Problem 241 - Calculus Volume 1

Question

[T] Find the equation of the tangent line to \(y=-\sin \left(\frac{x}{2}\right)\) at the origin. Use a calculator to graph the function and the tangent line together.

Step-by-Step Solution

Answer

The equation of the tangent line at the origin is \(y = -\frac{1}{2}x\).

Step 1: Differentiate the Function

The first step is to find the derivative of the function, which represents the slope of the tangent line. The function given is \[-y = -\sin\left(\frac{x}{2}\right)\]. Using the chain rule, the derivative of \[-y\] with respect to \(x\) is:\[y' = -\cos\left(\frac{x}{2}\right) \cdot \left(\frac{1}{2}\right) = -\frac{1}{2}\cos\left(\frac{x}{2}\right)\].

Step 2: Evaluate the Derivative at the Origin

To find the slope of the tangent line at the origin, evaluate the derivative at \(x = 0\):\[y'(0) = -\frac{1}{2}\cos\left(\frac{0}{2}\right) = -\frac{1}{2}\cos(0) = -\frac{1}{2}\].So, the slope of the tangent line at the origin is \(-\frac{1}{2}\).

Step 3: Find the Equation of the Tangent Line

Using the point-slope form of a line, \(y - y_1 = m (x - x_1)\), where \((x_1, y_1)\) is the point (origin) \((0, 0)\), and \(m\) is the slope \(-\frac{1}{2}\), the equation becomes:\[y - 0 = -\frac{1}{2}(x - 0)\],which simplifies to:\[y = -\frac{1}{2}x\].

Step 4: Graph the Function and Tangent Line

Use a graphing calculator to plot the original function \(y = -\sin\left(\frac{x}{2}\right)\) and the tangent line \(y = -\frac{1}{2}x\) on the same axes.Both should intersect at the origin, confirming that the tangent line is accurate at that point.

Key Concepts

Derivative of a Function

Understanding the derivative of a function is key to finding tangent lines. The derivative tells us how a function changes at any given point, which means it gives us the slope of the tangent line. For the function \( y = -\sin\left(\frac{x}{2}\right) \), its derivative is found using calculus techniques, including the chain rule. A derivative resembles a slope, but with precision, as it applies to curves rather than straight lines.

Chain Rule

The chain rule is a technique used to differentiate complex functions, especially when they are composed of simpler functions. In our function \( y = -\sin\left(\frac{x}{2}\right) \), the chain rule helps differentiate by recognizing the composition of \(-\sin \left( (x/2) \right)\). The rule states that you differentiate the outer function, \(-\sin(u)\), and multiply it by the derivative of the inner function, \(u=\frac{x}{2}\). This results in \(-\cos\left(\frac{x}{2}\right)\cdot\left(\frac{1}{2}\right)\), producing an elegant way to derive complex compositions.

Slope of a Line

A slope describes the inclination of a line, and in terms of tangent lines, it tells us how steep the line is at the point of tangency. The derivative gives us this slope. For the equation ossified by our derivative \( y' = -\frac{1}{2}\cos\left(\frac{x}{2}\right) \), evaluating at the origin \((0, 0)\) tells us exactly how steep the tangent line is there. This means the slope at the origin is \(-\frac{1}{2}\), and using the point-slope form of a line helps find the equation of the tangent line.

Graphing Functions

Graphing functions allows us to visualize mathematical relationships, clarifying how a derivative as a slope relates to the original function. Plotting \( y = -\sin\left(\frac{x}{2}\right) \) alongside its tangent \( y = -\frac{1}{2}x \) on a graph provides a visual confirmation. They intersect at the origin, verified by the tangent line’s slope and starting point, demonstrating how calculus translates graphically. Using graphing calculators helps to see this interaction and confirm the precision of our calculations.