/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Sketch a graph of the given func... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 26

Sketch a graph of the given function. $$f(x)=10 e^{-x / 3}$$

Short Answer

Expert verified
The graph of the function \(f(x) = 10e^{-x / 3}\) starts at the y-intercept (0, 10) and decreases from left to right, getting closer and closer to the asymptote at \(y = 0\) without actually reaching it. The slope of the function is decreasing whenever \(x > 0\).

Step by step solution

01

Understand the Function

The function given is \(f(x) = 10e^{-x/3}\). This is an exponential decay function. As \(x\) goes from negative infinity to positive infinity, the function starts from infinity, hits a high point at \(x = 0\), then decays towards 0 but never actually reaches 0.
02

Find the y-intercept

The y-intercept of the function is found by evaluating the function at \(x = 0\). So \(f(0) = 10e^{-0/3} = 10\). So, the y-intercept is (0, 10)
03

Identify the asymptote

The horizontal asymptote of the function is \(y = 0\). This is because, as \(x\) approaches infinity, the value of \(f(x)\) approaches 0.
04

Check for the slope of the function

The slope of the function is decreasing whenever \(x > 0\). This can be determined from the derivative of the function \(f'(x) = -10/3 * e^{-x / 3}\), which is always negative when \(x > 0\).
05

Sketch the Graph

Finally we have enough information to sketch the graph of this function. Start by drawing a horizontal line at \(y = 0\) for the asymptote. Plot the function starting at the y-intercept (0, 10), then draw the graph decreasing from left to right, getting closer and closer to \(y = 0\) without actually reaching it. The graph curves downward, showing that the slope is decreasing.

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Key Concepts

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

Graph Sketching
Sketching a graph of an exponential decay function involves understanding how the function behaves over its domain. For the function \(f(x) = 10e^{-x/3}\), it's crucial to grasp the key features that define the graph. The process involves a few simple steps:
  • Start by identifying the y-intercept, which is the point where the graph crosses the y-axis. For this function, it occurs at \(x = 0\), resulting in the point (0, 10).
  • Recognize the decay nature of the function. Since \(e^{-x/3}\) decreases as \(x\) increases, the function demonstrates an exponential decline from its initial high point.
  • Use the information about the horizontal asymptote to guide the shape of the curve. For \(f(x)\), the graph will approach but never actually reach \(y = 0\).
Once these features are understood, you can sketch the function by plotting the y-intercept and drawing a curve that descends from left to right, gradually leveling off as it nears the horizontal asymptote.
Exponential Function
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. The function \(f(x) = 10e^{-x/3}\) specifically represents exponential decay, where the value of the function reduces as \(x\) increases.Exponential functions can be identified by a few common elements:
  • A base (in this case, \(e\), the natural log base) raised to some power of \(x\).
  • A coefficient, such as 10, which influences the vertical stretch or compression of the graph.
  • A rate of change parameter, here \(-1/3\), that drives how quickly the function decays.
The decreasing nature is due to the negative exponent. As \(x\) rises, the exponent \(-x/3\) increases in magnitude negatively, causing the entire power of \(e\) to reduce towards zero. This behavior forms the signature downward sloping curve of the exponential decay on the graph.
Horizontal Asymptote
A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches or crosses as \(x\) moves to positive or negative infinity. For the function \(f(x) = 10e^{-x/3}\), the horizontal asymptote is \(y = 0\).Understanding horizontal asymptotes is key to graph sketching because they indicate the behavior of the function at extreme values of \(x\). Here's why \(y = 0\) is the asymptote in this case:
  • The exponential term \(e^{-x/3}\) approaches zero as \(x\) increases, causing \(f(x)\) to get closer to zero.
  • Since \(f(x)\) never actually becomes zero for any finite \(x\), the line \(y = 0\) is approached but not crossed or reached.
In a graph, the presence of a horizontal asymptote helps visualize the long-term behavior of the function, showing how it stabilizes and what value it tends towards without ever touching it.

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Most popular questions from this chapter

If \(y=a \cdot x^{m},\) show that \(\ln y=\ln a+m \ln x .\) If \(v=\ln y\) \(u=\ln x\) and \(b=\ln a,\) show that \(v=m u+b .\) Explain why the graph of \(v\) as a function of \(u\) would be a straight line. This graph is called the log-log plot of \(y\) and \(x\)

Use a graphing calculator or computer graphing utility to estimate all zeros. $$f(x)=x^{6}-4 x^{4}+2 x^{3}-8 x-2$$

Use a graphing calculator to graph \(y=x e^{-x}, y=x e^{-2 x}\) \(y=x e^{-3 x}\) and so on. Estimate the location of the maximum for each. In general, state a rule for the location of the maximum of \(y=x e^{-k x}\)

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=\frac{3 x}{\sqrt{x^{2}+4}}$$

Use a graphing calculator or computer graphing utility to estimate all zeros. $$f(x)=x^{4}-7 x^{3}-15 x^{2}-10 x-1410$$
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