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Lesson 9-1 Parametric Equations
Example 1. Plot points to sketch the curve described by the parametric equations. Mark the
orientation on the curve. 2 5x t
2
ty
3 2t
Example 2. Change the following to rectangular form by eliminating the parameter. Then graph.
1 and , 1
11
tx y t
tt
Example 3. Eliminate the parameter to sketch the curve.
cos and 3sin , 0 2x y
Even though equations are given in terms of a parameter,
it is possible to find 2
2 and
dy d y
dx dx by differentiating
and then dividing.
t
x
y
x
y
dy
dy dtdxdx
dt
and 2
2
d dy
dt dx
dx
dt
d y
dx
x
y
x
y
227
Example 4. If 2
2cos and 3sin find and .
dy d yx t y t
dx dx
Example 5. Find the slope and concavity of 21
4 and 1, 0x t y t t at the point 2,3 .
Example 6. Write an equation of a tangent line to the curve defined by 1
1 and 1t
x t y
at the point when 1t .
Arc Length: If a curve is smooth and does not intersect itself the length of an arc is given by
Example 7. Using the parametric equations from example 6, find the arc length on the interval
1 3t .
2 2
arc lengthb
a
dx dydt
dt dt
228
Assignment 9-1:
Using these parametric equations, eliminate the parameter to write the corresponding rectangular
equation. Sketch the curve indicating the orientation without using a calculator.
1. 2
32 3, 4x t y t 2. 3 2,x t y t 3. , 4x t y t 4. 4 , 4lnx t y t
5. Use a calculator set in parametric mode to graph the curve represented by the parametric
equations 3 4cos and 1 2sinx y . Then eliminate the parameter.
6. Given the parametric equations 32 1 and 3x t y t :
a. Find 2
2 and
dy d y
dx dx
b. Find an equation of the tangent line when t = 1.
c. Use concavity to determine if the tangent line is above the curve or below the curve.
7. Given the parametric equations 3cos and 3sinx y :
a. Find 2
2 and
dy d y
dx dx.
b. Find an equation of the tangent line when 4
.
c. Use concavity to determine if the tangent line is above the curve or below the curve.
8. Use a calculator to graph the curve represented by the parametric equations 3sin 2x t and
2siny t . The curve crosses itself at the point (0,0). Find equations of all tangent lines at
that point.
Find all points at which each curve has horizontal and vertical tangents.
9. 22 1,x t y t 10. 2 21, 4x t y t t
11. 2 33, 4 12x t t y t t 12. tan , secx y
Given the parametric equations 33 5 and 12 3x t y t t (without using a calculator):
13. Find 2
2 and
dy d y
dx dx in terms of t.
14. Use the second derivative test to determine if the curve has a local maximum, a local minimum,
or neither when t = 2.
15. Use the second derivative test to determine if the curve has a local maximum, a local minimum,
or neither at the point 1,19 .
16. Use the second derivative test to determine if the curve has a local maximum, a local minimum,
or neither at the point 8, 8 .
Show an integral setup and find the length of each arc on the given interval.
17. 3
2 23 , 4x t t y t 1 2t 18. cos , sinx t t y t t 0 t
19. 2arccos , ln 1x t y t 1
20 t
229
20. Find the length of the arc between the two y-intercepts of 2 1 and 2x t y t .
21. Differentiate 2sin( 2 ) 10y x x to find dy
dx.
Integrate each of the following:
22.
10
4tdt
t
23. 3 1y y dy
y
24.
23 2x xdx
x
25. 6 4
2 1
xdx
x
26.
0
23 tan
2
xdx
27. 2
3
6 18dx
x x
28. sin( )t te e dt
29. 2 5
1
(2 4 )
xdx
x x
30.
41
22
1 3xdx
x
31. 4
3 3
2If ( ) , (8) , and (27) 5, find ( ).f x x f f f x
32. Find the domain, vertical asymptote(s), hole(s) in the graph, x- and y-intercepts,
and end behavior for 2
4
4 16
16
xy
x
. Then sketch its graph without using a calculator.
The graph of the function y f x consists of line
segments and a semicircle as shown. Evaluate
the following using geometry formulas.
