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Section 5.6 Projects for Chapter 5

Project 5.2. The Logistic Function.

In this project, we investigate the graph of the logistic function.

  1. Graph the sigmoid function, \(s(t) = \dfrac{1}{1+e^{-t}}\text{,}\) in the window

    \begin{align*} {\text{Xmin}} \amp = -4 \amp\amp {\text{Xmax}} = 4\\ {\text{Ymin}} \amp = -1 \amp\amp {\text{Ymax}} = 2 \end{align*}

    What are the domain and range of the function? List the intercepts of the graph, as well as any horizontal or vertical asymptotes. Estimate the coordinates of the inflection point, where the graph changes concavity.

  2. Graph the two functions \(Y_1(t) = \dfrac{5}{1+4e^{-t}}\) and \(Y_2 = \dfrac{10}{1+9e^{-t}}\) in the window

    \begin{align*} {\text{Xmin}} \amp = -2 \amp\amp {\text{Xmax}} = 10\\ {\text{Ymin}} \amp = -1 \amp\amp {\text{Ymax}} = 11 \end{align*}

    How do the graphs of these functions differ from the sigmoid function? State the domain and range, intercepts, and asymptotes of \(Y_1\) and \(Y_2\text{.}\) Estimate the coordinates of their inflection points.

  3. The function \(P(t) =\dfrac{K P_0}{P_0 + (K - P_0)e^{-rt}}\) is called a logistic function. It is used to model population growth, among other things. It has three parameters, \(K\text{,}\) \(P_0\text{,}\) and \(r\text{.}\) The parameter \(K\) is called the carrying capacity. The functions \(Y_1\) and \(Y_2\) in part (b) are logistic functions with \(P_0 = 1 \) and \(r=1\text{.}\) What does the value of \(K\) tell you about the graph? What do you notice about the vertical coordinate of the inflection point?

  4. Graph the function \(P(t) = \dfrac{10P_0}{P_0 + (10 - P_0)e^{-t}}\) for \(P_0 = 3\text{,}\) \(4\text{,}\) and \(5\text{.}\) What does the value of \(P_0\) tell you about the graph?

  5. Graph the function \(P(t) = \dfrac{20}{2 + 8e^{-rt}}\) for \(r = 0.5\text{,}\) \(1\text{,}\) and \(2\text{.}\) What does the value of \(r\) tell you about the graph?

Project 5.3. Bell-shaped Curve.

In this project, we investigate the normal or bell-shaped curve.

  1. Graph the function \(f(x) = e^{-x^2} \text{,}\) in the window

    \begin{align*} {\text{Xmin}} \amp = -2 \amp\amp {\text{Xmax}} = 2\\ {\text{Ymin}} \amp = -1 \amp\amp {\text{Ymax}} = 2 \end{align*}

    What are the domain and range of the function? List the intercepts of the graph, as well as any horizontal or vertical asymptotes. Estimate the coordinates of the inflection point, where the graph changes concavity.

  2. Graph the function \(f (x) = e^{-(x-m)^2}\) for \(m = -1\text{,}\) \(0\text{,}\) \(1\text{,}\) and \(2\text{.}\) How does the value of \(m\) affect the graph?

  3. The function

    \begin{equation*} N(x) = \dfrac{1}{s\sqrt{2\pi}}e^{-(x-m)^2/2s^2} \end{equation*}

    is called the normal curve. It is used in statistics to describe the distribution of a variable, such as height, among a population. The parameter \(m\) gives the mean of the distribution, and \(s\) gives the standard deviation. For example, the distribution of height among American women has a mean of \(64\) inches and a standard deviation of \(2.5\) inches. Graph \(N(x)\) for these values.

  4. Graph the function

    \begin{equation*} N(x) = \dfrac{1}{s\sqrt{2\pi}}e^{-(x-m)^2/2s^2} \end{equation*}

    for \(s = 0.5\text{,}\) \(0.8\text{,}\) \(1\text{,}\) and \(1.2\text{.}\) (You may have to adjust the window to get a good graph.) How does the value of \(s\) affect the graph?

Project 5.4.

Do hedgerows planted at the boundaries of a field have a good or bad effect on crop yields? Hedges provide some shelter for the crops and retain moisture, but they may compete for nutrients or create too much shade. Results of studies on the microclimates produced by hedges are summarized in the figure, which shows how crop yields increase or decrease as a function of distance from the hedgerow. (Source: Briggs, David, and Courtney, 1985)

crop yield near hedgerow

  1. We will use trial-and-improvement to fit a curve to the graph. First, graph \(y_1 = xe^{-x}\) in the window \(\text{Xmin} = -2\text{,}\) \(\text{Xmax} = 5\text{,}\) \(\text{Ymin} = -1\text{,}\) \(\text{Ymax} = 1\) to see that it has the right shape.

