Problem 35 - University Calculus: Early Transcendentals

Question

Graph the curves over the given intervals, together with their tangents at the given values of \(x\). Label each curve and tangent with its equation. $$\begin{array}{l} y=\sin x, \quad-3 \pi / 2 \leq x \leq 2 \pi \\ x=-\pi, 0,3 \pi / 2 \end{array}$$

Step-by-Step Solution

Answer

Graph \(y = \sin x\) from \(-\frac{3\pi}{2}\) to \(2\pi\), with tangents \(y = -1(x + \pi)\), \(y = x\), and \(y = -1\).

Step 1: Understanding the Function

The function given is the sine function, specifically \(y = \sin x\). We are tasked with graphing this function over the interval \(-\frac{3\pi}{2} \leq x \leq 2\pi\).

Step 2: Identify the Points for Tangent Lines

We need to find the tangent lines at specified points: \(x = -\pi\), \(x = 0\), and \(x = \frac{3\pi}{2}\). This means we will calculate the derivative \(y'\) at these points.

Step 3: Calculating the Derivative

The derivative of \(y = \sin x\) is \(y' = \cos x\). We'll use this to find the slope of the tangent lines.

Step 4: Calculate the Tangent at \(x = -\pi\)

At \(x = -\pi\), \(y(-\pi) = \sin(-\pi) = 0\) and \(y'(-\pi) = \cos(-\pi) = -1\). Thus, the equation of the tangent is \(y = -1(x + \pi)\).

Step 5: Calculate the Tangent at \(x = 0\)

At \(x = 0\), \(y(0) = \sin(0) = 0\) and \(y'(0) = \cos(0) = 1\). Therefore, the equation of the tangent is \(y = 1(x - 0)\).

Step 6: Calculate the Tangent at \(x = \frac{3\pi}{2}\)

At \(x = \frac{3\pi}{2}\), \(y(\frac{3\pi}{2}) = \sin(\frac{3\pi}{2}) = -1\) and \(y'(\frac{3\pi}{2}) = \cos(\frac{3\pi}{2}) = 0\). Hence, the equation of the tangent is \(y = -1\).

Step 7: Graph the Function

On a graph, plot the function \(y = \sin x\) from \(-\frac{3\pi}{2}\) to \(2\pi\). Ensure that key points such as maximum, minimum, and intercept points are highlighted.

Step 8: Graph the Tangent Lines

Plot the tangent lines found at \(x = -\pi\) \([y = -1(x + \pi)]\), \(x = 0\) \([y = x]\), and \(x = \frac{3\pi}{2}\) \([y = -1)]\) on the same graph. Label each with its equation.

Key Concepts

Derivatives

A derivative is a fundamental concept in calculus. It measures how a function's output value changes as its input changes. In simpler terms, the derivative tells us the slope of a function at any given point. For the sine function, expressed as \(y = \sin x\), its derivative is \(y' = \cos x\). This derivative is crucial because it reveals how quickly the sine wave rises or falls at different points on the graph. Understanding derivatives is essential for tasks like finding tangent lines. The slope of the tangent line at a point on the curve can be found using the derivative. For example:

• At \(x = 0\), the derivative \(\cos 0 = 1\) tells us that the slope of the tangent line is 1, forming a line with a 45-degree angle relative to the x-axis.
• At \(x = -\pi\), the derivative \(\cos(-\pi) = -1\) indicates a negative slope, meaning the tangent line slopes downward.

Understanding and calculating derivatives help us delve deeper into the behavior of functions and their graphs.

Sine Function

The sine function, denoted as \(y = \sin x\), is one of the basic trigonometric functions. It produces a smooth, periodic wave that oscillates between -1 and 1. This function is often used to model wave-like phenomena, such as sound or light waves.In mathematics, the sine function is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) units along the x-axis. Some key properties of the sine wave include:

• The sine function is symmetric around the origin, making it an odd function. This means that \(\sin(-x) = -\sin(x)\).
• The sine graph crosses the x-axis at multiples of \(\pi\), such as \(-\pi, 0, \pi,\) etc.
• It reaches its maximum value of 1 at \(\pi/2\) and its minimum value of -1 at \(-\pi/2\).

The sine function is foundational in trigonometry and plays a significant role in many areas of science and engineering.

Graphing Functions

Graphing functions allows us to visualize mathematical relationships and understand their behavior over a range of values. This involves plotting points on a coordinate grid where the x-axis represents the input and the y-axis represents the output of the function. For the sine function \(y = \sin x\), graphing involves several steps to accurately depict its wavelike pattern:

• Identify critical points such as the intercepts, maxima, and minima. These points help guide the shape of the curve.
• Plot the key points and use the inherent properties of sine, such as periodicity and symmetry, to draw a smooth curve that oscillates between -1 and 1.
• Graph tangent lines at particular points to show how the sine function changes slope. Like at \(x = 0\), where the tangent line \(y = x\) shows a positive slope.

Graphing helps illustrate the connection between the algebraic representation of a function and its visual manifestation. Once students grasp graphing, it becomes easier to predict how changes in the function affect its graph.