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

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Chapter 5: 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 derivative $\frac{d^{2} y}{d x^{2}}$ for (i) is $- \frac{1}{t^{2}}$, for (ii) is $-sec^{2}\theta$ and for (iii) is $ \frac{b}{a} csc^{2}\theta$

Step by step solution

Calculating First Derivative For Part (i)

We start with the given equations $x=a t^{2}$ and $y=2 a t$. The first derivative is found by differentiating y with respect to t and dividing by the derivative of x with respect to t. This gives us $ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} $. Here $\frac{dy}{dt} = 2a$ and $\frac{dx}{dt} = 2at$, which gives $ \frac{dy}{dx} = \frac{1}{t}$.

Calculating Second Derivative For Part (i)

Second derivative is the derivative of the first derivative with respect to x. We can write it as $ \frac{d^{2} y}{d x^{2}} = \frac{d}{dx}(\frac{dy}{dx}) = - \frac{1}{t^{2}}$.

Calculating First Derivative For Part (ii)

We start with the given equations $x=a \cos ^{3} \theta$ and $y=a \sin ^{3} \theta$. Now, we differentiate y with respect to $ \theta $ and x with respect to $ \theta $. The first derivative is $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$. Here $\frac{dy}{d\theta} = 3a\sin^{2}\theta cos\theta$, $\frac{dx}{d\theta} = -3a\cos^{2}\theta sin\theta$, when simplified gives $ \frac{dy}{dx} = - tan\theta$.

Calculating Second Derivative For Part (ii)

The second derivative would be $ \frac{d^{2} y}{d x^{2}} = \frac{d}{dx}(\frac{dy}{dx}) = -sec^{2}\theta$.

Calculating First Derivative For Part (iii)

We start with the given equations $x=a \cos \theta$ and $y=b \sin \theta$. Now, we differentiate y with respect to $ \theta $ and x with respect to $ \theta $. The first derivative is $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$. Here $\frac{dy}{d\theta} = b cos\theta$, $\frac{dx}{d\theta} = -a sin\theta$, when simplified gives $ \frac{dy}{dx} = -\frac{b}{a} cot\theta$.

Calculating Second Derivative For Part (iii)

The second derivative would be $ \frac{d^{2} y}{d x^{2}} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{b}{a} csc^{2}\theta$.

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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

Calculating First Derivative For Part (i)

Calculating Second Derivative For Part (i)

Calculating First Derivative For Part (ii)

Calculating Second Derivative For Part (ii)

Calculating First Derivative For Part (iii)

Calculating Second Derivative For Part (iii)

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