Question

Suppose \(x\) and \(y\) are related by the equation \(\mathrm{x}=\int_{0}^{\mathrm{y}} \frac{1}{\sqrt{1+4 \mathrm{t}^{2}}} \mathrm{dt}\) Show that \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) is proportional to \(\mathrm{y}\) and find

Short answer

Question: Show that the second derivative of y with respect to x, \(\frac{d^2y}{dx^2}\), is proportional to y and find the constant of proportionality. Answer: The second derivative of y with respect to x, \(\frac{d^2y}{dx^2}\), is proportional to y with the constant of proportionality equal to \(k = \frac{4}{\sqrt{1+4y^2}}\).

Step-by-step solution

Find \(\frac{dy}{dx}\)

To find \(\frac{dy}{dx}\), we first need to differentiate the given equation with respect to y. Applying the Fundamental Theorem of Calculus to the integral, we get: \(\frac{dx}{dy} = \frac{1}{\sqrt{1+4y^2}}\) Now, we can find \(\frac{dy}{dx}\) by taking the reciprocal of \(\frac{dx}{dy}\): \(\frac{dy}{dx} = \sqrt{1 + 4y^2}\)

Find \(\frac{d^2y}{dx^2}\)

To find the second derivative, we differentiate \(\frac{dy}{dx}\) with respect to x. Since we have \(\frac{dy}{dx}\) in terms of y, we must use the chain rule: \(\frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dy}\left(\frac{dy}{dx}\right)\cdot\frac{dy}{dx}\) Differentiating \(\frac{dy}{dx} = \sqrt{1 + 4y^2}\) with respect to y, we get: \(\frac{d}{dy}\left(\frac{dy}{dx}\right) = \frac{4y}{\sqrt{(1+4y^2)^3}}\) Now, multiplying both sides by the chain rule, we get: \(\frac{d^2y}{dx^2} = \frac{4y}{\sqrt{(1+4y^2)^3}} \cdot \sqrt{1 + 4y^2}\)

Analyze the relationship between \(\frac{d^2y}{dx^2}\) and y

We want to show that \(\frac{d^2y}{dx^2}\) is proportional to y. This means that we can find a constant of proportionality, k, such that: \(\frac{d^2y}{dx^2} = ky\) Now let's substitute the expression for \(\frac{d^2y}{dx^2}\) we found earlier: \(\frac{4y}{\sqrt{(1+4y^2)^3}} \cdot \sqrt{1+4y^2} = ky\) Simplifying the left side of the equation: \(\frac{4y(1+4y^2)}{\sqrt{(1+4y^2)^3}} = ky\) We notice that the term \((1+4y^2)\) can be canceled: \(\frac{4y}{\sqrt{1+4y^2}} = ky\) Comparing both sides, we see that the constant of proportionality, k, is equal to: \(k = \frac{4}{\sqrt{1+4y^2}}\)

Write down the final answer

We have shown that \(\frac{d^2y}{dx^2}\) is proportional to y, and the constant of proportionality is: \(k = \frac{4}{\sqrt{1+4y^2}}\) Therefore, the second derivative of y with respect to x, \(\frac{d^2y}{dx^2}\), can be expressed as: \(\frac{d^2y}{dx^2} = \frac{4y}{\sqrt{1+4y^2}}\)

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with the accumulation of quantities and the areas under and between curves. It's instrumental in finding displacement, area, and volume from given functions or curves.

In the exercise, we're presented with the function \(x = \int_{0}^{y} \frac{1}{\sqrt{1+4t^{2}}} dt\) which describes a relationship between \(x\) and \(y\). The integral here represents the area under the curve \( \frac{1}{\sqrt{1+4t^{2}}} \) from \(t=0\) to \(t=y\). To extract meaningful information about the relationship between \(x\) and \(y\), such as rates of change, integral calculus is paired with its counterpart, differential calculus, making the Fundamental Theorem of Calculus a vital bridge between the two.

When solving such a problem, a good exercise improvement advice would be to consider the geometric interpretation of the integral and the resulting function to better understand their relationship.

Chain Rule

The chain rule is a principle in calculus used to find the derivative of a composition of functions. If one variable is dependent on another, which in turn is dependent on a third variable, the chain rule allows us to differentiate with respect to the third variable.

In our case, we have \(\frac{dy}{dx}\) expressed in terms of \(y\), and to find its derivative \(\frac{d^2y}{dx^2}\), we need to apply the chain rule. The chain rule states that \(\frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dy}\left(\frac{dy}{dx}\right)\cdot\frac{dy}{dx}\), effectively allowing us to differentiate a function of a function.

For students looking to enhance their grasp on this concept, practice applying the chain rule to various functions, especially where higher-order derivatives are involved, and always clarify the function relationships before starting.

Proportionality Constant

A proportionality constant is a fixed number that relates two variables that are directly proportional to each other. In other words, if one variable is always the product of the other and a constant, then the constant is the proportionality constant.

The final step in our exercise seeks to establish that \(\frac{d^2y}{dx^2}\) and \(y\) are directly proportional, implying that they can be related by a constant \(k\), or \(\frac{d^2y}{dx^2} = ky\). After some manipulation using integral and differential calculus, we find that \(k = \frac{4}{\sqrt{1+4y^2}}\). This \(k\) quantifies the precise relationship between the second derivative of \(y\) with respect to \(x\) and \(y\) itself.

One effective piece of advice for students is to pay careful attention to the algebraic manipulation when finding the proportionality constant, ensuring that all variables and their relationships are accounted for accurately. This will be an essential skill when solving similar problems where proportionality comes into play.