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Chapter 4: Problem 46

The functions \(f\) and \(g\) can be used to approximate \(e^{x}\) on the interval \([0,1] .\) Graph \(f, g,\) and \(y=e^{x}\) on the same coordinate plane, and compare the accuracy of \(f(x)\) and \(g(x)\) as an approximation to \(e^{x}\). $$f(x)=\frac{1}{2} x^{2}+x+1 ; \quad g(x)=0.84 x^{2}+0.878 x+1$$

Short Answer

Expert verified
g(x) is closer to e^x for 0 鈮� x 鈮� 1.

Step by step solution

01

Understand the Problem

We need to graph the given functions: \( f(x) = \frac{1}{2}x^2 + x + 1 \) and \( g(x) = 0.84x^2 + 0.878x + 1 \), along with \( y = e^x \), the exponential function, over the interval \([0, 1]\). After graphing, we will compare their accuracy in approximating \(e^x\).
02

Graph the Function f(x)

To graph \( f(x) = \frac{1}{2}x^2 + x + 1 \), calculate several points over the interval \([0, 1]\). For example, compute \( f(0) = 1 \), \( f(0.5) = \frac{1}{2}(0.5)^2 + 0.5 + 1 = 1.625 \), and \( f(1) = \frac{1}{2}(1)^2 + 1 + 1 = 2.5 \). Plot these points and draw a smooth curve connecting them.
03

Graph the Function g(x)

To graph \( g(x) = 0.84x^2 + 0.878x + 1 \), calculate several points over the interval \([0, 1]\). For instance, find \( g(0) = 1 \), \( g(0.5) = 0.84(0.5)^2 + 0.878(0.5) + 1 = 1.4595 \), and \( g(1) = 0.84(1)^2 + 0.878(1) + 1 = 2.718 \). Plot these points and draw a smooth curve connecting them.
04

Graph the Exponential Function y = e^x

Compute a few points on the curve \( y = e^x \) over the interval \([0, 1]\). For instance, use \( e^0 = 1 \), \( e^{0.5} \approx 1.6487 \), and \( e^1 \approx 2.7183 \). Plot these points and connect them with a smooth, increasing curve.
05

Compare the Graphs

Look at the three graphs together. Notice how close each of the approximation functions, \( f(x) \) and \( g(x) \), is to the actual exponential curve \( y = e^x \). \( g(x) \) is closer to \( y = e^x \) than \( f(x) \) over the interval \([0, 1]\), especially as it gets closer to \( x = 1 \).
06

Conclusion: Determine Which Function is More Accurate

Since \( g(x) \) produces values much closer to the exponential function \( y = e^x \) as seen on the graph and by the value comparisons, we conclude that \( g(x) \) is a better approximation of \( e^x \) compared to \( f(x) \) on the interval \([0, 1]\).

