Problems

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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02

Parametric concavity

Find at for the curve given parametrically by , .

Solution

03

Derivative functions:

Slope:

Second derivative:

At :

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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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04

Parametric arclength

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

Solution

07

Derivatives:

Arclength:

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05

Minimum speed of a particle

Suppose a travelling particle has position modelled by the parametric curve:

What is the slowest speed of the particle?

Solution

08

Derivatives:

Speed function:

Now we minimize this function as in Calc I.

Method 1:

Differentiate:

This equals zero if-and-only-if the numerator equals zero (assuming the denominator is not zero there):

Since is negative for and positive for , we may deduce that is the time of the minimal value of . So:

Method 2:

Instead of differentiating , we can look at its square , since the minimum of this will occur at the same time as the minimum of (because is a monotone increasing function). But becomes , and the rest of the solution proceeds as in Method 1.

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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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