Question

Find a linear approximation to \(f(x)=(1+x)^{\alpha}\) at \(x=0\), where \(\alpha\) is any number. For various values of \(\alpha\), plot \(f(x)\) and its linear approximation \(L(x)\). For what values of \(\alpha\) does the linear approximation always overestimate \(f(x) ?\) For what values of \(\alpha\) does the linear approximation always underestimate \(f(x)\) ?

Short answer

For \( \alpha > 1 \) or \( \alpha < 0 \), it underestimates; for \( 0 < \alpha < 1 \), it overestimates.

Step-by-step solution

Understand Linear Approximation Formula

The linear approximation or tangent line approximation of a function \( f(x) \) at \( x = a \) is given by \( L(x) = f(a) + f'(a) (x - a) \). In this problem, we're approximating at \( x=0 \), so \( a=0 \).

Calculate the Function Value at x=0

Evaluate \( f(x) = (1+x)^{\alpha} \) at \( x=0 \). We have \( f(0) = (1+0)^{\alpha} = 1 \).

Determine the Derivative of the Function

The derivative, \( f'(x) \), for \( f(x) = (1+x)^{\alpha} \) is found using the power rule. \( f'(x) = \alpha(1+x)^{\alpha-1} \).

Evaluate the Derivative at x=0

Now, we evaluate \( f'(x) \) at \( x=0 \). So, \( f'(0) = \alpha(1+0)^{\alpha-1} = \alpha \).

Write the Linear Approximation

Substitute \( f(0) \) and \( f'(0) \) into the formula for linear approximation: \( L(x) = 1 + \alpha \cdot x \). This is the linear approximation of \( f(x) = (1+x)^{\alpha} \) at \( x=0 \).

Analyze Overestimation or Underestimation

For different values of \( \alpha \), consider the second derivative \( f''(x) = \alpha(\alpha-1)(1+x)^{\alpha-2} \). The sign of \( f''(0) = \alpha(\alpha-1) \) determines the concavity: \( f(x) \) is concave up when \( f''(0) > 0 \) and concave down when \( f''(0) < 0 \).

Identify Overestimation and Underestimation Ranges

- If \( \alpha(\alpha-1) > 0 \) (i.e., \( \alpha > 1 \) or \( \alpha < 0 \)), \( f(x) \) is concave up, and \( L(x) \) underestimates \( f(x) \). - If \( \alpha(\alpha-1) < 0 \) (i.e., \( 0 < \alpha < 1 \)), \( f(x) \) is concave down, and \( L(x) \) overestimates \( f(x) \).

Key concepts

Taylor Series

The Taylor Series is a mathematical tool that helps us approximate complex functions with simpler polynomial expressions. To understand how linear approximation fits into this, think of linear approximation as the first step in the Taylor Series. The formula for a Taylor Series of a function \( f(x) \) centered at \( x = a \) is:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \, ... \, \]For a linear approximation, we only use the first two terms: \( f(a) + f'(a)(x-a) \). This represents the tangent line to the function at \( x = a \). It provides a close estimate of \( f(x) \) near \( a \). Though it's a simpler approximation, it's quite useful. It helps us get an idea of the function's behavior at points close to \( a \) without complicated computations. This is why the linear approximation is often used in calculus to estimate values quickly and effectively.

Derivative

A Derivative represents the rate at which a function changes. It's like the function's rate of progress or speed at a particular point. For the function \( f(x) = (1+x)^{\alpha} \), the derivative \( f'(x) \) can be calculated using the power rule: \( f'(x) = \alpha(1+x)^{\alpha-1} \). This tells us how \( f(x) \) changes as \( x \) changes.The first derivative \( f'(x) \) provides insights into the slope of the tangent line to the curve at any given point \( x \). For linear approximations, the derivative's value at a specific point (here, \( x = 0 \)) is crucial. We find that \( f'(0) = \alpha \). This means that at \( x = 0 \), the slope of the tangent, or our line of approximation, is \( \alpha \). Knowing this slope is key. It helps in building the linear approximation formula \( L(x) = 1 + \alpha \cdot x \).

Concavity

Concavity provides insights on how the function behaves beyond just its slope. It indicates whether a function curves upwards or downwards. This is determined using the second derivative \( f''(x) \). For \( f(x) = (1+x)^{\alpha} \), the second derivative is calculated as:\[ f''(x) = \alpha(\alpha-1)(1+x)^{\alpha-2} \]When \( f''(0) \) is positive, \( f(x) \) is concave up, forming a U-shape, and when it's negative, the curve is concave down, like an upside-down U.Understanding concavity helps in analyzing the linear approximation accuracy. When \( f(x) \) is concave up (\( f''(0)>0\), \( \alpha > 1 \text{ or } \alpha < 0\)), the approximation \( L(x) \) underestimates \( f(x) \). Conversely, while \( f(x) \) is concave down (\( f''(0)<0\), \( 0 < \alpha < 1 \)), the approximation tends to overestimate the function.

Function Analysis

Function Analysis involves examining different properties of a function systematically. This includes looking at various aspects such as the function’s values, its derivatives, and its concavity, among others. By thoroughly analyzing these properties, we understand the behavior and characteristics of the function.For the function \( f(x) = (1+x)^{\alpha} \), analysis begins with its value at specific points, like \( x=0 \), where \( f(0) = 1 \). We also compute the derivative \( f'(x) = \alpha(1+x)^{\alpha-1} \). This represents the function's rate of change at any point \( x \). Further, the second derivative \( f''(x) \) reveals concavity. By evaluating these, we see whether the function bends upwards or downwards, indicating underestimation or overestimation by the linear approximation.By understanding each of these components, we piece together a holistic view of \( f(x) \). This allows us to make informed predictions and adjustments in approximations. Each element of function analysis feeds into crafting more accurate and reliable mathematical models.