Question

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ G(x)=x+\sin 2 x, \text { at } a=\pi / 2,[0, \pi] $$

Short answer

Linear approximation: \( L(x) = \pi - x \).

Step-by-step solution

Understand Linear Approximation

Linear approximation is a method of finding an approximate value of a function near a given point using the tangent line at that point. The formula for the linear approximation of a function \( G(x) \) at \( x = a \) is given by \( L(x) = G(a) + G'(a)(x-a) \).

Compute \( G(a) \)

First, evaluate \( G(x) \) at \( x = a = \pi/2 \). Substitute \( x = \pi/2 \) into the function: \[ G(\pi/2) = \pi/2 + \sin(2 \times \pi/2) = \pi/2 + \sin(\pi) = \pi/2 + 0 = \pi/2. \]

Find the Derivative of \( G(x) \)

Determine \( G'(x) \) by differentiating \( G(x) = x + \sin(2x) \). Use the chain rule to find \( G'(x) \): \[ G'(x) = 1 + 2 \cos(2x). \]

Compute \( G'(a) \)

Evaluate \( G'(x) \) at \( x = a = \pi/2 \): \[ G'(\pi/2) = 1 + 2 \cos(\pi) = 1 + 2(-1) = 1 - 2 = -1. \]

Write the Linear Approximation

Using \( L(x) = G(a) + G'(a)(x-a) \), substitute \( G(a) = \pi/2 \) and \( G'(a) = -1 \): \[ L(x) = \frac{\pi}{2} - 1(x - \frac{\pi}{2}) = \frac{\pi}{2} - x + \frac{\pi}{2} = \pi - x. \]

Plot the Function and Linear Approximation

Graph \( G(x) = x + \sin(2x) \) and its linear approximation \( L(x) = \pi - x \) over the interval \([0, \pi]\). You should see the function and its tangent line intersecting at \( x = \pi/2 \), with the linear approximation close to the actual function near this point.

Key concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It provides tools to analyze changes in functions, which helps us understand their behavior. One of the core concepts of calculus is the study of rates of change and the accumulation of quantities. Calculus is divided into two major parts:

• Differential Calculus: This part focuses on rates of change, known as derivatives. It helps in finding the slope of a function at any point on its curve.
• Integral Calculus: This part deals with accumulation of quantities, which is represented by integrals. It is akin to finding the area under a curve.

Linear approximation, which we see in this exercise, is part of differential calculus. It's a technique to simplify complex functions by approximating them with a linear function near a specific point. This approximation aids in understanding the behavior of the function near that point.

Derivative

A derivative represents the instantaneous rate of change of a function with respect to one of its variables. Think of it as the slope of the tangent line to the function's graph at any given point. Derivatives can be seen as a measure of how a function's output value changes as its input value changes.

In the linear approximation problem, we calculated the derivative of the function \( G(x) = x + \sin(2x) \). Using differentiation rules, we found that its derivative is \( G'(x) = 1 + 2\cos(2x) \). This expression describes the rate at which the function \( G(x) \) changes at any point \( x \).

Evaluating the derivative at the point \( a = \pi/2 \) gave us \( G'(\pi/2) = -1 \). This value is key as it forms part of the equation used for linear approximation.

Function Graphing

Function graphing is a graphical representation of a function's behavior across an interval. By graphing a function, we can easily understand its shape, maxima, minima, and other characteristics. In calculus, graphing helps visualize concepts like continuity and differentiability, which are essential for understanding how a function behaves.

For the exercise, we graphed both the function \( G(x) = x + \sin(2x) \) and its linear approximation \( L(x) = \pi - x \) over the interval \([0, \pi]\). This plot shows the real function and its linear counterpart, highlighting how the linear approximation closely follows the curve near the point \( x = \pi/2 \). Function graphing thus provides a powerful tool for visualizing the effects of linear approximation.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that location. The concept of the tangent line is crucial in differential calculus and linear approximation.

For our function \( G(x) = x + \sin(2x) \), the tangent line at the point \( x = \pi/2 \) provides a linear estimate of the function's behavior near that point. In essence, the tangent line gives us the best straight-line "guess" of the function's direction at the point of tangency.

The slope of the tangent line is determined by the derivative \( G'(x) \), which we calculated to be \( -1 \) at \( x = \pi/2 \). The equation for the tangent line, or linear approximation, was determined to be \( L(x) = \pi - x \), which reflects this slope. Understanding tangent lines is key to mastering the process of linear approximation.