W13 - Homework

means: “eligible for Friday Presentation”

Stepwise problems - Mon. Nov 17, 11:59pm

Parametric curves

01

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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Calculus with parametric curves

02

01

Parametric curves: Points with given slope

Where on the image of does the tangent line have slope ?

Solution

02

Find a formula for the slope of the tangent line:

Solve for the where :

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03

02

Parametric concavity

Find at for the curve given parametrically by , .

Solution

03

Derivative functions:

Slope:

Second derivative:

At :

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Regular problems - Thu. Nov 20, 11:59pm

Parametric curves

04

02

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

04

(a)

Observe that and implies .

Therefore, all points on the curve satisfy and we set .

Since and covers the entire real line , the parametric curve is the entire line .

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(b)

Observe that and implies . Again, all points on the curve satisfy and so .

However, this time implies , and the entire range of is possible (set ) to find an inverse.

So the image of this parametric curve is for , and the origin is omitted.

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05

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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Calculus with parametric curves

06

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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07

04

Parametric arclength

Find the arclength of the curve given parametrically by , over the time interval .

Solution

07

Derivatives:

Arclength:

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