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Chapter 5: Problem 6

a. Sketch graphs of the functions \(f\) and \(g\) on the same axes. Shade the region(s) between the graphs. b. Solve for the input value(s) at which the graphs of \(f\) and \(g\) intersect. c. Calculate the difference in the area of the region between the graph of \(f\) and the horizontal axis and the area of the region between the graph of \(g\) and the horizontal axis from \(a\) to \(b\). d. Calculate the total area of the shaded region(s). $$ f(x)=e^{0.5 x} ; g(x)=\frac{2}{x} ; a=0.5 ; b=3 $$

Short Answer

Expert verified
Shaded region area is approximately 0.172.

Step by step solution

01

Graph Functions

To plot the graphs of the functions \( f(x) = e^{0.5x} \) and \( g(x) = \frac{2}{x} \), choose a range of \( x \) values from 0.5 to 3. For each function, calculate several corresponding \( y \) values (e.g., for \( x = 1, 2, 3 \)) and sketch their curves on the same coordinate plane. Both are continuous, but \( g(x) \) has a vertical asymptote at \( x = 0 \). Shade the region between where \( f(x) \) and \( g(x) \) intersect.
02

Find Intersection Points

To find where the graphs intersect, set \( f(x) = g(x) \), resulting in \( e^{0.5x} = \frac{2}{x} \). Numerically solve this equation (because it does not have an analytic solution), likely using software or iterative methods. The solutions occur approximately at \( x \approx 0.91 \) and \( x \approx 2.51 \).
03

Calculate Areas Under Curves

To find the area under each curve from \( a = 0.5 \) to \( b = 3 \), integrate each function separately. The area under \( f(x) \) is \( \int_{0.5}^{3} e^{0.5x} \, dx \), and the area under \( g(x) \) is \( \int_{0.5}^{3} \frac{2}{x} \, dx \). Evaluating these integrals yields approximately 3.436 and 3.772 respectively.
04

Calculate Difference in Areas

To find the difference in these areas, subtract the area under \( g(x) \) from the area under \( f(x) \). This results in \( 3.436 - 3.772 = -0.336 \).
05

Determine Total Shaded Area

The total area of the shaded region, which is the area between the curves \( f(x) \) and \( g(x) \) from \( x = 0.91 \) to \( x = 2.51 \), involves calculating the integral \( \int_{0.91}^{2.51} (e^{0.5x} - \frac{2}{x}) \, dx \). Evaluating this integral gives an area of approximately 0.172.

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Key Concepts

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

Graphing Functions
Sketching graphs of functions is a crucial step in visualizing how they behave in relation to one another. For the given functions, which are defined as \( f(x) = e^{0.5x} \) and \( g(x) = \frac{2}{x} \), it's important to choose appropriate values of \( x \) within the range from 0.5 to 3. Start by calculating the \( y \) values for \( x = 0.5, 1, 2, \) and \( 3 \).
  • For \( f(x) \), determine the exponential growth as \( x \) increases.
  • For \( g(x) \), note the hyperbolic decrease in values with increasing \( x \).
This visualization allows you to observe where the graphs intersect and helps you shade the region between these graphs. Remember, \( g(x) \) has a vertical asymptote at \( x = 0 \) making \( x = 0.5 \) the nearest point to consider.
Area between Curves
The area between the curves of two functions is found by integrating the difference between them over a specific interval. In our exercise, you're interested in the area between \( f(x) = e^{0.5x} \) and \( g(x) = \frac{2}{x} \) from \( x = 0.91 \) to \( x = 2.51 \).

To achieve this, set up the integral of the difference \( f(x) - g(x) \), which gives:e\[\int_{0.91}^{2.51} (e^{0.5x} - \frac{2}{x}) \, dx\]Evaluating this integral involves using standard integration techniques for both exponential and logarithmic functions. The calculated value represents the total area of the shaded region, approximately 0.172 square units.
Integration
Integration is a mathematical process used to find areas under curves, among other things. In this exercise, it's used to calculate areas under \( f(x) = e^{0.5x} \) and \( g(x) = \frac{2}{x} \) from a certain point to another. Let's break down the integration step:
  • For \( f(x) = e^{0.5x} \), integrate to find the area from \( 0.5 \) to \( 3 \), giving approximately 3.436 square units.
  • For \( g(x) = \frac{2}{x} \), integrate over the same interval to get about 3.772 square units.
The difference between these integrals is negative, which indicates the relative position of the curves concerning the horizontal axis. This calculation is crucial to understand how one curve lies above or below the other over a given range.
Intersection Points
Intersection points are values of \( x \) where two graphs meet. Finding these points helps to narrow down the specific interval to inspect for the area between the curves.

To find these points for \( f(x) \, \) and \( g(x) \), set their equations equal to each other:e\[e^{0.5x} = \frac{2}{x}\]This equation does not have a straightforward analytical solution, so numerical methods or graphing software are typically used. Through these methods, the intersection points have been calculated to be approximately \( x \approx 0.91 \) and \( x \approx 2.51 \). These points define the interval where the functions intersect and thus, the bounds of the integral to find the area between them.

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