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W13 - HW Solutions

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03 - Parametric concavity

Find at for the curve given parametrically by , .

Solution

Compute derivatives.

Compute .

Compute .

Evaluate at .

06 - Parametric concavity

Find the intervals of on which the parametric curve is concave up.

Solution

Compute derivatives.

Compute .

Compute .

Examine when second derivative is greater than 0.

The above expression is greater zero only when . So, the interval is .

07 - Parametric arclength

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

Solution

Compute derivatives.

Compute length.

08 - Minimum speed of a particle

Suppose a travelling particle has position modelled by the parametric curve for . What is the slowest speed that the particle experiences?

Solution

Compute derivatives.

Compute the speed.

Find critical points of the square of the speed function.

Evaluate for each of the critical points.

The minimum of these speeds is units per time unit.

09 - Cycloid - length 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

(a)

Compute derivatives.

Compute length.

(b)

Compute surface area.