Question

Local Linearity This is the graph of the function \(y=\sin x\) close to the origin. Since sin \(x\) is differentiable, this graph resembles a line. Find an equation for this line. \(y=x\)

Short answer

The equation for the line that the graph of the function \(y= \sin x\) resembles near the origin is \(y = x\).

Step-by-step solution

Identify the characteristics of the line

Since the function is \(y= \sin x\), and the given area is around the origin, the slope of the line would be the value of the derivative at that point. For \(y= \sin x\), the derivative is \(y'= \cos x\). At the origin , when \(x = 0\), \(y'= 1\). Therefore, the slope of the desired line is 1.

Apply the point-slope form of a line

The point-slope form of a line is \[y - y_1 = m(x - x_1)\], where \((x_1, y_1)\) are the coordinates of a point on the line and \(m\) is the slope. In this case, the point is the origin \((0,0)\) and the slope \(m\) is 1. Applying these values into the equation gives \[y - 0 = 1(x - 0)\], which simplifies to \(y = x\).

Key concepts

Differentiable Functions

When we delve into the world of calculus, one key term we frequently encounter is 'differentiable functions.' The idea of differentiability is foremost about smoothness. A function is said to be differentiable at a point if it has a derivative there. This means that as we zoom in on the graph of the function at that point, it starts to resemble a straight line, which is known as local linearity.

A function that is differentiable is smooth and continuous, without any sharp corners or breaks. For example, the sine function, represented as \(y = \sin(x)\), is differentiable because no matter how close we look at its graph, it always smoothly transitions from one point to the next with a consistent slope at each point. This smooth nature makes it easier to predict the function's behavior at points very close to where we zoom in, essentially because we can approximate it with a straight line at the point of tangency.

Derivative of the Sine Function

Understanding derivatives is another crucial part of calculus. The derivative measures how a function changes as its input changes. It gives us the slope of the tangent line to the function's graph at any particular point. The derivative of the sine function is particularly important because of its recurring presence in both pure and applied mathematics.

The sine function, given by \(y = \sin(x)\), has a derivative represented by \(y' = \cos(x)\). At any point on the sine curve, this derivative tells us the slope of the tangent line. For instance, if we want to find the derivative at the origin (when \(x = 0\)), we look at the cosine function value there, which is \(\cos(0) = 1\). Hence, the derivative of the sine function at the origin, indicating the slope of the tangent line at that point, is 1. This concept is vital in predicting and understanding the function's behavior near the origin and applies to a vast array of real-world situations where sine waves occur, like sound waves and alternating current.

Point-Slope Form

The point-slope form of a straight line is an invaluable tool for linear algebra. This form is specifically useful for finding the equation of a line when we know the slope and a single point on the line. It's written as \[y - y_1 = m(x - x_1)\], where \(m\) is the slope of the line, and \(x_1, y_1\) are the coordinates of the point through which the line passes.

For instance, if we apply this to the tangent line to the sine function at the origin, we already know that the derivative (slope) is 1, and the point is \(0,0\). Substituting these values into the formula gives us \[y - 0 = 1(x - 0)\], simplifying to the brief and elegant \(y = x\). This linear equation represents a direct proportionality between \(y\) and \(x\), which illustrates the tangent line's inclination at an angle of 45 degrees relative to the horizontal axis on the graph. Using the point-slope form is therefore a straightforward method to establish the equation of a line when dealing with problems involving tangency and linear approximations of nonlinear functions around specific points.