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Calculus 2 : Parametric, Polar, and Vector

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Example Questions

Example Question #11 : Graphing Polar Form

Describe the graph of $r=9-6cos\theta$ from $0\leq\theta\leq2\pi$ .

Possible Answers:

A limacon without a loop rotated $90^{\circ}$ left

A limacon without a loop rotated $90^{\circ}$ right

An upright limacon without a loop

An upside down limacon without a loop

Correct answer:

A limacon without a loop rotated $90^{\circ}$ right

Explanation:

Graphing polar equations is different that plotting cartesian equations. Instead of plotting an (x,y) coordinate, polar graphs consist of an $(r,\theta)$ coordinate where is the radial distance of a point from the origin and $\theta$ is the angle above the x-axis.

From our equation, we know the shape of our graph will be a limacon because our equation is in the form $r=a\pm bcos\theta$ where a>b . This limacon will have no loop because a/b>1 . Our limacon is symmetric about the x-axis because our equation includes the cosine function. The y-intercepts are at (0,a) and (0,-a) . The x-intercept is at -(a+b,0) .

We could also substitute values of $\theta$ (in radians) between and $2\pi$ into our expression, to find values of r. We then plot each ordered pair, $(r,\theta)$ , using the value as the radius and $\theta$ as the angle.

We get the graph below, an limacon turned $90^{\circ}$ right.

Fig10

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Example Question #19 : Graphing Polar Form

Describe the graph of $r=3+6cos\theta$ from $0\leq\theta\leq2\pi$ .

Possible Answers:

a limacon with a loop turned $90^{\circ}$ left

a limacon with a loop turned $90^{\circ}$ right

an upright limacon with a loop

an upside-down limacon with a loop

Correct answer:

a limacon with a loop turned $90^{\circ}$ left

Explanation:

From our equation, we know the shape of our graph will be a limacon because our equation is in the form $r=a\pm bcos\theta$ where a<b . This limacon will have a loop because a/b<1 . The length of the loop is b-a . Our limacon is symmetric about the x-axis because our equation includes the cosine function. The y-intercepts are at (0,a) and (0,-a) . The x-intercept is at (a+b,0) .

We get the graph below, a limacon with a loop turned $90^{\circ}$ left.

Fig11

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Example Question #20 : Graphing Polar Form

Describe the graph of $r=4-7sin\theta$ from $0\leq\theta\leq2\pi$ .

Possible Answers:

a limacon with a loop turned $90^{\circ}$ right

an upside-down limacon with a loop

a limacon with a loop turned $90^{\circ}$ left

an upright limacon with a loop

Correct answer:

an upright limacon with a loop

Explanation:

From our equation, we know the shape of our graph will be a limacon because our equation is in the form $r=a\pm bsin\theta$ where a<b . This limacon will have a loop because a/b<1 . The length of the loop is b-a . Our limacon is symmetric about the y-axis because our equation includes the sine function. The x-intercepts are at (a,0) and (-a,0) . The y-intercept is at -(0,a+b) .

We get the graph below, an upright limacon with a loop.

Fig12

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Example Question #281 : Parametric, Polar, And Vector

In which quadrant does the polar coordinate terminate?

(3,70^o)

Possible Answers:

III

Correct answer:

Explanation:

The coordinate (3,70^o) goes to the right units from the origin and is rotated 70^o counter-clockwise, terminating in

$Quadrant \,I$

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Example Question #282 : Parametric, Polar, And Vector

In which quadrant does the polar coordinate terminate?

(4,460^o)

Possible Answers:

III

Correct answer:

Explanation:

The coordinate (4,460^o) goes to the right units from the origin and is rotated 460^o counter-clockwise, terminating in

$Quadrant \,II$

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Example Question #283 : Parametric, Polar, And Vector

In which quadrant does the polar coordinate terminate?

(-2,45^o)

Possible Answers:

III

Correct answer:

III

Explanation:

The coordinate (-2,45^o) goes to the left units from the origin and is rotated 45^o counter-clockwise, terminating in

$Quadrant \,III$

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Example Question #284 : Parametric, Polar, And Vector

In which quadrant is the polar coordinate located?

(-3, 30^o)

Possible Answers:

III

Correct answer:

III

Explanation:

The polar coordinate

(-3, 30^o)

is graphed by moving units to the left of the origin and rotating 30^o counter-clockwise, resulting in

$Quadrant\,III$

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Example Question #285 : Parametric, Polar, And Vector

In which quadrant is the polar coordinate located?

(2, 45^o)

Possible Answers:

III

Correct answer:

Explanation:

The polar coordinate

(2, 45^o)

is graphed by moving units to the right of the origin and rotating 45^o counter-clockwise, resulting in

$Quadrant\,I$

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Example Question #286 : Parametric, Polar, And Vector

In which quadrant is the polar coordinate located?

(4, 100^o)

Possible Answers:

III

Correct answer:

Explanation:

The polar coordinate

(4, 100^o)

is graphed by moving units to the right of the origin and rotating 100^o counter-clockwise, resulting in

$Quadrant\,II$

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Example Question #287 : Parametric, Polar, And Vector

In which quadrant is the polar coordinate located?

(1,315^o)

Possible Answers:

III

Correct answer:

Explanation:

The polar coordinate

(1, 315^o)

is graphed by moving unit to the right of the origin and rotating 315^o counter-clockwise, resulting in

$Quadrant\,IV$

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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #11 : Graphing Polar Form

Example Question #19 : Graphing Polar Form

Example Question #20 : Graphing Polar Form

Example Question #281 : Parametric, Polar, And Vector

Example Question #282 : Parametric, Polar, And Vector

Example Question #283 : Parametric, Polar, And Vector

Example Question #284 : Parametric, Polar, And Vector

Example Question #285 : Parametric, Polar, And Vector

Example Question #286 : Parametric, Polar, And Vector

Example Question #287 : Parametric, Polar, And Vector

All Calculus 2 Resources

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