Problem 11 - Active Calculus Multivariable

Question

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 \(f_{c}(18,6)\) \(f_{c}(18,6) \approx\) __________.

Step-by-Step Solution

Answer

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

Step 1: 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, \(f_t(18,6) \approx -1.5\)

Step 2: 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 \(-1.22\))

Key Concepts

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.