Question

Write an equation for the tangent to the curve \(y=\sin m x\) at the origin.

Short answer

The equation for the tangent line to the curve \(y = \sin mx\) at the origin is \(y = mx\).

Step-by-step solution

Differentiate the function

The first step is to differentiate the function \(y = \sin mx\). Using the chain rule for differentiation gives \(y' = m\cos mx\).

Calculate the slope of tangent

Next, to find the slope of the tangent line to the curve at the origin, insert \(x=0\) into \(y'\), giving \(y'(0) = m\cos(0) = m\). So the slope of the tangent line to the curve at the origin is \(m\).

Use the point-slope form

The point-slope form of the equation for a straight line is \(y-y_1 = m(x - x_1)\). Now apply the x and y co-ordinates of the origin (0,0) and the slope \(m\), giving \(y - 0 = m(x - 0)\). This results in \(y = mx\).

Key concepts

Differentiation

When delving into calculus, differentiation is a fundamental concept that refers to the process of finding the derivative of a function. A derivative represents the rate at which a function changes at any point on its curve. Described as the function's slope, it effectively tells us how steep the graph is at any given point.

For example, if you're driving a car, the speedometer shows your speed — that's like the derivative of your position; it tells you how quickly your location is changing over time. In the context of our exercise, differentiating the function \(y = \text{sin} mx\) allows us to find the rate at which \(y\) changes with respect to \(x\), for any given value of \(x\). This is an essential step to determine the slope of the tangent line at a specific point, which in this case is the origin.

Chain Rule

The chain rule is a powerful tool in calculus used to differentiate composite functions — functions made up of two or more other functions.

The chain rule states that if you have a function \(g\) inside another function \(f\), the derivative of the composite function \(f(g(x))\) is \(f'(g(x))\cdot g'(x)\).

Imagine a factory assembly line where each worker depends on the previous worker's output to do their job — the rate at which the whole assembly line produces goods depends on the individual rates of all workers. Similarly, the chain rule combines the derivatives of each function within the composite to give the overall derivative. In our exercise, we used the chain rule to differentiate \(y = \text{sin} mx\), treating \(\text{sin}\) and \(mx\) as separate functions where \(y' = m\cos mx\).

Slope of Tangent

The slope of a tangent line to a curve is the slope of the line that just touches the curve at a single point, without crossing it. This slope is an actual value that expresses how sharp or flat the curve is at that exact point.

It's akin to touching a spherical object with one finger — the direction and steepness of your finger's motion at that contact point describe the slope of the object's surface. To determine this for our curve \(y = \text{sin} mx\) at the origin, we plug \(x = 0\) into the derivative, which simplifies to \(m\) since the cosine of zero is one. This value, \(m\), is the slope of the tangent line at the origin, an important component for writing the equation of the tangent line.

Point-Slope Form

The point-slope form is one of the standard ways to express the equation of a line. It is particularly useful when you know one point on the line and the slope. The general formula is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \(x_1, y_1\) are the coordinates of the known point.

The name comes from how the formula connects a single point's location (\(x_1, y_1\)) to every other point (\(x, y\)) on the line through the slope \(m\). In the context of the exercise, we have the origin (0,0) as our point and \(m\) as the slope, which gives us a simple equation \(y = mx\). It represents the tangent line that lightly kisses the curve at the origin, and it extends infinitely in both directions, maintaining a constant slope \(m\).