Question

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ h(x)=\sin x \text { at } a=0,[-\pi, \pi] $$

Short answer

The linear approximation to \(h(x) = \sin x\) at \(x = 0\) is \(L(x) = x\).

Step-by-step solution

Understand Linear Approximation

Linear approximation is a method of estimating the value of a function near a given point using the function's tangent line. This is useful for functions that are difficult to compute exactly.

Formula for Linear Approximation

The linear approximation of a function \(f(x)\) at a point \(a\) is given by \(L(x) = f(a) + f'(a)(x - a)\). Here \(f(a)\) is the value of the function at \(a\) and \(f'(a)\) is the derivative of the function evaluated at \(a\).

Calculate \(h(0)\)

We need to calculate \(h(0)\) since \(a = 0\). The function \(h(x) = \sin x\). Thus, \(h(0) = \sin(0) = 0\).

Calculate \(h'(x)\) and \(h'(0)\)

First, differentiate the function: \(h(x) = \sin x\) implies \(h'(x) = \cos x\). Now evaluate the derivative at \(a = 0\), which yields \(h'(0) = \cos(0) = 1\).

Write the Linear Approximation

Using the linear approximation formula, substitute \(f(a) = 0\), \(f'(a) = 1\), and \(x - a = x - 0 = x\). So, \(L(x) = 0 + 1 \cdot x = x\).

Plot the Function and Its Approximation

Plot \(h(x) = \sin x\) and \(L(x) = x\) over the interval \([-\pi, \pi]\). The plot will show \(\sin x\) as a wave-like function and \(x\) as a straight line passing through the origin. The line \(x\) should be a close approximation to \(\sin x\) near \(x = 0\).

Key concepts

Calculus

Calculus is a branch of mathematics focused on understanding how things change. Specifically, it deals with the study of limits, derivatives, integrals, and infinite series. This field is crucial in understanding and describing changes in mathematical functions and real-world scenarios. Linear approximation, for instance, stems from calculus.

In calculus, we often analyze complex functions that describe real phenomena. The ability to describe these functions in more manageable ways, like using derivatives, helps simplify problems. Calculus forms the foundation for various applied sciences, including physics, engineering, and economics.

• Limits help us understand behavior near a particular point.
• Derivatives provide information on the rate of change.
• Integrals offer a way to calculate areas under curves.

The concept of linear approximation comes into play when we desire a locally simplified version of a function — typically using the function's tangent line at that point.

Derivative

The derivative of a function measures how the function's value changes as its input changes. In simple terms, it represents the instantaneous rate of change, similar to speed in physics. The process of taking a derivative is called differentiation.

For instance, in our original problem, we need the derivative of the function \(h(x) = \sin x\). Differentiating \(\sin x\) gives us \(h'(x) = \cos x\). This derivative tells us how \(\sin x\) changes at any point \(x\).

• Derivatives help find the slope of a curve at any point.
• They are critical for optimizing functions — whether finding maxima or minima.
• They offer insights into the behavior of functions near a chosen point.

In our linear approximation exercise, the derivative \(h'(0) = \cos(0) = 1\) becomes a key part of determining the slope of the tangent line, providing the necessary rate of change for approximation.

Function Plotting

Function plotting visually represents mathematical functions, providing insights into their behavior over selected domains. For our exercise, plotting \(h(x) = \sin x\) and its approximation \(L(x) = x\) over the specified interval \([-\pi, \pi]\) shows us how well the linear approximation mimics the original function near \(x=0\).

With function plotting, you can observe:

• The periodic, wave-like nature of \(\sin x\).
• The straight line representation from \(L(x) = x\).
• How the approximation is accurate near its point of tangency (i.e., \(x=0\)).

This kind of visual understanding is essential in identifying patterns or characteristics of mathematical functions. It is crucial in determining whether simplifications or approximations give acceptable results for practical applications.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it, representing the instantaneous direction to which the curve is heading. In the context of linear approximation, the tangent line provides an easy-to-calculate linear function that approximates the curve near the tangent point.

In our task, the tangent line to \(h(x) = \sin x\) at \(a = 0\) helps approximate \(\sin x\) using the linear function \(L(x) = x\). The utility of the tangent line includes:

• Providing a simplified, linear representation of a nonlinear function.
• Supporting calculations where exact values of the function are challenging.
• Offering a glimpse into the function's behavior in a localized region.

This method of approximation helps make complex functions digestible and is especially valuable in engineering and physics for analyzing real-world systems.