33. 3
4f x dx
34. 2
22f x dx
Selected Answers:
1. 1
35y x 2.
2
3y x 4. lny x 5.
2 23 1
14 2
x y
6a. 2
2
2
3 3
2 2,
dy d yt t
dx dx b. 3
24 1y x c. below
7a. 2
3
2
1
3cot , csc
dy d y
dx dx b.
3 3
2 2y x
c. above 8.
1
3y x
9. H.T: (1,0) V.T: none 10. H.T: (5, 4 ) V.T: (1,0)
11. H.T: (3,-8), (5,8) V.T: 11 11
4 2, 12. H.T: 0, 1 V.T: none
13. 2
2
2
2
34,
dy d yt t
dx dx 14. min. 15. max. 17. 7.336 or 7.337
18. 3.678 19. .538 20. 4.591 22. 112
114t C
24. 5 3 1
2 2 26 2
5 34x x x C
x
y
230
More Selected Answers:
25. 32
34 2 1 2 1x x C or
3
24
36 4 2 1 2 1x x x C
26. 1
22ln ln 4 27.
3arctan + C
3
x 29.
421
162 4x x C
30. 1
21 1 8
13
2 31.
2
39 71
2 23x x 34. 8 2
Lesson 9-2 Polar Graphs
Plotting points in polar form
1. Use the polar grid to plot these polar points.
A 63,
B 4,
C 5
63,
D 3
22,
Point conversions 2. Write the rectangular point 1, 3 in polar form such that:
a. 0, 0r b. 0, 0r c. 0, 0r d. 0, 0r
3. Change the point 2, 4. Change the point 3,3
to rectangular form. to polar form.
Conversion Equations 2 2x y x
tan y
231
Equation conversions (rectangular to polar)
5. 4y 6. 3 2 0x y 7. 2 2 2 0x y x
Equation conversions (polar to rectangular)
8. 2r 9. 3cosr 10. 2cscr
Sketching polar graphs (use a calculator on those in bold)
11. Circles 12. Rose petal curves
a. 2cosr b. 5sinr a. 3cos 2r b. 4sin 3r
13. Limaçons 14. Lemniscate
a. 2 3cosr b. 3 3sinr 29sin 2r
232
Sketching polar graphs:
Circles: cosr d (x-axis symmetry)
sinr d (y-axis symmetry)
Rose petal curves: cosr a n (x-axis symmetry)
sinr a n (y-axis symmetry)
Limaçons: cosr a b (x-axis symmetry)
sinr a b (y-axis symmetry)
Tangent lines
15. Find an equation of the tangent line to the graph of 2 1 sinr at the point 2,0 .
16. Find the points at which the graph of 2 2cosr has horizontal tangents.
Tangent lines at the pole
17. Find the equations of the lines tangent to 4sin 3r at the pole.
is the diameterd
is the maximum ,
petals if is odd,
2 petals if is even
a r
n n
n n
may have inner loop
may or may not include the pole
233
Assignment 9-2
Without using a calculator, accurately plot each of the following polar coordinate points on a
separate graph. Give the rectangular coordinates of the point.
1. 25,
2. 3
43 2,
3. 3
4,
4. 7
61,
Without using a calculator, plot each of the following rectangular points and give two sets of polar
coordinates for 0 2 .
5. 5, 5 6. 1, 3 7. 5,0 8. 5, 5 3
9. Use a calculator to give rectangular coordinates for the polar point (-7.2,4.5).
10. Use a calculator to give two sets of polar coordinates ( 0 2 ) for the rectangular point
(-2,5).
Match each of the following equations with one of the descriptions given without using a calculator.
11. 3sin 2r 12. 4cosr 13. cos 4r 14. 4 2cosr
a. a circle with y-axis symmetry
b. a four petal rose
c. a vertical line
d. a limaçon with y-axis symmetry
e. a circle with x-axis symmetry
f. a horizontal line.
g. a limaçon with x-axis symmetry
Convert the following rectangular equations to polar (solve for r ) and sketch the graph.
15. 5y 16. 2 5 0x y 17. 2 2y x
Convert the following polar equations to rectangular and sketch the graph.
