Problem 130 Find $\frac{d^{2} y}{d x^{2}}$... [FREE SOLUTION]

91影视

differential calculus booster with problems and solutions for iit jee main and advanced rejaul-maksh

Rejaul Makshud

$Math Studyset 91影视 Explanations$ Math

2019 Edition

Chapter 6: Problem 130

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

The second derivatives of y with respect to x for the three cases are -1/t^2 , - sec^2 \theta and 0 respectively.

Step by step solution

Case (i) - Find $dy/dt$ and $dx/dt$

For $x=a t^{2}, y=2 a t$, we find the first derivative of y and x with respect to t by applying direct differentiation. So, $dy/dt = 2a$ and $dx/dt = 2at$.

Case (i) - Find $dy/dx$

Now, we calculate $dy/dx$ by substituting $dy/dt$ and $dx/dt$ in the formula $\frac{dy/dx} = \frac {dy/dt}{dx/dt}$. After simplifying, we find $ dy/dx = 1/t$.

Case (i) - Find second derivative $d^2y / dx^2$

Similarly, we calculate $ (d^2y / dx^2) = d/dx(dy/dx) = d(t)/dx(1/t) = -1/t^2$. So, the second derivative is found to be -1/t^2.

Case (ii) - Find $dy/dt$ and $dx/dt$

For $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$, applying direct differentiation, we find $dy/dt = 3a \sin^2 \theta \cos \theta$ and $dx/dt = -3a \cos^2 \theta \sin \theta$.

Case (ii) - Find $dy/dx$

Substituting $dy/dt$ and $dx/dt$ in the formula $\frac{dy/dx} = \frac {dy/dt}{dx/dt}$, we find $ dy/dx = -\tan \theta$.

Case (ii) - Find second derivative $d^2y / dx^2$

Doing the same confirmation as before, we find $ (d^2y / dx^2) = d/dx(dy/dx) = d\theta/dx(-\tan \theta) = - sec^2 \theta$. So, the second derivative is found to be - sec^2 \theta.

Case (iii) - Find $dy/dt$ and $dx/dt$

Now for $x=a \cos \theta, y=b \sin \theta$, we find $dy/dt = b \cos \theta$ and $dx/dt = -a \sin \theta$.

Case (iii) - Find $dy/dx$

Substititing $dy/dt$ and $dx/dt$ in the formula $\frac{dy/dx} = \frac {dy/dt}{dx/dt}$, we find $ dy/dx = - \frac{b}{a} \tan \theta$.

Case (iii) - Find second derivative $d^2y / dx^2$

Calculating $ (d^2y / dx^2) = d/dx(dy/dx) = d\theta/dx(-b/a) \tan \theta = 0 $. So, the second derivative is zero.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Step by step solution

Case (i) - Find \(dy/dt\) and \(dx/dt\)

Case (i) - Find \(dy/dx\)

Case (i) - Find second derivative \(d^2y / dx^2\)

Case (ii) - Find \(dy/dt\) and \(dx/dt\)

Case (ii) - Find \(dy/dx\)

Case (ii) - Find second derivative \(d^2y / dx^2\)

Case (iii) - Find \(dy/dt\) and \(dx/dt\)

Case (iii) - Find \(dy/dx\)

Case (iii) - Find second derivative \(d^2y / dx^2\)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.