Problem 5 - Calculus - AP Edition

Question

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\). $$(1+x)^{15} \approx 1+15 x$$

Step-by-Step Solution

Answer

Yes, the formula \((1+x)^{15} \approx 1 + 15x\) is the local linear approximation at \(x_0 = 0\).

Step 1: Understanding Local Linear Approximation

The local linear approximation, also known as the tangent line approximation, of a function at a point gives us a linear function that closely approximates the value of the function near that point. For the function \[(1+x)^{15},\]the local linear approximation at \(x_0=0\) takes the form \(L(x) = f(x_0) + f'(x_0) \cdot (x - x_0)\).

Step 2: Finding the Value of the Function at x=0

We begin by calculating \(f(x_0)\) which is the value of the function \((1+x)^{15}\) at \(x_0=0\). Substitute \(x=0\) into the function:\[f(0) = (1+0)^{15} = 1.\]

Step 3: Finding the Derivative at x=0

Next, we need to find \(f'(x_0)\), the derivative of the function at \(x_0=0\). First, compute the derivative of \((1+x)^{15}\):\[f'(x) = 15(1+x)^{14}.\]Now evaluate \(f'(x)\) at \(x=0\):\[f'(0) = 15(1+0)^{14} = 15.\]

Step 4: Constructing the Local Linear Approximation

With the values of \(f(x_0)\) and \(f'(x_0)\), construct the linear approximation:\[L(x) = f(x_0) + f'(x_0)(x - x_0) = 1 + 15(x - 0).\]This simplifies to:\[L(x) = 1 + 15x.\]

Step 5: Conclusion

The derived linear approximation \(L(x) = 1 + 15x\) matches the given formula \((1+x)^{15} \approx 1 + 15x\). Thus, the given formula is indeed the local linear approximation at \(x_0 = 0\).

Key Concepts

Tangent Line Approximation

Local linear approximation is widely utilized in calculus to estimate the behavior of complex functions near a given point. When you hear about the tangent line approximation, think of it as finding the tangent line to a curve at a particular point, which serves as the best linear estimate of the function at that location.
This concept preserves only the first derivative information about the curve, essentially capturing the slope at that point. For example, when we approximate \((1+x)^{15}\) at \(x_0 = 0\), we use its tangent line to estimate values of the function near \(x=0\).
This is not a substitute for the function over large intervals, but it offers an efficient and simple way to understand and compute values close to \(x_0\).

• Local linear approximations are expressed as \(L(x) = f(x_0) + f'(x_0) \cdot (x - x_0)\).
• It relies heavily on the function and its derivative being continuous and differentiable at the point of interest.

Approximations work best for small changes in \(x\), near the point of tangency, and their accuracy decreases as we move away from the point.

Derivative Calculation

Derivative calculation is a critical step in determining the local linear approximation. The derivative at a particular point tells us the slope of the tangent line to the graph of the function at that specific point. This slope serves as the rate of change of the function’s value with respect to small changes in \(x\).
For our function, \(f(x) = (1+x)^{15}\), the derivative is calculated using the power rule of derivatives, which tells us how to derive powers of a variable:

• The derivative of \((1+x)^{15}\) is \[f'(x) = 15(1+x)^{14}\].
• Evaluating this at \(x_0 = 0\) yields \[f'(0) = 15(1+0)^{14} = 15\].

Knowing this slope, we can visualize the tangent line's steepness and direction. This information directly contributes to constructing the linear approximation by influencing how much \(L(x)\) changes as \(x\) changes. The derivative is thus crucial for providing an accurate local behavior description of the function in question.

Function Evaluation

Before we can make any approximations, we must first evaluate the function at the point of interest, \(x_0\). Function evaluation provides the baseline starting point on the graph from which all approximations begin.
For the function \((1+x)^{15}\) at \(x_0 = 0\), we would substitute 0 into the function:

• The computation results in \(f(0) = (1+0)^{15} = 1\).

This value is particularly significant because it represents the y-intercept of our tangent line and becomes the initial value in the formal expression for the tangent line approximation (local linear approximation).
Accurate function evaluation at \(x_0\) is essential because any error at this stage will propagate through all subsequent calculations, possibly affecting the reliability of the linear approximation. Hence, obtaining the exact function value at \(x_0\) is critical for constructing a dependable approximation.