Problem 135 - Thomas Calculus in SI Units

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

The linear approximation is \(e^x \approx 1+x\). Error at \(x=0.2\) is \approx 0.02443. It overestimates for \(x < 0\) and underestimates for \(x > 0\).

Step 1: Calculating the Derivative

The first step is to calculate the derivative of the function. The function is \( f(x) = e^x \). Its derivative is \( f'(x) = e^x \).

Step 2: Linearization Formula

The formula for the linearization of a function \( f(x) \) at a point \( x = a \) is given by \( L(x) = f(a) + f'(a)(x-a) \). For our case, \( a = 0 \), \( f(0) = e^0 = 1 \), \( f'(0) = e^0 = 1 \).

Step 3: Linear Approximation

Substitute \( a = 0 \), \( f(a) = 1 \), and \( f'(a) = 1 \) into the linearization formula: \( L(x) = 1 + 1(x-0) = 1 + x \). Hence, the linear approximation of \( e^x \) at \( x = 0 \) is \( e^x \approx 1 + x \).

Step 4: Calculate Error at x = 0.2

To estimate the error, use the concept of a Taylor Series with the next term: error can be approximated by \(\frac{1}{2}(x-c)^2\cdot \text{{max}}|f''(x)|\) for \(x \in [0,0.2]\), where \( f''(x) = e^x \). The maximum of \( e^x \) on [0,0.2] is \( e^{0.2}\approx 1.2214 \). The error is therefore approximately \(\frac{1}{2}(0.2)^2 \times 1.2214 \approx 0.024428 \). Hence, the magnitude of the error is roughly 0.02443.

Step 5: Plotting Functions

Plot both \( e^x \) and the linear approximation \( 1 + x \) in the graph for \( -2 \leq x \leq 2 \) using different colors. Visually analyze the intervals for overestimation and underestimation: \( 1+x \) overestimates \( e^x \) for \( x < 0 \) and underestimates for \( x > 0 \).

Step 6: Conclusion on Approximation Intervals

Based on the graph, the linear approximation \(1+x\) underestimates \(e^x\) for any \(x > 0\) and on the contrary, it overestimates when \(x < 0\).

Key Concepts

Exponential Function

The exponential function is a powerful mathematical tool, commonly represented as \( e^x \). It is defined as the mathematical constant \( e \) raised to the power of a variable \( x \). This function has unique properties that make it prevalent in various fields such as finance, biology, and physics.

Some key features of the exponential function include:

• It is always positive, meaning \( e^x > 0 \) for any real number \( x \).
• Its rate of change is equal to its value— a unique property that makes it a solution to the differential equation \( \frac{dy}{dx} = y \).
• It has a horizontal asymptote at \( y = 0 \) as \( x \) approaches negative infinity.

These properties highlight the significance of the exponential function, supporting its use in linearization and other mathematical concepts.

Taylor Series

The Taylor series is a mathematical tool used to approximate complex functions with a series of polynomial terms. This approximation is achieved by expanding a function around a specific point, known as the center, and using its derivatives.

The general form of the Taylor series for a function \( f(x) \) about the point \( a \) is:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]For the exponential function \( e^x \), the series becomes:\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]This series is crucial in deriving approximations like linearization, where the first few terms are used to yield simple polynomial expressions.

Function Approximation

Function approximation involves finding a simple expression to represent a more complex function. In the context of linearization, it provides a means to approximate functions using linear polynomials. Linearization focuses around a point, often yielding quick estimates with minimal computational efforts.

For example, the linear approximation of \( e^x \) at \( x = 0 \) is given as \( e^x \approx 1 + x \). This approximation is derived from the Taylor series by ignoring higher-order terms, achieving a balance between simplicity and accuracy for small \( x \) values.

While it simplifies calculations, approximation comes with some degree of error, often estimated using subsequent terms from the series.

Graphing Functions

Graphing functions visually represents mathematical expressions and their behavior over a range of values. This helps in understanding trends, identifying patterns, and visually comparing different functions.

Consider the exponential function \( e^x \) and its linear approximation \( 1 + x \): their graphs illustrate how well the approximation performs over a specified range. For the interval [-2, 2], both functions reveal insightful patterns:

• For \( x < 0 \), the graph of \( 1 + x \) lies above \( e^x \), indicating overestimation.
• For \( x > 0 \), \( 1 + x \) falls below \( e^x \), showing underestimation.

This visual analysis helps students grasp the intervals where linear approximations align closely with the actual function plot.