11.2E: Exercises for Section 11.2
- Last updated
- Dec 12, 2022
- Save as PDF
- Page ID
- 120859
( \newcommand{\kernel}{\mathrm{null}\,}\)
In exercises 1 - 4, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.
1)
2)
- Answer
3)
4)
- Answer
In exercises 5 - 9, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.
5)
6)
- Answer
- Slope
7)
8)
- Answer
- Slope is undefined;
9)
In exercises 10 - 13, find all points on the curve that have the given slope.
10)
- Solution
Setting this derivative equal towe obtain the equation,
Note also that this pair of parametric equations represents the circle
By substitution, we find that this curve has a slope ofat the points:
and
11)
12)
- Answer
- No points possible; undefined expression.
13)
In exercises 14 - 16, write the equation of the tangent line in Cartesian coordinates for the given parameter
14)
- Answer
15)
16)
- Answer
17) For
18) For
- Answer
- A vertical tangent line exists at
19) Find all points on the curve
20) Find
- Answer
21) Find the equation of the tangent line to
22) For the curve
- Answer
23) For the parametric curve whose equation is
24) Find the slope and concavity for the curve whose equation is
- Answer
25) Find all points on the curve
26) Find all points on the curve
- Answer
- No horizontal tangents. Vertical tangents at
and .
In exercises 27 - 29, find
27)
28)
- Answer
29)
In exercises 30 - 31, find points on the curve at which tangent line is horizontal or vertical.
30)
- Answer
- Horizontal
;
Vertical
31)
In exercises 32 - 34, find
32)
- Answer
33)
34)
- Answer
In exercises 35 - 36, find
35)
36)
- Answer
37) Find intervals for
38) Determine the concavity of the curve
- Answer
- Concave up on
39) Sketch and find the area under one arch of the cycloid
40) Find the area bounded by the curve
- Answer
41) Find the area enclosed by the ellipse
42) Find the area of the region bounded by
- Answer
In exercises 43 - 46, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.
43)
44) [T]
- Answer
45) [T]
46) [T]
- Answer
In exercises 47 - 52, find the arc length of the curve on the indicated interval of the parameter.
47)
48)
- Answer
49)
50)
- Answer
51)
52)
- Answer
53) Find the length of one arch of the cycloid
54) Find the distance traveled by a particle with position
- Answer
units
55) Find the length of one arch of the cycloid
56) Show that the total length of the ellipse
57) Find the length of the curve
In exercises 58 - 59, find the area of the surface obtained by rotating the given curve about the
58)
- Answer
59)
60) [T] Use a CAS to find the area of the surface generated by rotating
- Answer
61) Find the surface area obtained by rotating
62) Find the area of the surface generated by revolving
- Answer
63) Find the surface area generated by revolving
