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Chapter 12: Problem 3

Calculate \(\frac{d^{2} y}{d x^{2}}\). \(y=\frac{2}{x}\)

Short Answer

Expert verified
The second derivative of the function \(y = \frac{2}{x}\) is \(\frac{d^2y}{dx^2} = \frac{4}{x^3}\).

Step by step solution

01

Write down the given function

The given function is: \(y = \frac{2}{x}\)
02

Calculate the first derivative

To find the first derivative, we will use the power rule. We can first rewrite the function as: \(y = 2x^{-1}\) Now, we use the power rule to calculate the first derivative: \(\frac{dy}{dx} = -1 \times 2x^{-2} = -2x^{-2}\) So, the first derivative is: \(\frac{dy}{dx} = -\frac{2}{x^2}\)
03

Calculate the second derivative

Now that we have the first derivative, we can find the second derivative. We start by rewriting the first derivative as: \(\frac{dy}{dx} = -2x^{-2}\) Now, we use the power rule again to calculate the second derivative: \(\frac{d^2y}{dx^2} = -2 \times (-2)x^{-3} = 4x^{-3}\) So, the second derivative is: \(\frac{d^2y}{dx^2} = \frac{4}{x^3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a fundamental concept in calculus differentiation. It is especially useful when differentiating polynomial functions or expressions where a variable is raised to a power. The general form of the power rule states:
  • If you have a function in the form of \( f(x) = ax^n \), where \( a \) is a constant and \( n \) is a real number, the derivative \( f'(x) \) is obtained by multiplying the power \( n \) by the coefficient \( a \), and then reducing the power by one.
The derivative is thus given by: \[f'(x) = n imes a x^{n-1}\]When applied to functions like \( y = 2x^{-1} \), as seen in our exercise, the power rule allows us to easily differentiate and find the first derivative. This rule simplifies the process and can be applied repeatedly to compute higher-order derivatives, assisting in understanding changes in slope and curvature.
First Derivative
The first derivative of a function represents the rate of change or the slope of the original function. It is a measure of how the function's output values vary with changes in input values. For our given function \( y = \frac{2}{x} \), we first rewrote it in terms of a power of \( x \) to apply the power rule without confusion:
  • We transformed it into \( y = 2x^{-1} \).
This transformation allows direct application of the power rule:
  • By calculating \( \frac{dy}{dx} = -1 \times 2x^{-2} \), we got \( -2x^{-2} \).
  • This simplifies to \( \frac{dy}{dx} = -\frac{2}{x^2} \).
Graphically, the first derivative shows us how steep or flat the graph of the function \( y \) is at any given point \( x \). Understanding this is key in solving problems involving rates and optimizing functions in real-world contexts.
Second Derivative
The second derivative provides information about the concavity and curvature of the original function. It indicates how the slope given by the first derivative changes. From the first derivative \( \frac{dy}{dx} = -\frac{2}{x^2} \), one can find the second derivative by applying the power rule once more:
  • Rewriting \( \frac{dy}{dx} \) as \( -2x^{-2} \), we differentiate again.
  • Applying the power rule yields \( \frac{d^2y}{dx^2} = 4x^{-3} \).
  • This means the second derivative is \( \frac{4}{x^3} \).
Understanding the second derivative is essential for analyzing the behavior of functions with respect to their concave or convex nature; it's a core tool in fields such as dynamics and optimization. Analyzing this curvature helps assess when a function is at a maximum or minimum point in its graph, a crucial element in practical calculus applications.

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Most popular questions from this chapter

Sketch the graph of the given function, indicating (a) \(x\) - and \(y\) -intercepts, (b) extrema, (c) points of inflection, \((d)\) behavior near points where the function is not defined, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. \(f(x)=-x^{2}-2 x-1\)

You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best selling price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form \(q=m p+b\), where \(p\) is the price per book, \(q\) is the demand in annual sales, and \(m\) and \(b\) are certain constants you'll have to figure out. a. Your market studies reveal the following sales figures: when the price is set at \(\$ 50.00\) per book, the sales amount to 10,000 per year; when the price is set at \(\$ 80.00\) per book, the sales drop to 1000 per year. Use these data to calculate the demand equation. b. Now estimate the unit price that maximizes annual revenue and predict what Big Book Publishing, Inc.'s annual revenue will be at that price.

A right circular conical vessel is being filled with green industrial waste at a rate of 100 cubic meters per second. How fast is the level rising after \(200 \pi\) cubic meters have been poured in? The cone has a height of \(50 \mathrm{~m}\) and a radius of \(30 \mathrm{~m}\) at its brim. (The volume of a cone of height \(h\) and crosssectional radius \(r\) at its brim is given by \(V=\frac{1}{3} \pi r^{2} h .\).)

A time- series study of the demand for higher education, using tuition charges as a price variable, yields the following result: $$\frac{d q}{d p} \cdot \frac{p}{q}=-0.4$$ where \(p\) is tuition and \(q\) is the quantity of higher education. \(\mathbf{2 5}\). Which of the following is suggested by the result? (A) As tuition rises, students want to buy a greater quantity \(\quad \mathbf{2 6}\). of education. (B) As a determinant of the demand for higher education, income is more important than price. (C) If colleges lowered tuition slightly, their total tuition receipts would increase. (D) If colleges raised tuition slightly, their total tuition receipts would increase. (E) Colleges cannot increase enrollments by offering larger scholarships.

Daily sales of Kent's Tents reached a maximum in January 2002 and declined to a minimum in January 2003 before starting to climb again. The graph of daily sales shows a point of inflection at June 2002 . What is the significance of the point of inflection?
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