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Chapter 6: Problem 91

Are the statements is true or false? Give an explanation for your answer. An antiderivative of \(e^{-x^{2}}\) is \(-e^{-x^{2}} / 2 x\)

Short Answer

Expert verified
False; the derivative of \(-e^{-x^{2}} / 2x\) is not \(e^{-x^{2}}\).

Step by step solution

01

Understand the Statement

The statement is asking whether \(-e^{-x^{2}} / 2x\) is an antiderivative of \(e^{-x^{2}}\). This means we need to check if the derivative of \(-e^{-x^{2}} / 2x\) is \(e^{-x^{2}}\).
02

Differentiate the Given Function

To determine if \(-e^{-x^{2}} / 2x\) is an antiderivative, calculate its derivative. Use the quotient rule: If \(f(x) = \frac{u(x)}{v(x)}\), then \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \].In this case, \(u(x) = -e^{-x^{2}}\) and \(v(x) = 2x\).
03

Apply the Quotient Rule

Find \(u'(x)\) and \(v'(x)\):- \(u'(x) = \frac{d}{dx}[-e^{-x^{2}}] = 2xe^{-x^{2}}\) (using the chain rule),- \(v'(x) = \frac{d}{dx}[2x] = 2\).Substitute these into the quotient rule formula: \[ f'(x) = \frac{2xe^{-x^{2}}(2x) - (-e^{-x^{2}})(2)}{(2x)^2} = \frac{4x^2e^{-x^{2}} + 2e^{-x^{2}}}{4x^2}\].
04

Simplify the Expression

Simplify the expression for \(f'(x)\):\[ f'(x) = \frac{4x^2e^{-x^{2}} + 2e^{-x^{2}}}{4x^2} = \frac{2e^{-x^{2}}(2x^2 + 1)}{4x^2} = \frac{e^{-x^{2}} (2x^2 + 1)}{2x^2}\].The result is not \(e^{-x^{2}}\), which would be necessary for the original statement to be true.
05

Conclusion

The derivative of \(-e^{-x^{2}} / 2x\) is not \(e^{-x^{2}}\). Therefore, the statement that \(-e^{-x^{2}} / 2x\) is an antiderivative of \(e^{-x^{2}}\) is false.

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Key Concepts

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

Understanding the Chain Rule
The chain rule is a fundamental concept in calculus that is used to find the derivative of composite functions. A composite function is one where a function is applied inside another function, like \(e^{-x^2}\). To use the chain rule effectively, you identify the outer function and the inner function.

In the expression \(e^{-x^2}\), the outer function is the exponential function \(e^u\), where \(u = -x^2\), and the inner function is \(-x^2\). The chain rule states that the derivative of a composite function \(f(g(x))\) is given by the formula \[f'(g(x)) \, \cdot \, g'(x)\].
  • Determine \(f(u)\). Here, \(f(u) = e^u\). The derivative is \(f'(u) = e^u\).
  • Determine \(g(x)\). For this, \(g(x) = -x^2\). Its derivative is \(g'(x) = -2x\).
  • Apply the chain rule: the derivative of \(e^{-x^2}\) is \((-2x) \, \cdot \, e^{-x^2}\).
Using the chain rule allows you to handle derivatives of more complicated functions more efficiently.
Utilizing the Quotient Rule
The quotient rule is another essential tool in differentiation, especially useful when dealing with division of two functions. If you have a function \(f(x) = \frac{u(x)}{v(x)}\), the derivative is given by \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \].

When considering the problem's function \(-\frac{e^{-x^2}}{2x}\), employ the quotient rule.
  • Identify \(u(x) = -e^{-x^2}\) and \(v(x) = 2x\).
  • Find the derivatives \(u'(x) = 2xe^{-x^2}\) (using the chain rule) and \(v'(x) = 2\).
  • Substitute into the quotient rule formula: \[\frac{2xe^{-x^2} \cdot 2x + e^{-x^2} \cdot 2}{4x^2} = \frac{4x^2e^{-x^2} + 2e^{-x^2}}{4x^2} \].
Understanding how to apply the quotient rule ensures you correctly find the derivatives of functions represented as fractions.
The Process of Differentiation
Differentiation is the method of finding how a function changes as its input changes. It is a foundational concept in calculus that deals with calculating the derivative of a function. A derivative represents the rate of change or slope of the function at any given point.

In calculus, when you differentiate a function, you determine its behavior and how it reacts to changes. This is particularly significant in finding how a composite or a quotient function behaves, which requires techniques like the chain rule or quotient rule.
  • The derivative expresses how function outputs vary with small changes in inputs, giving insights into rates of change.
  • It is vital in solving practical problems involving motion, growth rates, and optimizations.
  • In examining if a function \(-\frac{e^{-x^2}}{2x}\) is an antiderivative, you calculate its derivative to determine if it matches \(e^{-x^2}\).
Differentiation is crucial in verifying if given functions are antiderivatives of other functions by confirming through their derivatives.

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Most popular questions from this chapter

Find the given quantities. The error filnc. tion, \(\operatorname{erf}(x),\) is defined by $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$. $$\frac{d}{d x}\left(\int_{x}^{x^{3}} e^{-t^{2}} d t\right)$$

Newton's law of gravity says that the gravitational force between two bodies is attractive and given by $$F=\frac{G M m}{r^{2}}$$, where \(G\) is the gravitational constant, \(m\) and \(M\) are the masses of the two bodies, and \(r\) is the distance between them. This is the famous inverse square law. For a falling body, we take \(M\) to be the mass of the earth and \(r\) to be the distance from the body to the center of the earth. So, actually, \(r\) changes as the body falls, but for anything we can easily observe (say, a ball dropped from the Tower of Pisa), it won't change significantly over the course of the motion. Hence, as an approximation, it is reasonable to assume that the force is constant. According to Newton's second law, acceleration is caused by a force and Force \(=\) Mass \(\times\) Acceleration. (a) Find the differential equation for the position, \(s\) of a moving body as a function of time. (b) Explain how the differential equation shows the acceleration of the body is independent of its mass.

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{3}\left(x^{2}+4 x+3\right) d x$$

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{1} 2 e^{x} d x$$

Explain what is wrong with the statement. Two solutions to a differential equation \(d y / d x=f(x)\) have graphs which cross at the initial value.
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