18. 5r 19. 3
4
20. 3secr
Use a calculator to graph. Determine if the interval 0 2 produces a complete graph.
21. 3
5sin2
r
22. 5 6cosr 23. r
24. Without using a calculator graph 2 4sinr . Find an equation of the line tangent to the
curve at the point 2,0 .
25. Without using a calculator graph 4cosr . Find an equation of the line tangent to the
curve at the polar point 62 3,
.
26. Without using a calculator graph 1 sinr . Find the points at which the curve has horizontal
tangent lines.
Without using a calculator graph each equation and find equations of each tangent at the pole.
27. 5sinr 28. 3 3cosr 29. 4cos 3r
234
30. Find the length of the arc of the curve defined by ln and 1x t y t on the interval 1 6t .
31. Find an equation of the line tangent to the curve defined by ln and 1x t y t when t = 1.
32. Is the curve defined by ln and 1x t y t concave upward or downward when t = 2?
33. Give a rectangular equation for the curve defined by ln and 1x t y t .
Lesson 9-3 Polar Area
Example 1. Find the area of one petal of the curve 3cos 3r .
Polar area = 21
2r d
Selected Answers:
show graphs for 1-8 1. (0,5) 2. (3,-3) 3. 2, 2 3 5. 7 3
4 45 2, , 5 2,
7. 5, , 5,0 8. 4
3 310, , 10,
9. (1.517 or 1.518 , 7.038)
11. b 13. c 15. 5cscr 16. 5
2cos sinr
17.
2
2cos
sinr
19. y x 20. 3x
21. 23.
0 4 shows a complete graph. No interval shows a complete graph.
24. 1
22y x 25.
1
33 3y x 26. 3 1 1 5
2 2 6 2 62, , , , ,
27. 0 29. 5
6 2 6, ,
30. 5.384 31. 2y x 33. 1xy e
5
5
5
577
7
10
235
Intersections of Polar Graphs
Example 2. Find the points of intersection of the graphs of 1 2cosr and 1r .
Area between two curves
Example 3. Find the area of the region common to the two regions bounded by 6cosr and
2 2cosr .
Example 4. Find the area between the loops of 2 1 2sinr .
236
Assignment 9-3
Without using a calculator, graph the following polar curves and find the points of intersection.
1. 1 sin and 1 sinr r 2. 1 sin and 1 cosr r
3. Use a calculator to graph the following curves. Then find the points of intersection.
6 8sin and 2r r
Graph the following polar curves without using a calculator. Set up a definite integral for the area
of the indicated region. Use a calculator to evaluate the integral.
4. the interior of 1 cosr 5. one petal of 4sin 3r
6. one petal of 3cos 2r 7. the common interior of 3 2cos and r
3 2cosr
Use a calculator to graph the following curves. Set up a definite integral for the area
of the indicated region. Use a calculator to evaluate the integral.
8. between the loops of 1 2sinr 9. inside 3cos and outside 2 cosr r
10. common interior of 3 and r 11. region bounded by sin 3r and
6sin 2r the x-axis for 0
12. Given the parametric equations 4 1 and 8 4x t y t , eliminate the parameter to write the
corresponding rectangular equation. Sketch the curve indicating the orientation without using a
calculator.
13. Without using a calculator given the parametric equations 23 5 and 8 4x t y t , find an
equation of the line tangent to the curve when x = 2.
14. Without using a calculator given the parametric equations 4cos and 8sinx y , determine
the concavity on an interval containing 7
6
.
15. Given the parametric equations 2 2cos and 1 sinx y , show work to determine the
points of horizontal and vertical tangency. Graph with a calculator to see if your answers appear
correct.
16. Without a calculator convert the polar point 3
23,
to rectangular form.
17. Without a calculator convert the polar point 2
34,
to rectangular form.
18. Without a calculator convert the rectangular point 5, 5 to polar form. Give two answers
such that 0 2 .
19. Using a calculator convert the rectangular point 1.372 , 5.164 to polar form. Give two
answers such that 0 2 .
