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Chapter 10: Problem 11

An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, \(P=f(t, c),\) of rats surviving an exposure to formaldehyde at a concentration of \(c\) (in parts \(f\). $$\begin{array}{|l|l|l|l|l|l|l|}\hline & t=14 & t=16 & t=18 & t=20 & t=22 & t=24 \\\\\hline c=0 & 100 & 100 & 100 & 99 & 97 & 95 \\\\\hline c=2 & 100 & 99 & 98 & 97 & 95 & 92 \\\\\hline c=6 & 96 & 95 & 93 & 90 & 86 & 80 \\\\\hline c=15 & 96 & 93 & 82 & 70 & 58 & 36 \\\\\hline\end{array}$$\ (a) Estimate \(f_{t}(18,6):\) \(f_{t}(18,6) \approx\) __________. (b) Estimate \(f_{c}(18,6)\) \(f_{c}(18,6) \approx\) __________.

Short Answer

Expert verified
(a) \(f_t(18,6) \approx -1.5\) (b) \(f_c(18,6) \approx -1.\bar{2}\) (approximately \(-1.22\))

Step by step solution

01

Approximate \(f_t(18,6)\) using forward difference method

To approximate \(f_t(18,6)\), we will use the data points at \((t, c) = (18, 6)\) and \((t, c) = (20, 6)\). The forward difference method is given by: \[f_t(18,6) \approx \frac{f(20,6) - f(18,6)}{20 - 18}\] Using the table, we have: \(f(18, 6) = 93\%\) (percent survival at time \(18\) and concentration \(6\)) \(f(20, 6) = 90\%\) (percent survival at time \(20\) and concentration \(6\)) Now, we can apply the forward difference method: \[f_t(18,6) \approx \frac{90 - 93}{20 - 18} = \frac{-3}{2} \] Therefore, \(f_t(18,6) \approx -1.5\)
02

Approximate \(f_c(18,6)\) using forward difference method

To approximate \(f_c(18,6)\), we will use the data points at \((t, c) = (18, 6)\) and \((t, c) = (18, 15)\). The forward difference method is given by: \[f_c(18,6) \approx \frac{f(18,15) - f(18,6)}{15 - 6}\] Using the table, we have: \(f(18, 6) = 93\%\) (percent survival at time \(18\) and concentration \(6\)) \(f(18, 15) = 82\%\) (percent survival at time \(18\) and concentration \(15\)) Now, we can apply the forward difference method: \[f_c(18,6) \approx \frac{82 - 93}{15 - 6} = \frac{-11}{9} \] Therefore, \(f_c(18,6) \approx -1.\bar{2}\) (approximately \(-1.22\)) In conclusion, we have: (a) \(f_t(18,6) \approx -1.5\) (b) \(f_c(18,6) \approx -1.\bar{2}\) (approximately \(-1.22\))

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Key Concepts

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

Forward Difference Method
The forward difference method serves as a numerical technique to estimate the rate of change of a function, which can be particularly useful when dealing with discrete data or when an analytic approach is difficult. This method is based on the concept of a derivative in calculus, which measures how a function changes as its input changes.

In the context of our exercise, the forward difference method is used to approximate the partial derivatives of the function describing the percent survival of rats, denoted by f(t, c), with respect to time t and concentration c. The approximation is done by taking the function values at two nearby points and dividing the difference in function values by the difference in input values.

It's important to remember that the closer the points are, the better the approximation; however, when working with data from experiments, we are often limited to using the given data points. This method provides a quick and straightforward estimation but comes with the understanding that it's not as accurate as an exact derivative calculation would be.
Partial Differentiation
Partial differentiation is a fundamental concept in multivariable calculus used to determine the rate of change of a function with respect to one of its variables, while holding the other variables constant. In our example, having a function f(t, c) representing the survival percentage of rats depending on time t and concentration c, the partial derivative with respect to time, denoted by f_t, tells us how the survival percentage changes as time progresses, keeping concentration constant.

