Problem 15 - Calculus for Biology and Medicine

Question

(a) Use the Intermediate Value Theorem to show that \(e^{x}=2-x\) has a solution in \((0,2) .\) (b) Find this solution to an accuracy of \(10^{-4}\) using the bisection search method, implemented as a spreadsheet.

Step-by-Step Solution

Answer

There is a solution for the equation on the interval \((0,2)\). Using the bisection method, the approximate solution is \( x \approx 0.4428 \).

Step 1: Define the Function

Let's define the function we are interested in:\[ f(x) = e^x - (2 - x) \]We need to find values \( x \) in the interval \( (0, 2) \) where this function equals zero.

Step 2: Apply the Intermediate Value Theorem

First, calculate \( f(0) \) and \( f(2) \):\[ f(0) = e^0 - (2 - 0) = 1 - 2 = -1 \]\[ f(2) = e^2 - (2 - 2) = e^2 > 0 \]Since \( f(0) = -1 \) and \( f(2) > 0 \), by the Intermediate Value Theorem, there must be a root in the interval \( (0, 2) \) because the function changes sign.

Step 3: Initial Setup for Bisection Method

For the bisection method, our initial interval is \( a = 0 \) and \( b = 2 \) with \( f(a) = -1 \) and \( f(b) > 0 \). This method will repeatedly halve the interval and select the subinterval in which the function changes sign.

Step 4: Implement Bisection Method

1. Compute the midpoint: \( c = \frac{a + b}{2} \).2. Evaluate \( f(c) \). If \( f(c) = 0 \) or the interval \( b - a \) is sufficiently small (less than \( 10^{-4} \)), stop.3. Determine which subinterval has a sign change: - If \( f(c) \) has the same sign as \( f(a) \), set \( a = c \). - If \( f(c) \) has the same sign as \( f(b) \), set \( b = c \).4. Repeat the process until the interval is sufficiently small.

Step 5: Spreadsheet Implementation

Create a spreadsheet with columns for \( a \), \( b \), \( c \), \( f(c) \), and the interval size \( b-a \). 1. In the first row, input \( a = 0 \), \( b = 2 \).2. Calculate \( c \) as \( \frac{a + b}{2} \) and \( f(c) \).3. Adjust \( a \) or \( b \) based on the sign of \( f(c) \) as described previously.4. Continue to fill out the spreadsheet until \( b-a < 10^{-4} \).

Step 6: Verify the Solution

Upon completion of the spreadsheet, the value of \( c \) in the final row is the approximate root of the equation \( e^x = 2-x \) accurate to \( 10^{-4} \).

Key Concepts

Bisection Method

The Bisection Method is a popular numerical technique used to find roots of continuous functions. It's based on the Intermediate Value Theorem, which guarantees that if a continuous function changes sign over an interval \(a, b\), then there is at least one root in that interval. The essence of the bisection method is to halve the interval repeatedly to zero in on a root.
Here is how it works:

• Start with an initial interval \(a, b\) where the function changes sign.
• Calculate the midpoint \(c = \frac{a + b}{2}\).
• Evaluate the function at \(c\); if \(f(c) = 0\), you found a root.
• If not, check if the sign change occurs between \(a\) and \(c\) or \(c\) and \(b\).
• Set the new interval as the half that contains the sign change and repeat until the interval is sufficiently small, indicating proximity to a root.

This method is especially beneficial in scenarios where the exact root cannot be easily determined algebraically but can be approximated to a desired accuracy level, like \(10^{-4}\) as in our problem.
The method is not only straightforward but also reliable, as it ensures convergence to a root within the specified tolerance level.

Numerical Analysis

Numerical Analysis is the field of mathematics and computer science that focuses on devising algorithms to obtain numerical solutions to mathematical problems. It is particularly useful in cases where exact solutions are impossible or impractical to determine, such as solving equations involving detailed functions like \(e^x = 2 - x\).
Key aspects to keep in mind about numerical analysis include:

• The precision of numerical methods, like the Bisection Method, which helps refine an estimate for a solution.
• Their stability, which refers to the method's sensitivity to small changes or errors in data.
• Their efficiency, which concerns how quickly and resourcefully a method reaches a solution.

Much of numerical analysis is about balancing these properties to ensure a method is not only accurate but also operationally viable. It involves iterating on approximations to converge closer and closer to a solution, rather than expecting an immediate exact answer.
In practical applications, like those tackled with spreadsheets or programming, these methods can be automated, lending great utility in scientific computations and real-world problem-solving tasks.

Root Finding

Root Finding is a type of problem in numerical analysis where we seek values of \(x\) such that a given function \(f(x)\) equals zero. This is crucial since many real-world problems can be reduced to equations that require solving for these critical points. Finding roots tells us about the behavior of functions in different contexts.
This process is vital for:

• Identifying points where a system transitions, like during zero crossing in signal processing or finding equilibrium points in applied physics.
• Solving mathematical equations that can't be rearranged to isolate \(x\) easily.
• Testing for conditions or constraints in optimization problems.

The Bisection Method, as described in our exercise, is one of many techniques used for root finding. What makes it special is its robustness, ensuring a root in an interval with sign change without requiring derivative calculations or other complexities. In practical contexts, like using spreadsheets, root-finding techniques allow for efficient calculation and quick iterations to pinpoint these zeros with desired accuracy, serving many fields from engineering to finance.