237
20. Give the rectangular form of the polar equation 2sinr .
21. Given the polar equation 1 2sinr :
a. Use a calculator to sketch a graph.
b. Find equations of tangents at the pole.
c. Find the equation of the line tangent to the curve at the point (1,0).
d. Find the points at which the curve has vertical tangents. You may use a calculator.
Selected Answers:
1. 1,0 , 1, , 0,0 2. 1 3 1 7
4 42 21 , , 1 , , 0,0
4. 4.712
5. 4.188 or 4.189 6. 3.534 7. 10.557 or 10.558 8. 8.337 or 8.338 9. 5.196
10. 22.110 or 22.111 11. 7.000 12. 2 2y x 13. 16
312 2y x
15. V.T: 4,1 , 0,1 H.T: 2,2 , 2,0 17. 2,2 3 18. 5
4 45 2, , 5 2,
19. 5.343,1.830 , 5.343,4.972 21b. 6
c.
1
20 1y x
Lesson 9-4 Polar Arc Length, Vector Definitions
Arc Length
Example 1. Find the length of the arc from
0 to 2 for the curve 2 2cosr .
Vectors (definitions):
Vector-
Magnitude-
Direction-
Equivalent Vectors-
Component form-
Arc Length 2
2 dr
dr d
238
Example 2. The initial point of a vector is 1,3 and its terminal point is 2,7
a. Graph the vector. b. Graph the vector c. Give the component form
in standard position of the vector.
on the same axes.
d. Find the magnitude e. Find the direction
of the vector. of the vector.
Example 3. Find the direction of a vector given by 3 , 5 .
Example 4. If the magnitude of a vector v is 6v and its direction is 2
3
, write the
vector in component form.
Assignment 9-4
1. Given the polar curve 4sinr .
a. Graph without using a calculator.
b. Find the circumference using a geometry formula.
c. Find the circumference showing a polar arc length integral setup and integrate without
using a calculator.
2. Graph 4cos 2r without a calculator. Then use a calculator to find the length of the arc
forming one petal.
3. Use a calculator to graph 2r e
on the interval 3
20
and find the length of the curve.
4. The region shown is bounded by the polar curve 1 sinr
and the line 6
.
a. Find the area of the region.
b. Find the perimeter of the region.
x
y
239
Find the component form of the vector and sketch it with the initial point at the origin.
5. 6.
7. Find the component form of the u and v vectors whose initial and terminal points are given.
Show that u and v are equivalent. : 3, 2 , 5,2 : 1, 4 , 1,0u v
8. The initial and terminal points of a vector are 1,3 and 3,6 .
a. Sketch the vector.
b. Write the component form.
c. Sketch the vector with the initial point at the origin.
9. If the initial point of vector v is 5, 2 and 2,4v , find the terminal point.
10. Find the magnitude of the vector 4,3v .
Find the component form of each vector given the magnitude and the direction without using a
calculator.
11. 5, 0.v 12. 4
6, .3
v
13. 7
4, .4
v
14. Use a calculator to find the magnitude and the direction of the vector 8, 13v .
15. The graph at the right shows the polar curve sin 3r on the
interval 2
.
a. Find the area of the region bounded by the curve, the x-axis , and
the y-axis.
b. Find dr
d at
3
4
without using a calculator.
c. Use your answer to part b to determine if r is increasing or decreasing on an interval
containing 3
4
.
d. Find the value of on 2
at which the curve is closest to the pole.
e. Find the x-coordinate of the point on the curve when 3
4
.
f. Find dx
d at
3
4
using a calculator.
g. Use your answer to part f to determine if x is increasing or decreasing on an interval
containing 3
4
.
240
16. ( ) 4v t t represents the velocity equation of an object moving along a vertical path for
4.t Let ( )a t represent the acceleration and ( )y t represent the position of the object at
time t . Find:
a. an equation for the acceleration of the object at time t.
b. an equation for the position of the object at time t if 10
30y .
c. (5)y d. (5)v e. (5)a f. the speed of the object at time t =5.
17. 2
1( )
1
tv t
t
is the velocity equation for an object moving along a horizontal path when 0t .
Use a calculator to find:
a. the velocity of the object at 2.3t b. the acceleration of the object at 2.3t
c. the displacement from 0 to 3t t . d. the total distance traveled from 0 to 3t t .
