Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Let \(E(t)=10^{t}\) a. Approximate \(E^{\prime}(0)\) using the centered difference quotient on [-0.0001,0.0001] . b. Use your value for \(E^{\prime}(0)\) and \(E^{\prime}(t)=E^{\prime}(0) E(t)\) to approximate \(E^{\prime}(-1), E^{\prime}(1),\) and \(E^{\prime}(2)\). c. Sketch the graphs of \(E(t)\) and \(E^{\prime}(t)\). d. Repeat a., b., and c. for \(E(t)=8^{t}\).

Step-by-Step Solution

Answer

a. \(E'(0) \approx 2.302\) for 10^t, \(1.842\) for 8^t. b. Use formula with \(E'(0)\). c. Sketch exponential graphs for both \(E(t)\) and \(E'(t)\).

Step 1: Approximate E'(0) using centered difference for E(t)=10^t

The centered difference quotient is given by:\[ E'(t) \approx \frac{E(t+h) - E(t-h)}{2h} \]For our specific case,\[ E'(0) \approx \frac{E(0.0001) - E(-0.0001)}{2 \times 0.0001} \]Calculate:\[ E(0.0001) = 10^{0.0001} \approx 1.0002303 \]\[ E(-0.0001) = 10^{-0.0001} \approx 0.9997697 \]Then,\[ E'(0) \approx \frac{1.0002303 - 0.9997697}{0.0002} = 2.302 \]

Step 2: Calculate E'(t) using E'(0) for E(t)=10^t

Given \(E'(t) = E'(0)E(t)\) and \(E'(0)\approx2.302\):- \(E'(-1) = 2.302 \cdot E(-1) = 2.302 \cdot 10^{-1} = 0.2302\)- \(E'(1) = 2.302 \cdot E(1) = 2.302 \cdot 10^{1} = 23.02\)- \(E'(2) = 2.302 \cdot E(2) = 2.302 \cdot 10^{2} = 230.2\)

Step 3: Graph the functions for E(t)=10^t

The graph of \(E(t) = 10^t\) is an exponential curve starting near zero and increasing rapidly. The derivative \(E'(t)\) also shows exponential growth, following \(E(t)\) but scaled. At points -1, 1, and 2, \(E'(t)\) values are significantly smaller at negative \(t\), much larger as \(t\) increases.

Step 4: Approximate E'(0) using centered difference for E(t)=8^t

Using the centered difference quotient:\[ E'(0) \approx \frac{E(0.0001) - E(-0.0001)}{2 \times 0.0001} \]Calculate:\[ E(0.0001) = 8^{0.0001} \approx 1.0001842 \]\[ E(-0.0001) = 8^{-0.0001} \approx 0.9998159 \]Thus,\[ E'(0) \approx \frac{1.0001842 - 0.9998159}{0.0002} = 1.842 \]

Step 5: Calculate E'(t) using E'(0) for E(t)=8^t

Given \(E'(t) = E'(0)E(t)\) and \(E'(0)\approx1.842\):- \(E'(-1) = 1.842 \cdot E(-1) = 1.842 \cdot 8^{-1} = 0.23025\)- \(E'(1) = 1.842 \cdot E(1) = 1.842 \cdot 8^{1} = 14.736\)- \(E'(2) = 1.842 \cdot E(2) = 1.842 \cdot 8^{2} = 117.888\)

Step 6: Graph the functions for E(t)=8^t

The graph of \(E(t) = 8^t\) is characteristic of an exponential function, increasing slowly at first and then more rapidly. \(E'(t)\) follows the same pattern as \(E(t)\), but is influenced by the computed \(E'(0)\), showing exponential behavior through similar trends as examined points.

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in calculus and life sciences, often used to model growth and decay processes. An exponential function is of the form \( E(t) = a^t \), where \( a \) is a positive constant and \( t \) is a variable. The base \( a \) determines the rate of growth or decay:

• If \( a > 1 \), the function represents exponential growth.
• If \( 0 < a < 1 \), it shows exponential decay.

For instance, if we consider \( E(t) = 10^t \), it represents a rapid growth rate, useful in describing phenomena like population growth or radioactive decay, depending on the context. The shape of the graph of such a function is typically a smooth curve starting near zero and rising steeply as \( t \) increases.
Exponential functions play a vital role in life sciences, modeling processes such as cell growth, metabolism, and drug concentration over time.

Derivative Approximation

The derivative of a function provides insight into its rate of change. In calculus, exact derivatives can be complex or unsolvable analytically, prompting the use of approximation methods like the centered difference quotient.
Derivative approximation allows us to estimate the slope of a function's tangent at a given point. By calculating derivatives approximately, we can analyze how quickly or slowly quantities change in real-world applications. This approach is essential in life sciences where the exact growth rate or decay cannot always be precisely measured. For instance, finding \( E'(0) \) for \( E(t) = 10^t \) involves estimating this change near the point \( t=0 \) using small increments.
This makes it possible to make real-time predictions without requiring detailed derivatives. Understanding how to use these methods accurately provides a practical toolset for scientific and mathematical studies.

Centered Difference Quotient

The centered difference quotient is a practical technique used for approximating derivatives. It's particularly useful when dealing with functions where precise differentiation is complicated or numerical methods are more feasible.
The centered difference quotient for a function \( E(t) \) is expressed as:
\[ E'(t) \approx \frac{E(t+h) - E(t-h)}{2h} \]
Here, \( h \) represents a small increment. This formula averages the rate of change over a tiny interval around the point \( t \), giving a close approximation of \( E'(t) \).
For example, using \( E(t) = 10^t \), we can calculate \( E'(0) \) by plugging in \( t=0 \) and a small \( h \), such as 0.0001, to get \( E'(0) \approx 2.302 \).
This method is advantageous because it balances numerical stability and accuracy, making it suitable for computational tasks and simulations in life sciences.

Graph Sketching

Graph sketching is a valuable skill in calculus, providing visual insights into the behavior of functions and their derivatives. It allows us to conceptualize how functions evolve and change over different intervals.
For an exponential function like \( E(t) = 10^t \), the graph typically depicts a steadily increasing trend, starting from a lower bound and escalating rapidly as \( t \) grows.
When sketching its derivative \( E'(t) \), we observe a similar pattern, reflecting exponential growth but scaled by the derivative value. For instance, if \( E'(0) \approx 2.302 \), this influences the overall gradient of the derivative graph.
By sketching both \( E(t) \) and \( E'(t) \), we gain a comprehensive understanding of the mathematical relationships in life sciences models, such as predicting biology-driven reactions or interpreting statistical data from experiments. These visual tools are essential for exploratory data analysis and hypothesis testing.