/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 127 Find \(\frac{d^{2} y}{d x^{2}}\)... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 127

Find \(\frac{d^{2} y}{d x^{2}}\), if (i) \(x=a t^{2}, y=2 a t\) (ii) \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\) (iii) \(x=a \cos \theta, y=b \sin \theta\)

Short Answer

Expert verified
(i) Second derivative, \(d^2y/dx^2 = -1/(at^2)\), (ii) Second derivative, \(d^2y/dx^2 = 1/(a cos胃 sin胃)\), (iii) Second derivative, \(d^2y/dx^2 = b sec^2胃 / (a虏 cos胃)\).

Step by step solution

01

Differentiate y and x with respect to 't' or '胃'

(i) For \(x=a t^{2}, y=2 a t\): Differentiate given y -> \(dy/dt = 2a\) and x -> \(dx/dt = 2at\). (ii) For \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\): Differentiate given y -> \(dy/d胃 = 3a sin^2胃 cos胃\) and x -> \(dx/d胃 = -3a cos^2胃 sin胃\). (iii) For \(x=a \cos \theta, y=b \sin \theta\): Differentiate given y -> \(dy/d胃 = b cos胃\) and x -> \(dx/d胃 = -a sin胃\).
02

Find dy/dx

Using chain rule, dy/dx = dy/d't' or dy/d胃 / dx/d't' or dx/d胃. (i) dy/dx = \(2a / 2at = 1/t\). (ii) dy/dx = \(-3a sin^2胃 cos胃 / -3a cos^2胃 sin胃 = tan胃\). (iii) dy/dx = \(b cos胃 / -a sin胃 = -b cot胃 / a\).
03

Compute second derivative d^2y/dx^2

To find d^2y/dx^2, we differentiate dy/dx with respect to 'x'. The important part to remember while differentiating is to keep 'x' constant. (i) \(d^2y/dx^2 = d/dx(1/t) = -1/(at^2)\). (ii) \(d^2y/dx^2 = d/dx(tan胃) = sec^2胃 / (3a cos^2胃 sin胃) = 1/(a cos胃 sin胃)\). (iii) \(d^2y/dx^2 = d/dx(-b cot胃/a) = b sec^2胃 / (a虏 cos胃)\).

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Most popular questions from this chapter

If \(x=a \cos \theta, y=b \sin \theta\), find \(\frac{d^{2} y}{d x^{2}}\)

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