Problem 23 - Calculus for Biology and Medicine

Question

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=e^{x}, a=2, n=3 ; x=2.1 $$

Step-by-Step Solution

Answer

The Taylor polynomial approximates \( e^{2.1} \) as \( e^2 \times 1.1051667 \).

Step 1: Recall the Taylor Polynomial Expansion

The Taylor polynomial of degree 3 for a function \( f(x) \) centered at \( a \) is given by:\[P_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3\]For this problem, \( f(x) = e^x \), \( a = 2 \). We need to find the derivatives.

Step 2: Evaluate the Function and Its Derivatives at a

For \( f(x) = e^x \), all derivatives are \( f^{(k)}(x) = e^x \). Evaluating these at \( a = 2 \):\[\begin{align*}f(2) &= e^2, \f'(2) &= e^2, \f''(2) &= e^2, \f'''(2) &= e^2.\end{align*}\]

Step 3: Substitute into the Third-Degree Taylor Polynomial

Substitute the values into the Taylor polynomial formula:\[P_3(x) = e^2 + e^2(x-2) + \frac{e^2}{2}(x-2)^2 + \frac{e^2}{6}(x-2)^3\]

Step 4: Calculate the Taylor Polynomial at x=2.1

Evaluate \( P_3(x) \) at \( x = 2.1 \):\[\begin{align*}P_3(2.1) &= e^2 + e^2(2.1-2) + \frac{e^2}{2}(2.1-2)^2 + \frac{e^2}{6}(2.1-2)^3 \&= e^2 + 0.1e^2 + \frac{e^2}{2}(0.1)^2 + \frac{e^2}{6}(0.1)^3 \&= e^2(1 + 0.1 + 0.005 + 0.0001667) \&= e^2(1.1051667).\end{align*}\]Compute numerically if needed.

Step 5: Compare with the Exact Value

Calculate the exact value of \( f(2.1) = e^{2.1} \) using a calculator for numerical comparison. Compare this with the polynomial approximation \( P_3(2.1) \) to observe how closely the Taylor polynomial approximates \( e^{2.1} \).

Key Concepts

Derivatives

In calculus, derivatives represent how a function changes as its input changes. They are essentially the rates of change or the slopes of the function at any given point. Understanding derivatives is crucial when working with Taylor polynomials, as these polynomials are formed using derivatives of different orders.The first derivative indicates the slope of the tangent line to the curve of the function. For our exercise with the function \( f(x) = e^x \), the first derivative is simply \( f'(x) = e^x \). The same goes for the second and third derivatives: \( f''(x) = e^x \) and \( f'''(x) = e^x \), respectively.When computing a Taylor polynomial, these derivatives are evaluated at the center point \( a \). For this problem, being centered at \( a = 2 \) means computing each of these derivatives at that point. This results in \( f(2) = f'(2) = f''(2) = f'''(2) = e^2 \), forming the foundation of the Taylor polynomial.

Approximation

Approximation is a fundamental concept in mathematics, especially when exact answers are difficult or impossible to find analytically. When using a Taylor polynomial, we create a polynomial function that closely follows the shape of our original function over a small interval.By using derivatives evaluated at a specific center \( a \), we construct a Taylor polynomial that approximates the function near that point. For our exercise, we constructed a third-degree Taylor polynomial to approximate the exponential function \( e^x \) near \( x = 2 \).The Taylor polynomial \( P_3(x) \) provides a practical means to approximate values like \( e^{2.1} \) without needing a calculator or undefined numbers. This is done by plugging \( x = 2.1 \) into our polynomial. The approximation quality typically improves with higher-degree polynomials, but even a third-degree can offer a reasonably accurate estimation.

Exponential Function

The exponential function \( f(x) = e^x \) is a critical mathematical function with unique properties. It grows rapidly, and its derivative is identical to the function itself, making it a common choice for demonstrating differentiation and Taylor series.This function arises naturally in many areas of applied mathematics, science, and engineering due to its powerful properties. It describes processes involving constant relative growth and decay, such as population growth, radioactive decay, and compounding interest.When dealing with polynomials like Taylor series, the exponential function's behavior is captured succinctly through its consistent derivatives. This creates a smooth and accurate approximation curve. Using this, we can estimate functions like \( e^{2.1} \) easily and observe the closeness of our Taylor approximation \( P_3(2.1) \) to the actual value through numerical comparisons. The simple derivative makes constructing Taylor polynomials straightforward, forming a powerful tool for approximations.