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Key Concepts

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

Exponential Functions
Exponential functions describe growth or decay at a constant relative rate. One of the most well-known exponential functions is the natural exponential function, which is defined by the expression \( y = e^x \). This function is special because it increases at a rate proportional to its current value.
  • The base \( e \) (approximately 2.718) is an irrational number roughly equal to 2.7182818. It is unique because it appears naturally in many areas of mathematics.
  • The graph of \( y = e^x \) is an upward-sloping curve that becomes steeper as \( x \) increases, representing exponential growth.
Understanding this function is crucial in various fields such as finance, biology, and physics, where growth patterns are similar to those described by \( e^x \). Being familiar with its properties also helps in approximating the function using polynomials, which are simpler mathematical expressions.
Approximation
In mathematics, approximation involves finding a simpler or less accurate form of a complex equation that still provides acceptable results. When you approximate \( e^x \), you use polynomials like \( f(x) \) and \( g(x) \). These polynomials are easier to handle because they only involve addition, subtraction, and multiplication.
  • Polynomial approximations are often used because they provide a balance between simplicity and accuracy.
  • They are particularly useful in intervals, like [0,1], where the behavior of the function can be more easily predicted.
  • The challenge is in choosing coefficients for the polynomial that give the best fit for the interval of interest.
A good approximation will closely match the real function over a specific range. This is usually assessed by graphing both the original function and its approximation together.
Quadratic Functions
Quadratic functions, which have the form \( ax^2 + bx + c \), are a type of polynomial function where the highest degree is 2. Both \( f(x) = \frac{1}{2}x^2 + x + 1 \) and \( g(x) = 0.84x^2 + 0.878x + 1 \) are quadratic functions used to approximate \( e^x \) over the interval [0, 1].
  • Their graphs are parabolas, which are symmetrical U-shaped curves. The "U" can also be inverted (\(\land\)) if the leading coefficient is negative.
  • Quadratic functions are widely used in approximation because they offer a good balance between complexity and computing simplicity.
  • The accuracy of the approximation depends on how closely the parabola can mimic the exponential curve within the specified range.
In the context of approximating \( e^x \), the shape and position of the parabola are adjusted by tweaking the coefficients \( a, b, and c \) in the quadratic equations.
Graphical Analysis
Graphical analysis involves comparing the shapes and positions of graphs to understand relationships and differences between functions. By graphing the functions \( f(x), g(x), \) and \( y = e^x \) together, you can visually compare their similarities and differences.
  • When you look at their graphs over the interval [0, 1], you're able to see which quadratic function is closer to \( e^x \).
  • By measuring the vertical distance between the curves of \( y = e^x \) and the approximations, you can assess the accuracy of each function.
  • Typically, a smaller distance indicates a better approximation.
Graphical analysis not only helps in visually identifying accurate approximations but also gives insight into how modifications to function parameters affect their shapes. Observing these changes is useful in refining and developing better approximation methods.

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Most popular questions from this chapter

The loudness of a sound, as experienced by the human ear, is based on its intensity level. A formula used for finding the intensity level \(\alpha\) (in decibels) that corresponds to a sound intensity \(I\) is \(\alpha=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is a special value of \(I\) agreed to be the weakest sound that can be detected by the ear under certain conditions. Find \(\alpha\) if. (a) \(I\) is 10 times as great as \(I_{0}\) (b) \(I\) is 1000 times as great as \(I_{0}\) (c) \(I\) is \(10,000\) times as great as \(I_{0}\) (This is the intensity level of the average voice.)

According to Newton's law of cooling, the rate at which an object cools is directly proportional to the difference in temperature between the object and the surrounding medium. The face of a household iron cools from \(125^{\circ}\) to \(100^{\circ}\) in 30 minutes in a room that remains at a constant temperature of \(75^{\circ} .\) From calculus, the temperature \(f(t)\) of the face after \(t\) hours of cooling is given by \(f(t)=50(2)^{-2 t}+75\) (a) Assuming \(t=0\) corresponds to 1: 00 PM., approximate to the nearest tenth of a degree the temperature of the face at 2: 00 P.M., 3: 30 P.M., and 4: 00 P.M. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 4\)

For manufacturers of computer chips, it is important to consider the fraction \(F\) of chips that will fail after \(t\) years of service. This fraction can sometimes be approximated by the formula \(F=1-e^{-c t},\) where \(c\) is a positive constant. (a) How does the value of \(c\) affect the reliability of a chip? (b) If \(c=0.125,\) after how many years will 35 \(\%\) of the chips have failed?

Approximate the function at the value of \(x\) to four decimal places. (a) \(f(x)=2^{3 / 1-x}, \quad x=2.5\) (b) \(g(x)=\left(\frac{2}{25}+x\right)^{-3 x}, \quad x=2.1\) \(h(x)=\frac{3^{-x}+5}{3^{x}-16}, \quad x=\sqrt{2}\)

Graph \(f\) and \(g\) on the same coordinate plane, and estimate the solution of the equation \(f(x)=g(x)\) $$f(x)=x ; \quad g(x)=3 \log _{2} x$$
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