Similarly, the partial derivative with respect to concentration, denoted by f_c, indicates how the survival rate is affected when the concentration of formaldehyde changes while time is held fixed. The concept of partial derivatives is crucial in any field that involves multiple variables - from physics and engineering to economics and beyond. It allows researchers and analysts to isolate the influence of a single factor in a multivariable system.
Rate of Change
The rate of change is a measure of how a quantity, such as the survival rate of rats in our study, changes in response to changes in another quantity, like time or concentration of formaldehyde. In the realm of calculus, this is expressed as the derivative of a function. In a multivariable function, the rate of change with respect to a single variable is given by a partial derivative.

A positive rate of change indicates an increasing function, while a negative rate of change signifies that the function is decreasing with respect to the variable. For example, in our exercise, the negative values of the estimated partial derivatives f_t(18,6) and f_c(18,6) suggest that the percentage of rats surviving decreases as time increases or as the concentration of formaldehyde rises, respectively. Understanding the rate of change is crucial in predicting and controlling various phenomena in natural and social sciences.

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Most popular questions from this chapter

The airlines place restrictions on luggage that can be carried onto planes. \- A carry-on bag can weigh no more than 40 lbs. \- The length plus width plus height of a bag cannot exceed 45 inches. \- The bag must fit in an overhead bin. Let \(x, y,\) and \(z\) be the length, width, and height (in inches) of a carry on bag. In this problem we find the dimensions of the bag of largest volume, \(V=x y z,\) that satisfies the second restriction. Assume that we use all 45 inches to get a maximum volume. (Note that this bag of maximum volume might not satisfy the third restriction.) a. Write the volume \(V=V(x, y)\) as a function of just the two variables \(x\) and \(y\) b. Explain why the domain over which \(V\) is defined is the triangular region \(R\) with vertices \((0,0),(45,0),\) and (0,45) . c. Find the critical points, if any, of \(V\) in the interior of the region \(R\). d. Find the maximum value of \(V\) on the boundary of the region \(R\), and then determine the dimensions of a bag with maximum volume on the entire region \(R\). (Note that most carry-on bags sold today measure 22 by 14 by 9 inches with a volume of 2772 cubic inches, so that the bags will fit into the overhead bins.)

Given \(F(r, s, t)=r\left(9 s^{4}-t^{5}\right),\) compute: $$F_{r s t}=$$ _________.

Design a rectangular milk carton box of width \(w\), length \(l\), and height \(h\) which holds \(500 \mathrm{~cm}^{3}\) of milk. The sides of the box cost 1 cent \(/ \mathrm{cm}^{2}\) and the top and bottom cost \(2 \mathrm{cent} / \mathrm{cm}^{2}\). Find the dimensions of the box that minimize the total cost of materials used. dimensions = __________.

Shown in Figure 10.3 .9 is a contour plot of a function \(f\) with the values of \(f\) labeled on the contours. The point (2,1) is highlighted in red. a. Estimate the partial derivatives \(f_{x}(2,1)\) and \(f_{y}(2,1)\). b. Determine whether the second-order partial derivative \(f_{x x}(2,1)\) is positive or negative, and explain your thinking. c. Determine whether the second-order partial derivative \(f_{y y}(2,1)\) is positive or negative, and explain your thinking. d. Determine whether the second-order partial derivative \(f_{x y}(2,1)\) is positive or negative, and explain your thinking. e. Determine whether the second-order partial derivative \(f_{y x}(2,1)\) is positive or negative, and explain your thinking. f. Consider a function \(g\) of the variables \(x\) and \(y\) for which \(g_{x}(2,2)>0\) and \(g_{x x}(2,2)<0 .\) Sketch possible behavior of some contours around (2,2) on the left axes in Figure 10.3 .10 . g. Consider a function \(h\) of the variables \(x\) and \(y\) for which \(h_{x}(2,2)>0\) and \(h_{x y}(2,2)<0\). Sketch possible behavior of some contour lines around (2,2) on the right axes in Figure \(10.3 .10 .\)

Find the maximum and minimum values of the function \(f(x, y, z, t)=\) \(x+y+z+t\) subject to the constraint \(x^{2}+y^{2}+z^{2}+t^{2}=100 .\) Maximum value is ________, occuring at points (positive integer or "infinitely many"). Minimum value is ________,, occuring at points (positive integer or "infinitely many").
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