Problem 4 - Thomas Calculus

Question

In Exercises \(1-4,\) find the linearization \(L(x)\) of \(f(x)\) at \(x=a\) $$ f(x)=\sqrt[3]{x}, \quad a=-8 $$

Step-by-Step Solution

Answer

The linearization \( L(x) \) of \( f(x)=\sqrt[3]{x} \) at \( x=-8 \) is \( L(x) = \frac{1}{12}x - \frac{4}{3} \).

Step 1: Understand Linearization

Linearization is the process of approximating a function near a given point using a tangent line. The formula for linearization at a point \( x = a \) is given by \[ L(x) = f(a) + f'(a)(x-a) \], where \( f(a) \) is the value of the function at \( x = a \) and \( f'(a) \) is the derivative of the function evaluated at \( a \).

Step 2: Calculate \( f(a) \)

Find the value of the function \( f(x) = \sqrt[3]{x} \) at \( x = -8 \).\[ f(-8) = \sqrt[3]{-8} = -2 \].

Step 3: Find the Derivative \( f'(x) \)

Differentiate the function \( f(x) = \sqrt[3]{x} \) using the power rule.\[ f(x) = x^{1/3} \rightarrow f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}} \].

Step 4: Evaluate \( f'(a) \)

Substitute \( x = -8 \) into \( f'(x) = \frac{1}{3\sqrt[3]{x^2}} \) to find \( f'(-8) \).\[ f'(-8) = \frac{1}{3\sqrt[3]{(-8)^2}} = \frac{1}{3\sqrt[3]{64}} = \frac{1}{12} \].

Step 5: Formulate the Linearization \( L(x) \)

Use the linearization formula \( L(x) = f(a) + f'(a)(x-a) \) to find \( L(x) \) at \( a = -8 \).\[ L(x) = -2 + \frac{1}{12}(x + 8) \].

Step 6: Simplify the Linearization

Simplify the expression for \( L(x) \).\[ L(x) = -2 + \frac{1}{12}x + \frac{8}{12} = -2 + \frac{1}{12}x + \frac{2}{3} \].\[ L(x) = \frac{1}{12}x - \frac{4}{3} \].

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, focusing on how functions change, or their rate of change. It is essentially about finding the derivative of a function. This tells us how the function behaves at any given point. The derivative can be thought of as the slope of the tangent line to the function at a particular point. For a function like \( f(x) = \sqrt[3]{x} \), differentiation helps us understand how steep the curve is at different points. By differentiating this function, we know precisely how quickly it grows or decreases as \( x \) changes, which is crucial for understanding its behavior.

Tangent Line

A tangent line is a straight line that touches a curve at one point, without crossing it. At this point, the line has the same slope as the curve. This means the tangent line is a great way to approximate the behavior of a curve very close to the touching point. For instance, if you have a curve representing \( f(x) = \sqrt[3]{x} \) and you look at the point where \( x = -8 \), the tangent line offers a snapshot, mimicking the function's behavior precisely at that point. Knowing the equation of this tangent line enables us to accurately estimate function values near our point of interest.

Power Rule

The Power Rule is a quick technique for finding the derivative of a function of the form \( x^n \). According to the Power Rule, the derivative of \( x^n \) is \( nx^{n-1} \). This is particularly useful because it simplifies the differentiation process, allowing us to efficiently find how a function changes. Let's apply this rule to our example, \( f(x) = x^{1/3} \). By applying the Power Rule, we find the derivative: \( f'(x) = \frac{1}{3}x^{-2/3} \). Here, the exponent "1/3" becomes the coefficient, and we subtract one from the exponent to yield "-2/3". This calculation helps determine the slope of the tangent line, which is vital for linear approximation.

Function Approximation

Function approximation is a technique used to find simple expressions that can closely predict or estimate values of a more complex function. Linearization is one of the methods used for this purpose, employing the equation of the tangent line as the approximation tool.By computing \( L(x) = f(a) + f'(a)(x-a) \), we take the exact function, like \( \sqrt[3]{x} \), and replace it with a linear function that gives similar results near the point of interest, \( x = -8 \). This form uses both the function value \( f(a) \) and the derivative \( f'(a) \), simplifying complex calculations into easier linear ones, accurately reflecting the local behavior of the function. Linear approximations are especially beneficial when evaluating functions that are difficult to compute directly.