Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/178

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In the following seven examples the equations are given in parametric form.

Find $\tfrac{dy}{dx}$ and $\tfrac{d^2 y}{dx^2}$ in each case:

16. $x = 7 + t^2, y = 3 + t^2 - 3t^4$.	Ans. $\frac{dy}{dx} = 1 - 6 t^2, \frac{d^2 y}{dx^2} = -6$.
17. $x = \cot t, y = \sin^3 t$.	Ans. $\frac{dy}{dx} = -3 \sin^4 t \cos t, \frac{d^2 y}{dx^2} = 3 \sin^5 t (4 - 5 \sin^2 t)$.
18. $x = a(\cos t + \sin t), y = a(\sin t - t \cos t)$.	Ans. $\frac{dy}{dx} = \tan t, \frac{d^2 y}{dx^2} = \frac{1}{at \cos^3 t}$.
19. $x = \frac{1 - t}{1 + t}, y = \frac{2t}{1 + t}$.
20. $x = 2 t, y = 2 - t^2$.
21. $x = 1 - t^2, y = t^3$.
22. $x = a \cos t, y = b \sin t$.
23. Transform $\frac{ x \frac{dy}{dx} - y}{ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} }$ by assuming $x = \rho \cos \theta, y = \rho \sin \theta$.
	Ans. $\frac{\rho^2}{ \sqrt{ \rho \left( \frac{d\rho}{d\theta} \right)^2 } }$.
24. Let $f(x, y) = 0$ be the equation of a curve. Find an expression for its slope $\left( \frac{dy}{dx} \right)$ in terms of polar coördinates.
	Ans. $\frac{dy}{dx} = \frac{ \rho \cos \theta + \sin \theta \frac{d\rho}{d\theta} }{ -\rho \sin \theta + \cos \theta \frac{d\rho}{d\theta} }$.

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