (a) to this case, we obtain that

.

31. (a) Show that the curvature at each point of a straight line is .

(b) Show that the curvature at each point of a circle of radius is .

(a) For a straight line, we parametrize . Since is a straight line,

is a constant, and hence is zero. By #30-(b), we have .

(b) . Then,

and . By problem 30 (a), we have .