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Chapter 3: Problem 42

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=2 e^{-0.5 x}$$

Short Answer

Expert verified
To graph the exponential function \( f(x) = 2e^{-0.5x} \), first create a table of values using a variety of x-values. Then, plot these points on a graph and sketch a smooth curve going through the points. The shape of the function's graph is a downward sloping curve above the x-axis.

Step by step solution

01

Construct a table of values

Using a graphing utility, calculate the value of \(f(x)\) for various x-values. Make sure to include negative, positive and zero values for x. These calculated values will create a table of values that look something like \[\begin{align*} x & f(x) \\ -2 & 14.7781122 \\ -1 & 5.43656366 \\ 0 & 2 \\ 1 & 0.735758882 \\ 2 & 0.270670566 \end{align*}\]
02

Sketch the graph

Using the table of values calculated in step 1, plot the points on a graph. Once the points have been plotted, draw a smooth curve that fits those points. Note that as x increases, the function \(f(x)\) approaches 0, and as x decreases, \(f(x)\) increases without limit. The curve is decreasing and always above the x-axis.

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Most popular questions from this chapter

At 8: 30 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9: 00 A.M. the temperature was \(85.7^{\circ} \mathrm{F}\), and at 11: 00 A.M. the temperature was \(82.8^{\circ} \mathrm{F}\). From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula is derived from a general cooling principle called Newton's Law of Cooling. It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death, and that the room temperature was a constant \(70^{\circ} \mathrm{F}\).) Use the formula to estimate the time of death of the person.

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