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Chapter 3: Problem 40

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=e^{-x}$$

Short Answer

Expert verified
The graph of the function \( f(x) = e^{-x} \) would start at the y-intercept at (0,1) and then decrease as 'x' increases, getting increasingly close to 0 but never actually reaching it. The table of values will vary depending on what 'x' values are chosen, but they should follow this general trend.

Step by step solution

01

Understand the Function

The function given is \( f(x) = e^{-x} \), which is an exponential decay function. The base of this function is \( e \), which is Euler's number and it's approximately equal to 2.71828. The input value 'x' is in the exponent and it's negated, so as 'x' increases, the function value will decrease.
02

Construct a Table of Values

Using a graphing utility, we can input the function \( f(x) = e^{-x} \) to construct a table of values. Depending on the exact graphing utility, the specifics will vary. However, pick varied 'x' values, such as -2, -1, 0, 1, and 2, plug these into the function to get the corresponding 'y' values.
03

Sketch the Graph

With the table of values from Step 2, plot those points on a graph. Continue sketching the graph using those points as a guide. Pay attention to the general trends of the function. The graph of \( f(x) = e^{-x} \) will have a y-intercept at (0,1), and as 'x' moves to the right (increases), the function will be decreasing, approaching but never actually reaching 0.

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Most popular questions from this chapter

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution in which \(\mathrm{pH}=3.2\).

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln \left(\frac{1}{x}\right)-x=0$$

The demand equation for a limited edition coin set is \(p=1000\left(1-\frac{5}{5+e^{-0.001 x}}\right)\) Find the demand \(x\) for a price of (a) \(p=\$ 139.50\) and (b) \(p=\$ 99.99\).

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Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$2 \ln (x+3)=3$$
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