18. If the acceleration of an object is given by sin( ) ta t e and at time t = 3 seconds the velocity of
the object is 15 feet per second, find the object’s velocity at t = 10 seconds.
19. If the graph of ( )f x is concave upward on ( , )a b , does the Trapezoidal Rule give a value
larger or smaller than the actual value of ( )b
af x dx ?
20. If ( ) 0f x on ( , )a b , does the Trapezoidal Rule give a value larger or smaller than the
actual value of ( )b
af x dx ?
21. If ( ) 0f x on ( , )a b , what must be true about the value found using the Trapezoidal Rule
and the actual value of ( )b
af x dx ?
Selected Answers:
1a. b. 4 2. 9.688 3. 21.356
4a. .596 or .597 b. 3.499 or 3.500
5. 6,1 6. 6,0
7. 2,4u v 8b. 4,3 9. 3, 2 10. 5v 11. 5,0v
12. 3, 3 3v 13. 2 2, 2 2v 14. 15.264v , 4.160 or 4.161
15a. 3.756 b. 3
21 d. 2.504 or 2.505 e. -1.166 f. -.373
241
Lesson 9-5 Calculus of Vector-Valued Functions
Position vector- Speed-
Velocity vector- Distance traveled-
Acceleration vector- “at rest”
Example 1. Given a position vector 2 3 23 , 3 4t t t for a particle moving in the xy-plane find the
following.
a. graph the path of the particle b. the velocity vector c. the speed of the particle
on the interval 0 2t at time 1t at time 1t
d. the distance traveled e. the time(s) when the f. the acceleration vector
between 0 and 3t t particle is at rest at time 2t
g. the direction of the particle at time 1t and when 2t
x
y
More Selected Answers
16. a. 1
21
24a t t
b.
3
22
34 2y t t c. 16 d. 3 e.
1
6 f. 3
17a. 2.3 .206 or .207v b. 2.3 .007 or .008a c. disp. = .097 or .098
17d. T. D. = .779 or .780 or .781 18. 23.547 19. larger 20. smaller
21. They are equal.
242
Example 2. A particle is moving in the xy-plane with acceleration vector 2cos , 3sina t t t .
At time 0t its velocity vector is 0 0,3v and its position vector is 2
20 2,s . Find :
a. the velocity vector when 4
t
b. the position vector when 4
t
c. the speed when 4
t
d. the time(s) when the e. the direction of the
particle is at rest particle when 4
t
f. the distance traveled between 0 and 2t t
Assignment 9-5
1. The position of a particle in the xy-plane is given by 24 and x t y t . Find the following:
a. the velocity vector at 4t
b. the acceleration vector at 4t
c. the speed of the particle at 4t
d. the distance the particle moves between 0t and 4t
e. the direction of the particle at 4t
2. The position of a particle is given by 2 3 and x t y t . Find the following:
a. the speed of the particle at 2t
b. the direction of the particle at 2t
c. the distance the particle moves between 1t and 4t
d. the velocity vector at 3t
e. the acceleration vector at 4t
243
3. A particle moves in the xy-plane so that at time t its velocity vector is 3 ,cosv t t t
and the particle’s position vector at time 0t is 2,1 .
a. What is the position vector of the particle when 2t ?
b. What is the acceleration vector of the particle when 2t ?
c. What is the direction of the particle when 1t ?
d. What is the distance the particle travels between 0 and 2t t ?
e. When is the particle at rest?
f. What is the speed of the particle when 2t ?
4. A particle moves on the xy-plane so that at time t its coordinates are 3x t t and
5 22y t t .
Find its velocity vector at time 2t .
5. A calculator is allowed on this problem.
The position of an object moving on a curve is ,x t y t at time t.
Given 2 and cos 12
dx t dyt
dt t dt
. At time 1t , the position of the object is 2,4 .
a. Find the position of the object at time t = 3.
b. Find the speed of the object at time 3.t
c. Find the total distance traveled by the object over the interval 1 3t .
d. Find an equation of the line tangent to the curve at time 3.t
e. Find the acceleration vector at time 3.t
6. Given a parametric curve defined by and 1tx e y t .
a. Find the length of the arc of the curve on the interval 1 6t .
b. Find an equation of the line tangent to the curve when t = 1.
c. Is the curve concave upward or downward when t = 2?
d. Give a rectangular equation for the curve .
