Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find all points on the graph of \(y=9 \sin x \cos x\) where the tangent line is horizontal.

Step-by-Step Solution

Answer

The tangent is horizontal at points \( x = \frac{\pi}{4} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \).

Step 1: Understand the Problem

To find where the tangent line to the graph is horizontal, we need to determine where the derivative of the function is equal to zero.

Step 2: Differentiate the Function

The given function is \[ y = 9 \sin x \cos x \]We use the product rule to differentiate. Recall that the product rule states:\[(uv)' = u'v + uv',\]where \(u = 9 \sin x \) and \(v = \cos x \). The derivatives of \( u \) and \( v \) are:\[ u' = 9 \cos x, \quad v' = -\sin x. \]Now apply the product rule:\[ y' = (9 \sin x)' \cos x + 9 \sin x (\cos x)' = 9 \cos x \cos x + 9 \sin x (-\sin x) = 9 \cos^2 x - 9 \sin^2 x. \]

Step 3: Simplify Using a Trigonometric Identity

Notice that:\[ 9 \cos^2 x - 9 \sin^2 x = 9(\cos^2 x - \sin^2 x). \]Recognize the trigonometric identity \(\cos^2 x - \sin^2 x = \cos 2x\). Thus,\[ y' = 9 \cos 2x. \]

Step 4: Solve for Horizontal Tangents

Set the derivative equal to zero to find horizontal tangents:\[ 9 \cos 2x = 0. \]This simplifies to:\[ \cos 2x = 0. \]The solutions for \(\cos 2x = 0\) are:\[ 2x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}. \]Solve for \(x\):\[ x = \frac{\pi}{4} + \frac{n\pi}{2}, \quad n \in \mathbb{Z}. \]

Step 5: List the Points on the Graph

Each of these \(x\) values corresponds to a point on the graph. Substituting back into the original function gives the points:- At \( x = \frac{\pi}{4} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \), substitute back to find:\[ y = 9 \sin\left(\frac{\pi}{4} + \frac{n\pi}{2}\right) \cos\left(\frac{\pi}{4} + \frac{n\pi}{2}\right). \]Calculate these values for specific \(n\) if needed.

Key Concepts

Tangent Line

When studying calculus, one important concept involves the tangent line to a curve at a particular point.

A tangent line is a straight line that touches a curve at one point without crossing it nearby. Think of it as just brushing the curve.

• At any given point on a curve, the tangent line has the same direction as the curve.
• Its slope at that point is the same as the slope of the curve.

This slope is given by the derivative of the function that defines the curve, evaluated at the specific point.

Finding tangent lines helps us approximate values and understand the behavior of functions near a given point. A horizontal tangent line occurs when this slope is zero.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for any value of the involved variables.

They are key tools in simplifying complex trigonometric expressions. For instance, in calculus, they are handy for finding derivatives or simplifying them.

• The identity \( \cos^2 x - \sin^2 x = \cos 2x \) is known as a double angle formula and is incredibly useful for problem-solving.
• This specific identity simplifies expressions and calculations involving squares of sine and cosine functions.

By using trigonometric identities, solving calculus problems can become much simpler and more intuitive.

Derivative

The derivative of a function is a fundamental concept in calculus, representing the rate of change of the function with respect to a variable.

In simpler terms, it's how much the function's value changes as the input changes. To find a horizontal tangent line, you first need to find the derivative.

• The process of differentiation involves finding this derivative, which is essentially the function's slope.
• An important rule for differentiation is the product rule, useful when dealing with products of functions, as seen in the example: if \( y = 9 \sin x \cos x \), you apply the product rule to differentiate.

The derivative of a function is crucial for understanding the function's behavior and solving optimization problems.

Horizontal Tangent

A horizontal tangent line on a graph is a line that runs parallel to the x-axis at a particular point on the function.

This occurs where the slope of the tangent line, or the derivative of the function, is zero.

• Finding where the derivative equals zero helps in locating these horizontal tangents.
• In our exercise, by setting the derivative of the function \( y = 9 \sin x \cos x \) to zero, we determined the points of horizontal tangency.

In many scenarios, finding where the slope is zero helps identify maxima, minima, or points of inflection in the function, providing deep insights into the function's overall behavior.