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Chap 10 HW -- Parametric and Polar (11708646)

Due: Fri, May 31, 2019 12:00 AM MST

1. -Question Details SCalcET8 10.1.AE.002. [3798588]

Video Example

EXAMPLE 2 What curve is represented by the following parametric equations?

SOLUTION If we plot points, it appears that the curve is a circle. We can confirm this impressionby eliminating t. Observe that

.

Thus the point moves on the circle Notice that in this example theparameter t can be interpreted as the angle (in radians) shown in the figure. As t increases from 0to 2π, the point moves once around the circle in the

---Select--- direction starting at the point

x = 3 cos(t) y = 3 sin(t) 0 ≤ t ≤ 2π

x2 + y2 = 9 cos2(t) + =

(x, y) x2 + y2 = .

(x, y) = (3 cos(t), 3 sin(t))

(x, y) = .

2. -Question Details SCalcET8 10.1.512.XP.MI.SA. [3945508]

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and youwill not be able to come back to the skipped part. Consider the following.

Exercise (a)

Eliminate the parameter to find a Cartesian equation of the curve.

Exercise (b)

Sketch the curve and indicate with an arrow the direction in which the curve is traced as theparameter increases.

x = ln(t), y = , t ≥ 4t



3. -Question Details SCalcET8 10.1.510.XP. [3945524]

Consider the following.

(a) Eliminate the parameter to find a Cartesian equation of the curve.

(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.

x = sin(θ), y = cos(θ), 0 ≤ θ ≤ π



4. -Question Details SCalcET8 10.1.509.XP. [3945533]

Consider the parametric equations below.

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.

(b) Eliminate the parameter to find a Cartesian equation of the curve.

for

x = t2 − 5, y = 2 − 5t, −3 ≤ t ≤ 4

−18 ≤ y ≤ 17

5. -Question Details SCalcET8 10.1.046. [3945485]

If a projectile is fired with an initial velocity of v0 meters per second at an angle α above the horizontal and air resistance is assumed to be negligible, then its

position after t seconds is given by the parametric equations

where g is the acceleration due to gravity (9.8 m/s2). (Round your answers to the nearest whole number.)

(a) If a gun is fired with α = 30° and v0 = 600 m/s. When will the bullet hit the ground?

s How far from the gun will it hit the ground?

m What is the maximum height reached by the bullet?

m (b) Find the equation of the parabolic path by eliminating the parameter.

x = (v0 cos(α))t y = (v0 sin(α))t − gt212




If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as theparameter. (Enter your answer as a comma-separated list of equations. Let x and y be in terms of θ.)

Eliminate the parameter.

Identify the curve.

ellipse

circle

parabola

hyperbola




Match the graphs with their parametric equations.

I

II

III

IV

V VI

---Select---

---Select---

---Select---

---Select---

---Select---

(a) x = t4 − t + 2, y = t2

(b) x = t2 − 3t, y = t

(c) x = sin(2t), y = sin(t + sin(2t))

(d) x = cos(5t), y = sin(2t)

(e) x = t + sin(4t), y = t2 + cos(3t)

(f) sin(2t) cos(2t)



---Select---

(f) x = , y = s ( t)4 + t2

cos( t)4 + t2


Find dy/dx.

x = , y = t9 + t

9 + t

= dydx


Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

x = t3 + 1, y = t4 + t; t = −1

y =


Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

x = t cos(t), y = t sin(t); t = π

y =


Find and

For which values of t is the curve concave upward? (Enter your answer using interval notation.)

dy/dx d2y/dx2.

x = t2 + 3, y = t2 + 3t

=

=

dydx

d2ydx2


Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. (Enter your answers as acomma-separated list of ordered pairs.)

x = t3 − 3t, y = t2 − 5

horizontal tangent (x, y) =

vertical tangent (x, y) =




Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.

(a)

(b)

(2, 3π/2)

(x, y) =

(2 , π/4)2



(c)

(x, y) =

(−9, −π/6)

(x, y) =




The Cartesian coordinates of a point are given.

(a)

(i) Find polar coordinates of the point, where and

(ii) Find polar coordinates of the point, where and

(b)

(i) Find polar coordinates of the point, where and

(ii) Find polar coordinates of the point, where and

(−6, 6)

(r, θ) r > 0 0 ≤ θ < 2π.

(r, θ) =

(r, θ) r < 0 0 ≤ θ < 2π.

(r, θ) =

(3, 3 )3

(r, θ) r > 0 0 ≤ θ < 2π.

(r, θ) =

(r, θ) r < 0 0 ≤ θ < 2π.

(r, θ) =


Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

1 < r < 5, 3π/2 ≤ θ ≤ 5π/2




Find a Cartesian equation for the curve and identify it.

r2 = 11

hyperbola

parabola

circle

ellipse

limaÃ§on


Find a Cartesian equation for the curve and identify it.

r = 9 cos(θ)

ellipse

circle

parabola

hyperbola

limaçon


For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.

(a) A circle with radius 5 and center (1, 1).

(b) A circle centered at the origin with radius 1.




Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.

r = θ, θ ≥ 0




Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.

r = 3 cos(4θ)

21. -Question Details SCalcET8 10.4.AE.001. [3798865]

Video Example

EXAMPLE 1 Find the area enclosed by one loop of the four-leafed rose SOLUTION The curve is sketched in the figure to the left. Notice from the figurethat the region enclosed by the right loop is swept out by a ray that rotates from to

Therefore this formula gives

r = 5 cos(2θ).

r = 5 cos(2θ)θ = −π/4

θ = π/4.

A =

=

= 25

= 25

=

= .

r2 dθπ/4 1

2−π/4

12

dθπ/4

−π/4

dθπ/4

0

(1 + cos(4θ)) dθπ/4 1

20

252

π/4

0

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Name (AID): Chap 10 HW -- Parametric and Polar (11708646)Submissions Allowed: 15Category: HomeworkCode:Locked: YesAuthor: Bird, Brian ( [email protected] )Last Saved: May 31, 2019 07:32 AM MSTPermission: ProtectedRandomization: PersonWhich graded: Last

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22. -Question Details SCalcET8 10.4.JIT.001. [3799146]

This exercise involves the formula for the area of a circular sector. Find the area A of the sector shown in each figure.

(a)

A =

(b)

A =


Find the area of the region enclosed by one loop of the curve.

r = sin(2θ)


Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval.

One loop of the curve r = cos(2θ)

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