7. Find all points of horizontal and vertical tangency to the curve 2 2cos , 2sin 2x y .
8. Find the tangents at the pole for 1 2cosr .
9. Find the area common to the interiors of 4cos and 2r r .
10. Given the function ln 1f x x :
a. Write a power series for f showing four terms and the general term.
b. Find the interval of convergence of this power series.
c. Approximate ln(1.2) by using a fourth degree Taylor polynomial of f.
d. Using your answer for part c and the alternating series remainder, give an upper and lower
limit for the actual value of ln(1.2)
244
11. The figure at the right shows a shaded region
bounded by the polar curves 4r and
8sinr .
a. Find the area of the shaded region.
b. Find the perimeter of the shaded region.
c. Convert the two polar equations to rectangular form.
d. Set up an integral with respect to the variable x and find the area of the shaded region.
Assignment 9-6 Review
1. Given the parametric equations 23,x t y t , eliminate the parameter to write the
corresponding rectangular equation. Sketch the curve indicating the orientation without using
a calculator.
2. Given the parametric equations 2 3 21 1 1
2 3 2,x t y t t :
a. Find points at which the curve has a horizontal tangent.
b. Use the second derivative test to determine if the curve has a local maximum, a local
minimum, or neither at the point found in part a.
c. Find an equation of the tangent line when t = 6.
3. Given the parametric equations 3cos , 2sinx y , find the length of the arc on the interval
2
0
,
4. Convert the polar equation 4sinr to rectangular form. Sketch a graph without using a
calculator.
Selected Answers:
1a. 1
44 32,v b.
1
324 8,a c. 4v = 32.001 d. dist. = 64.413 e. .031
2a. 2v = 12.649 b. 1.107 c. dist. = 64.949 d. 3 6,27v e. 4 2,24a
3a. 2 6,1s b. 2 12,0a c. .418 d. dist. = 4.567 or 4.568 e. never
3f. 2 8.062v 4. 13,72 5a. 3 3.394,4.280 or 4.281s b. 3 .788v
5c. dist. = 1.960 or 1.961 d. 4.280 .187 3.394 or 4.281 .188 3.394y x y x
5e. 3 .051 or .052, 5.936a 6a. 400.891 b. 1
2e
y x e c. concave down
6d. ln 1y x 7. Horiz. 2 2 , 2 , Vert. 0,0 , 4,0 8. 2 4
,3 3
9. 4.913 10a.
12 3 4 1
2 3 4
n nxx x xx
n
b. 1,1 c. .18227
11a. 19.653 or 19.654 b. 16.755 c. 2 2 16x y , 2 2 8x y y d. 19.653 or 19.654
4r
8sinr
245
5. Convert the polar equation 3 3cosr to rectangular form. Sketch a graph without using a
calculator.
6. Graph 2 3cosr without using a calculator.
7. Without using a calculator, find an equation of the line tangent to the graph of 2 3cosr at
2
.
8. Find all points of intersection of 2 3cosr and 5cosr .
9. Use a calculator to graph 2 10sin 2 and 5r r and find the area common to the two
interiors.
10. Set up an integral for the length of the arc in the first quadrant on the curve 2 10sin 2r .
Why can this integral not be evaluated on a calculator?
11. The position of a particle moving in the xy-plane at any time t , 0 2t , is given by the
parametric equations 4cos 2 and 4sinx t y t . Do not use a calculator.
a. Find the velocity vector for the particle at any time t.
b. Find the velocity vector for the particle at time t = .
c. Find the acceleration vector for the particle at time t = .
d. Find the speed when y = 4 .
e. For what values of t is the particle at rest?
f. Find the direction of the particle when t =2
.
g. Set up an integral for the distance traveled on the interval 1 5t .
12. A particle moves along the graph of siny x . If the x-component of acceleration is always 3
and at time 0t , the position of the particle is the point ,0 and the velocity vector of the
particle is 2, 2 . Without using a calculator. Find the x- and y-coordinates of the position of
the particle in terms of t.
