Unit 04 - Essential problems

Parametric curves

01

Convert parametric curve to function graph

Write the following curves as the graphs of a function . (Find for each case.)

(a) , and

(b) , and

Sketch each curve.

Solution

01

(a) From the first equation, .

Plug that in for in :

The sketched curve should be the portion of the line with .

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(b)

For this one, best not to solve for . Instead, notice the trig identity:

Therefore the points on the curve satisfy the equation . Solve this for the function:

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03

Convert function graph to parametric curve

Find parametric curves whose images are the following graphs:

(a) and

(b) and

Solution

05

(a)

First choose a function , then set to ensure the equation is satisfied.

When choosing , we want to cover the whole domain of which is . We also need to satisfy the initial condition.

Start by trying :

But then . Since we should have . We can arrange for this by setting and solving for :

Therefore we define . Then:

So we use:

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(b)

Same method but different condition:

Therefore we define . Then:

So we use:

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03

Parametric concavity

Find the interval(s) of on which the parametric curve is concave up.

Solution

06

First derivative:

Second derivative:

This is positive if-and-only-if . (Numerator always positive, denominator same sign as .) So:

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06

Cycloid - Arclength and surface area of revolution

Consider the cycloid given parametrically by .

(a) Find the length of one arch of the cycloid.

(b) Suppose one arch of the cycloid is revolved around the -axis. Find the area of this surface of revolution.

Solution

09

(a)

One arch is formed from the range .

Compute :

Therefore .

Now recall a power-to-frequency formula, and use it in reverse:

Therefore:

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(b)

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Polar curves

01

Convert points: Cartesian to Polar

Convert the Cartesian (rectangular) coordinates for these points into polar coordinates:

(a) (b) (c) (d)

Solution

01

(a)

Q1, SAFE, .

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(b)

Q1, SAFE, .

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(c)

Q2, UNSAFE, and we add to this angle.

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(d)

Q2, UNSAFE, (use 30-60-90 triangle) and we add to this angle.

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01

Polar curve - Vertical or horizontal tangent lines

Find all points on the given curve where the tangent line is horizontal or vertical.

Hint: First determine parametric Cartesian coordinate functions using as the parameter.

Solution

03

Observe that this parametric curve is a circle centered at with radius . So we expect vertical tangents at and horizontal tangents at .

Treat as the parameter. We always have . This equals here because . Since we can further simplify to .

Then .

Also .

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To find vertical tangents, solve for :

Check that is not also zero at these points, else they would be stationary points:

Now find the Cartesian coordinates for these points:

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To find the horizontal tangents, solve for :

Check that is not also zero at these points, else they would be stationary points:

Now find the Cartesian coordinates for these points:

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04

Convert equations: Cartesian to Polar

Convert the Cartesian equation to a polar equation. Be sure to simplify.

(a) (b) (c)

Solution

05

(a) (b) (c)

(a)

Insert and :

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(b)

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(c)

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04

Polar coordinates - lunar areas

(a) Find the area of the green region.

(b) Find the area of the yellow region.

(You can find these in either order.)

Solution

04

(a) Find the angle of the line from the origin to the point of intersection of the two curves (in Quadrant I):

Compute the area below this line, inside the larger circle, and above the -axis:

(This circular sector is also just of the whole disk area, which is .)

Compute the area above the line and inside the smaller circle:

Combined area in green above the -axis is . Double this for the total green area:

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(b) Notice that green and yellow combine to give the area of the smaller circle. The area of the smaller circle is .

Therefore, the yellow region has area:

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Note: It is also reasonable to find the yellow region first, using this formula:

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06

Area of an inner loop

A limaçon is given as the graph of the polar curve .

Find the area of the inner loop of this limaçon.

Solution

07

Solve for consecutive (in ) solutions to to get the starting and ending for a single loop:

The interval corresponds to the inner loop. To see this, draw a graph of the limaçon:

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Complex numbers

01

Complex forms - exponential to Cartesian

Write each number in the form .

(a) (b)

Solution

01

(a) Use Euler’s Formula:

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(b)

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04

Complex products and quotients using polar

For each pair of complex numbers and , compute:

(a)

(b)

(Use polar forms with .)

Solution

05

(a)

Product:

Dividend:

Reciprocal:

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(b)

Product:

Dividend:

Reciprocal:

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05

Complex powers using polar

Using De Moivre’s Theorem, write each number in the form .

(a) (b)

(First convert to polar/exponential, then compute the power, then convert back.)

Solution

06

(a)

Convert to polar:

De Moivre’s Theorem:

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(b)

Convert to polar:

De Moivre’s Theorem:

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02

Complex roots using polar

Find each of the indicated roots.

(a) The four roots of .

(b) The three cube () roots of .

Try to write your answer in form if that is not hard, otherwise leave it in polar form.

Solution

07

(a)

For roots, always start by writing in exponential:

Now use the roots formula:

Write out these roots by evaluating at :

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(b)

For roots, always start by writing in exponential:

Now use the roots formula:

Write out these roots by evaluating at :

(The polar answers are acceptable too.)

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