Problem 20 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose that in the problem of Example \(3.5 .6,\) the work of building a web is proportional to \(d^{4},\) the fourth power of the diameter, \(d\). Then the energy available to the spider is $$E=k_{1} d^{2}-k_{2} d^{4}$$ Assume that \(k_{1}=0.01\) and \(k_{2}=0.000001\). a. Draw a graph of \(E=0.01 d^{2}-0.000001 d^{4}\) for \(-10 \leq d \leq 110\). b. Find \(E^{\prime}(d)\) for \(E(d)=0.01 d^{2}-0.000001 d^{4}\). c. Find two numbers, \(d,\) for which \(E^{\prime}(d)=0\). d. Find the highest point of the graph between \(d=0\) and \(d=100\).

Step-by-Step Solution

Answer

The highest energy point is at \(d = 70.71\) with energy \(E = 25\). Critical points are at \(d = 0\) and \(d = 70.71\).

Step 1: Set Up the Equation

The energy available is given by the equation \[ E(d) = 0.01d^2 - 0.000001d^4. \] This is a polynomial equation we will explore graphically and analytically.

Step 2: Graph the Equation

To draw the graph of the function, plot the values of \(E\) for \(d\) in the range \(-10 \leq d \leq 110\). Note that since \(d\) represents a diameter, it realistically starts from zero. Use a graphing tool or software to visualize this, where the graph should show a parabola opening downward due to the negative \(d^4\) term.

Step 3: Derive the Function

To find the derivative \(E'(d)\), apply the power rule to each term: \[ E'(d) = \frac{d}{d} (0.01d^2) - \frac{d}{d} (0.000001d^4) = 0.02d - 0.000004d^3. \]Thus, \(E'(d) = 0.02d - 0.000004d^3.\)

Step 4: Solve for Critical Points

To find where \(E'(d) = 0\), set the derivative equal to zero: \[ 0.02d - 0.000004d^3 = 0. \]Factor out a \(d\): \[ d(0.02 - 0.000004d^2) = 0. \]This gives us \(d = 0\) or \(0.02 = 0.000004d^2\). Solving the equation \(0.02 = 0.000004d^2\), we get:\[ d^2 = \frac{0.02}{0.000004} = 5000. \]\(d = \sqrt{5000} = 70.71.\)Thus, the critical points are \(d = 0\) and \(d \approx 70.71\).

Step 5: Determine the Highest Point

To find the highest point of the graph between \(d = 0\) and \(d = 100\), examine both critical points and endpoints: - Calculate \(E(0) = 0.01(0)^2 - 0.000001(0)^4 = 0. \)- Calculate \(E(70.71) = 0.01(70.71)^2 - 0.000001(70.71)^4.\) Evaluate to verify: \[E(70.71) = 0.01(5000) - 0.000001(25 \times 10^6) = 50 - 25 = 25.\]- Calculate \(E(100) = 0.01(100)^2 - 0.000001(100)^4 = 100 - 100 = 0.\)The highest point (maximum energy available) is \(d \approx 70.71\) with \(E \approx 25.\)

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that involve powers of variables with constant coefficients. For example, the energy function for our spider is a polynomial because it combines terms like the square of diameter and its fourth power, represented as:

• \(E(d) = 0.01d^2 - 0.000001d^4\).

Polynomials are very versatile in modeling real-world situations, like the energy needed to build a web. In our case, polynomial forms can show how quantities increase rapidly, like the negative effect of larger diameters on energy. Understanding these terms helps in graphing and analyzing the behavior of the function. These functions are defined over a set of real numbers, which is why we can explore the results graphically for various values of diameter, \(d\).

Derivative Calculation

Derivatives represent rates at which things change and they play a crucial role in calculus, especially in the life sciences for modeling dynamic systems. To find the derivative of a polynomial function, like our energy function, we use the power rule. For each term of the polynomial, we:

• Multiply the exponent by the coefficient.
• Reduce the exponent by one.

For the energy function, after applying the power rule we get:

• The derivative is \(E'(d) = 0.02d - 0.000004d^3\).
• This tells us how quickly energy changes in response to changes in diameter.

Calculating derivatives allow us to find critical points where changes are momentarily zero and aid in understanding maximum and minimum values, vital in biophysical modeling.

Critical Points

Critical points occur in a function where the derivative is zero or undefined. In our exercise, they are calculated to understand when the energy function's output stops increasing or decreasing with respect to the diameter. For a polynomial like ours:

• We set \(E'(d) = 0\) to find potential points of interest.
• This yields the equation \(0.02d - 0.000004d^3 = 0\), leading to \(d = 0\) and \(d \approx 70.71\).

These figures are insightful: \(d = 0\) naturally corresponds to no diameter, while \(d \approx 70.71\) indicates a significant shift in energy dynamics. At these points, the slope of the tangent to the energy function graph is zero, showing where energy levels off before changing direction.

Graphical Analysis

Graphical analysis involves translating equations into visual representations. This is essential for interpreting how polynomial functions behave over intervals. When graphing:

• We plot the values of our function \(E(d) = 0.01d^2 - 0.000001d^4\) against diameter \(d\).
• By examining the graph, we can visualize critical points, such as where \(d \approx 70.71\), indicating a local maximum energy.

The graph shows a downward-opening parabola due to the negative sign attached to the \(d^4\) term. This shape confirms the polynomial behavior where energy initially increases with diameter but then declines sharply past the critical point. Such graphical insights allow us to predict behavior without performing extensive calculations, making it a powerful tool in life sciences.