  2. Graph \(y_2 = (x - 2)e^{-(x-2)}\) on the same axes. How is the graph of \(y_2\) different from the graph of \(y_1\text{?}\)

  3. Next we'll find the correct scale by trying functions of the form \(y = a(x - 2)e^{-(x-2)/b}\text{.}\) Experiment with different whole number values of \(a\) and \(b\text{.}\) How do the values of \(a\) and \(b\) affect the curve?

  4. Graph \(y = 5(x - 2)e^{-(x-2)/4}\) in the window \(\text{Xmin} = -5\text{,}\) \(\text{Xmax} = 25\text{,}\) \(\text{Ymin} = -20\text{,}\) \(\text{Ymax} = 25\text{.}\) This function is a reasonable approximation for the curve in the figure. Compare the area of decreased yield (below the \(x\)-axis) with the area of increased yield (above the \(x\)-axis). Which area is larger? Is the overall effect of hedgerows on crop yield good or bad?

  5. About how far from the hedgerow do the beneficial effects extend? If the average hedgerow is about \(2.5\) meters tall, how large should the field be to exploit their advantages?

Project 5.5. Carbon Content.

Organic matter in the ground decomposes over time, and if the soil is cultivated properly, the fraction of its original organic carbon content is given by

\begin{equation*} C(t)=\dfrac{a}{b}-\dfrac{a-b}{b}e^{-bt} \end{equation*}

where \(t\) is in years, and \(a\) and \(b\) are constants. (Source: Briggs, David, and Courtney, 1985)

  1. Write and simplify the formula for \(C(t)\) if \(a = 0.01\text{,}\) \(b = 0.028\text{.}\)

  2. Graph \(C(t)\) in the window \(\text{Xmin} = 0\text{,}\) \(\text{Xmax} = 200\text{,}\) \(\text{Ymin} = 0\text{,}\) \(\text{Ymax} = 1.5\text{.}\)

  3. What value does \(C(t)\) approach as \(t\) increases? Compare this value to \(\dfrac{a}{b}\text{.}\)

  4. The half-life of this function is the amount of time until \(C(t)\) declines halfway to its limiting value, \(\dfrac{a}{b} \text{.}\) What is the half-life?

Project 5.6. Change of Base.

This project derives the change of base formula.

  1. Follow the steps below to calculate \(\log_8 20\text{.}\)

    Step 1

    Let \(x = \log_8 20\text{.}\) Write the equation in exponential form.

    Step 2

    Take the logarithm base \(10\) of both sides of your new equation.

    Step 3

    Simplify and solve for \(x\text{.}\)

  2. Follow the steps in part (a) to calculate \(\log_8 5\text{.}\)

  3. Use part (a) to find a formula for calculating \(\log_8 Q\text{,}\) where \(Q\) is any positive number.

  4. Find a formula for calculating \(\log_b Q\text{,}\) where \(b\gt 1\) and \(Q\) is any positive number.

  5. Find a formula for calculating \(\ln Q\) in terms of \(\log_{10} Q\text{.}\)

  6. Find a formula for calculating \(\log_{10} Q\) in terms of \(\ln Q\text{.}\)

Project 5.7. Log Equations.

In this project, we solve logarithm equations with a graphing calculator. We have already used the Intersect feature to find approximate solutions for linear, exponential, and other types of equations in one variable. The same technique works for equations that involve common or natural logarithms.

  1. Solve \(\log_{10}(x + 1) + \log_{10}(x - 2) = 1\) using the Intersect feature by setting \(Y_1 = \log(x + 1) + \log(x - 2)\) and \(Y_2 = 1\text{.}\) What about logarithmic equations with other bases? The calculator typically does not have a log key for bases other than \(10\) or \(e\text{.}\) However, by using the change of base formula from Project 5, we can rewrite any logarithm in terms of a common or natural logarithm.

  2. Use the change of base formula to write \(y = \log_2 x\) and \(y = \log_2(x - 2)\) in terms of common logarithms.

  3. Solve \(\log_2 x + \log_2(x - 2) = 3\) by using the Intersect feature on your calculator.

  4. Solve \(\log_3(x - 2) - \log_3(x + 1) = 3\) by using the Intersect feature on your calculator.