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Chapter 6: Problem 79

Sketch the graph of \(g(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) and determine \(g^{\prime}(0)\).

Short Answer

Expert verified
The sketched graph of the function would display a horizontal asymptote at y=0, with the graph lying above the x-axis but never touching or crossing it. The derivative of g at x=0 is 0.

Step by step solution

01

Analyze the function

Understand the nature of the given function. The function takes two distinct forms: \(e^{-1 / x^{2}}\) for x 鈮� 0 and 0 for x = 0.
02

Sketch the function.

For x 鈮� 0, the function \(e^{-1 / x^{2}}\) will always give positive values close to zero since the exponent is a negative reciprocal. As x gets close to 0 from both sides, the output of the function tends towards 0, which agrees with the definition of the function at x = 0. So, sketch a graph that displays this behavior, with a horizontal asymptote at y=0.
03

Find the derivative of \(e^{-1 / x^{2}}\)

Firstly, let's recognize that we are dealing with the function of a function here (Chain rule applies). So, differentiate \(e^{-1 / x^{2}}\) with respect to x, by applying the chain rule. The chain rule dictates that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. Therefore the derivative of \(e^{-1 / x^{2}}\) is \(-2e^{-1 / x^{2}}/x^{3} = -2g(x)/x^{3}\) for \(x \neq 0\). Note: this result is not valid for x=0.
04

Determine the derivative at x=0

To derive the function at x=0, we need to verify if the limit \( \lim_{{x \to 0}} {g^{'}(x)}\) exists. The function g(x) has the same value, 0, when approached from both sides of x=0, therefore the limit exists. This implies that g'(0) = 0.

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Most popular questions from this chapter

Determine all values of \(p\) for which the improper integral converges. $$ \int_{0}^{1} \frac{1}{x^{p}} d x $$

(a) Use a graphing utility to graph the function \(y=e^{-x^{2}}\). (b) Show that \(\int_{0}^{\infty} e^{-x^{2}} d x=\int_{0}^{1} \sqrt{-\ln y} d y\).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\cos a t $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) converges

Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. (c) Use a graphing utility to verify the result of part (b). FOR FURTHER INFORMATION For a geometric approach to this exercise, see the article "A Geometric Proof of \(\lim _{l \rightarrow 0^{+}}(-d \ln d)=0\) " by John H. Mathews in the College Mathematics Journal. To view this article, go to the website www.matharticles.com.
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