Calculus-Sec-10-2-Solution

10.2    Calculus with Parametric Curves           by SGLee - HSKim - JHLee

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 .   <-- y=-2+3/5(x+2)=3/5x-4/5

4.

At the point with parameter value , the slope is .    <-- 역수로 고쳐야 함

Slope of the tangent at  is .

Hence, the equation of the tangent line is .      <-- y=sqrt(3)/2 x - 1/2

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.

At the point with parameter , the slope is .

If . It’s slope of the tangent at  is .

Hence, the equation of the tangent line is .

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 .

Note that this curve is an ellipse.

10.

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

11.

so

so

The curve has horizontal tangents at ,

and vertical tangents at .

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.

From the figure it is clear that at (0, 0) the curve does cross itself.

Moreover,  is 0 when .

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 ?

Given latter function’s slope is ,

so we should find  when the former function’s slope is .

. We can get  is  or .

When  is  or , the former function’s tangent parallel to the latter’s.

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

When  or .

When , x=-3/2. When , x=3/2 .

∵

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

The curve  intersects the -axis when .

Then, the corresponding values of  are .

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

∴()

19. Find the arc length of the curve defined by

.

. Then, .

Hence, .

20. Find the arc length of the curve defined by

.

,

=sqrt(2)*e^pi - sqrt(2)

21. Find the arc length of the curve defined by

.

22. Find the arc length of the curve defined by

.

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.  ;

,

= 2/1215*(247*sqrt(13) + 64)*pi

28.  ;

29.  ;