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

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. ${\frac {dy}{dx}}=1-6t^{2},{\frac {d^{2}y}{dx^{2}}}=-6$.
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. $x=2t,y=2-t^{2}$.
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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