13. Calculator Allowed
A particle moving along a curve in the xy-plane has position ,x t y t with
22sin and tdx dy
t edt dt
. The position of the particle is 1,4 at time 3t .
a. Find the acceleration vector at time 2t .
b. Find the position point of the particle at time 0t .
c. At what time does the speed of the particle reach 15 when 0t .
14. Calculator Allowed
Let g be a function that has derivatives of all orders for all real numbers. Assume 0 2,g
0 1, 0 4, and g 0 3.g g
a. Write the third-degree Taylor polynomial for g about x = 0.
b. Write the fourth-degree Taylor polynomial for 2g x .
c. Write the third-degree Taylor polynomial for h, where 0
, about 0.x
h x g t dt x
246
15. Euler’s Formula
a. Use the elementary series for xe to write at least six terms of the series for ie .
b. Simplify your answer using the relationship for powers of i. ( 1 2 3, 1, , i i i i i= = − = −
4 5 1, , etc.)i i i= =
c. Separate the terms of your simplified series into two infinite series (one with the odd power
terms and one with the even power terms). Use elementary series for sine and cosine to
evaluate.
Note: This result is sometimes written as the equation 1 0ie + = called Euler’s Formula. This
mathematically elegant equation contains the five most “important” numbers in mathematics.
Answers:
1. ( )2
3y x= − 2a. ( )1 1
2 6,− b. local minimum c. ( )54 5 18y x− = −
2 23. 3.966 4. 4 x y y+ =
5. 6.
7. ( )3
22 0y x− = − −
8. ( ) ( )5, , 0,0 9. 6.575 or 6.576
10. ( )( )
( )
2
2
0
10cos 210sin 2
10sin 2d
+
This integral is improper since dr
dis undefined at both endpoints.
11a. ( ) ( )8sin 2 , 4cosv t t t= − b. ( ) 0, 4v = − c. ( ) 16,0a = − d. ( )3
20v
=
11e. 3
2 2,t
= f.
3
4
= g. ( )( ) ( )
5 2 2
18sin 2 4cost t dt− +
12. ( ) ( ) ( )2 23 3
2 22 , sin 2x t t t y t t t = + + = + + 13a. ( )2 2.614 or 2.615, 218.393a = − −
13b. ( )1.773 or 1.774, 1440.545− − − c. t = 1.645
14a. ( ) 2 31
22 2g x x x x − + + b. ( )2 2 42 2g x x x − + c. ( ) 2 31 2
2 32h x x x x − +
15a. ( )( ) ( ) ( ) ( ) ( ) ( )
2 3 4 5 6 7
12! 3! 4! 5! 6! 7!
ii i i i i i
e i • • •= + + + + + + + +
b. ( )2 3 4 5 6 7
12! 3! 4! 5! 6! 7!
i i i ie i
• • •= + − − + + − − +
c. -1
x
y
x
y
x
y
− −
x
y
− −
247
Unit 9 Summary
Parametric derivatives:
dy
dy dtdxdx
dt
2
2
d dy
dt dx
dx
dt
d y
dx
Arc Length 2 2
1 1
2 2 22 21b t
a t
dx dy dr
dt dt df x dx dt r d
Polar Conversions: 2 2 2 , cos , sinr x y x r y r , arctan
y
x
Sketching polar graphs:
Circles: cosr d (x-axis symmetry)
sinr d (y-axis symmetry)
Rose petal curves: cosr a n (x-axis symmetry)
sinr a n (y-axis symmetry)
Limaçons: cosr a b (x-axis symmetry)
sinr a b (y-axis symmetry)
Polar Area2
1
21
2r d
Tangent lines at the pole: let 0r
Vectors:
Magnitude = 2 2x y Direction = arctan
y
x
Speed = 2 2
dx dy
dt dtv t Total Dist. = 2 2
1 1
2 2t t
t t
dx dy
dt dtv t dt dt
is the diameterd
is the maximum ,
petals if is odd,
2 petals if is even
a r
n n
n n
may have inner loop
may or may not include the pole
parametric polar
