Problem 77 - Thomas Calculus

Question

The linearization of \(e^{x}\) at \(x=0\) a. Derive the linear approximation \(e^{x} \approx 1+x\) at \(x=0\) . b. Estimate to five decimal places the magnitude of the error involved in replacing \(e^{x}\) by \(1+x\) on the interval \([0,0.2] .\) c. Graph \(e^{x}\) and \(1+x\) together for \(-2 \leq x \leq 2 .\) Use different colors, if available. On what intervals does the approximation appear to overestimate \(e^{x} ?\) Underestimate \(e^{x} ?\)

Step-by-Step Solution

Answer

a. Linear approximation is \(1+x\). b. Error is approximately 0.0202. c. It overestimates for \( x < 0 \) and underestimates for \( x > 0 \).

Step 1: Find the Derivative of the Function

To find the linear approximation, we first need the derivative of the function. The function is \( f(x) = e^x \), and its derivative is also \( f'(x) = e^x \).

Step 2: Evaluate the Function and its Derivative at x=0

Next, we evaluate both the function \( f(x) \) and its derivative \( f'(x) \) at \( x = 0 \). Thus, \( f(0) = e^0 = 1 \) and \( f'(0) = e^0 = 1 \).

Step 3: Use the Linear Approximation Formula

The linear approximation of a function \( f(x) \) at \( x = a \) is given by the formula: \( f(x) \approx f(a) + f'(a)(x-a) \). Substituting \( a = 0 \), \( f(0) = 1 \), and \( f'(0) = 1 \), we get: \[ f(x) \approx 1 + 1 \cdot (x-0) = 1 + x \].

Step 4: Estimate the Error on the Interval [0, 0.2]

To approximate the error, we use the next term in the Taylor series expansion. The second derivative of \( f(x) \) is \( e^x \), evaluated at \( x=0 \) the maximum error is at \( x = 0.2 \). The error formula is: \[ |R(x)| = \frac{|f''(c)|\cdot x^2}{2!} \] where \( f''(x) = e^x \) and \( |R(0.2)| = \frac{e^{0.2} \cdot 0.2^2}{2} \). Calculating this gives \[ |R(0.2)| \approx 0.0202 \].

Step 5: Graph the Functions

Plot the functions \( e^x \) and \( 1+x \) together over the interval \([-2, 2]\). Use different colors for clarity. Observe that \( 1+x \) overestimates \( e^x \) where \( x \) is negative and underestimates for positive \( x \) values.

Key Concepts

Linear Approximation

In mathematics, linear approximation is a technique used to estimate the value of a function at a given point by using a linear function. This method simplifies complex functions to straight lines, making them easier to deal with. In this context, consider the function of interest, which is the exponential function denoted as \( e^x \). To find the linear approximation at a point, say \( x = 0 \), we make use of the function value and its derivative at that point.
This means we use the formula \( f(x) \approx f(a) + f'(a)(x-a) \), where \( f(a) \) is the function value at \( x = a \), and \( f'(a) \) is the derivative value at the same point.
So, for \( e^x \) at \( x = 0 \), our approximation is \( e^x \approx 1 + x \). This creates a tangent line that touches the curve of \( e^x \) at \( x = 0 \) and gives a simple linear representation over a small interval.

Taylor Series

The Taylor series is a powerful mathematical tool for approximating complex functions with a sum of polynomial terms. This series provides a way to express functions as infinite sums of their derivatives at a single point.
For a function \( f(x) \) centered at \( x=a \), the Taylor series is given by:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \)

By using only the first term and the first derivative (first two terms), we arrive at the linear approximation, a simplified version of the Taylor series. This is used when higher precision is not needed, and calculations need to be simplified.

Derivative

A derivative represents the rate at which a function changes at any given point. Think of it as the slope of the tangent line to the curve of the function at that point.
In the context of linear approximation for \( e^x \), knowing the derivative, \( f'(x) = e^x \), is crucial. Evaluating it at \( x=0 \) gives us a value of 1, hence making the linear approximation \( e^x \approx 1+x \) valid at that point.
Calculating the derivative correctly is always the first step in finding linear approximations and is central to concepts such as the Taylor series.

Error Estimation

When approximating functions using linear approximations, understanding the potential error involved is essential. This error tells you the difference between your approximation and the actual function value.
In our example, the actual function \( e^x \) is approximated by \( 1+x \). The error can be approximated using the next term in the Taylor series that's been omitted.

• For this approximation, we use the formula: \( |R(x)| = \frac{|f''(c)| \cdot (x-a)^2}{2!} \), where \( f''(c) \) is the maximum value of the second derivative over the interval.

So, at \( x=0.2 \), the error \( |R(0.2)| \) is approximately \( 0.0202 \), indicating it's a small error for small values of \( x \).