A skateboarder’s path along a half-pipe is modeled for (time in seconds) by x=2t- t

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A skateboarder’s path along a half-pipe is modeled for

0\le t\le2\pi

(time in seconds) by

x=2t-\sin t,\qquad y=1-\cos t\quad\text{(units in meters)}.

Use this model to analyze changes in the path’s concavity. (a) Compute

\dfrac{dy}{dx}

as a function of

t

. (b) Determine

\dfrac{d^{2}y}{dx^{2}}

in terms of

t

. (c) Find all values of

t

[0,2\pi]

at which the curvature of the path potentially changes. Provide exact values. (d) Verify the concavity on each subinterval of

t

and identify the coordinates of every inflection point within the given time domain. (e) Briefly interpret, in the context of the skateboarder’s motion, what an inflection point on this path signifies about the rider’s change in direction along the vertical dimension.

Related theory

Curve Analysis
Parametric Functions
Implicit and parametric differentiation
Derivatives of parametric functions
Continuity and Differentiability

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