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 .