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Magazine Chapter 2: Problem 43
Find an equation for the tangent line to the graph at the specified value of \(x\). \(y=x \cos 3 x, x=\pi\)
Short Answer
Expert verified
The equation of the tangent line is \(y = -x\).
Step by step solution
01
Identify the Function and Point
The function given is \(y = x \cos 3x\). We are asked to find the tangent line at \(x = \pi\). The point on the graph is \((\pi, y(\pi))\). Calculate \(y(\pi) = \pi \cos(3\pi) = \pi \times (-1) = -\pi\). Thus, the point is \((\pi, -\pi)\).
02
Differentiate the Function
To find the slope of the tangent line, we need the derivative of the function. Using the product rule for differentiation, where if \(y = u \cdot v\), then \(y' = u'v + uv'\), let \(u = x\) and \(v = \cos 3x\). Then \(u' = 1\) and \(v' = -3\sin 3x\). So, \(y' = 1\cdot \cos 3x + x\cdot (-3)\sin 3x = \cos 3x - 3x \sin 3x\).
03
Evaluate the Derivative at the Given Point
Substitute \(x = \pi\) into the derivative to get the slope at the point. \(y'(\pi) = \cos 3\pi - 3\pi \sin 3\pi\). We already know \(\cos 3\pi = -1\) and \(\sin 3\pi = 0\), so \(y'(\pi) = -1\). This slope is \(-1\).
04
Write the Equation of the Tangent Line
Using the point-slope form of the equation for a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point \((\pi, -\pi)\) and \(m\) is the slope \(-1\), we write: \(y + \pi = -1(x - \pi)\) or \(y + \pi = -x + \pi\). Simplifying, we get \(y = -x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative: The Rate of Change
A derivative represents the rate at which a function is changing at any given point. It's akin to finding the speed of a moving object at an exact moment. In calculus, the derivative is used to determine the slope of a function at a specific point, which is vital when finding equations of tangent lines.
For the function in question, \( y = x \cos 3x \), we need to find how this function changes as \( x \) changes鈥攖his is captured by its derivative. The process of differentiation gives us a new function that describes these changes. This new function provides the slope of the original function at any point \( x \). Here, differentiation reveals the slope of the tangent line at the specified point, \ \( x = \pi \).
Derivatives are essential in many fields, allowing us to understand how quantities change and interact in complex systems.
For the function in question, \( y = x \cos 3x \), we need to find how this function changes as \( x \) changes鈥攖his is captured by its derivative. The process of differentiation gives us a new function that describes these changes. This new function provides the slope of the original function at any point \( x \). Here, differentiation reveals the slope of the tangent line at the specified point, \ \( x = \pi \).
Derivatives are essential in many fields, allowing us to understand how quantities change and interact in complex systems.
Product Rule: Multiplying Derivatives
The product rule is a critical tool in differentiation when dealing with functions that are products of two or more functions. It states that if you have a function \( y = u \cdot v \), the derivative \( y' \) is given by \( u'v + uv' \).
This rule is invaluable when working with composite functions, like the one given here.
In our example, let \( u = x \) and \( v = \cos 3x \). Following the product rule, we find the derivative of each part:\[ y' = 1 \cdot \cos 3x + x \cdot (-3)\sin 3x \].
Utilizing the product rule ensures we accurately account for both parts of the function when differentiating, giving us a cohesive answer that describes the whole function's behavior.
This rule is invaluable when working with composite functions, like the one given here.
In our example, let \( u = x \) and \( v = \cos 3x \). Following the product rule, we find the derivative of each part:\[ y' = 1 \cdot \cos 3x + x \cdot (-3)\sin 3x \].
Utilizing the product rule ensures we accurately account for both parts of the function when differentiating, giving us a cohesive answer that describes the whole function's behavior.
Point-Slope Form: Building Tangent Lines
The point-slope form of a linear equation is incredibly useful when you know a point and the slope of a line. It provides a blueprint for writing the equation of a line based on this information.
The general formula is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope. In our exercise, we found \( x_1 = \pi \), \( y_1 = -\pi \), and \( m = -1 \).
Using these values, our line's equation becomes \( y + \pi = -1(x - \pi) \).
This form is straightforward and powerful, providing a direct way to derive the equation of the tangent line from the derivative and the given point.
The general formula is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope. In our exercise, we found \( x_1 = \pi \), \( y_1 = -\pi \), and \( m = -1 \).
Using these values, our line's equation becomes \( y + \pi = -1(x - \pi) \).
This form is straightforward and powerful, providing a direct way to derive the equation of the tangent line from the derivative and the given point.
Trigonometric Functions: Waves of Change
Trigonometric functions are fundamental in understanding cycles and oscillations experienced in many natural and man-made phenomena. In this problem, the function \( \cos 3x \) plays a significant role.
These functions are periodic, meaning they repeat at regular intervals.
This cyclic nature of sine and cosine functions makes them crucial in studying wave-like phenomena. The derivatives of trigonometric functions also follow specific patterns, with \( \frac{d}{dx}(\cos x) = -\sin x \) and \( \frac{d}{dx}(\sin x) = \cos x \).
Incorporating trigonometric rules with derivatives enhances our ability to model and predict changes within wave-related problems, showcasing their widespread utility across various scientific and engineering disciplines.
These functions are periodic, meaning they repeat at regular intervals.
This cyclic nature of sine and cosine functions makes them crucial in studying wave-like phenomena. The derivatives of trigonometric functions also follow specific patterns, with \( \frac{d}{dx}(\cos x) = -\sin x \) and \( \frac{d}{dx}(\sin x) = \cos x \).
Incorporating trigonometric rules with derivatives enhances our ability to model and predict changes within wave-related problems, showcasing their widespread utility across various scientific and engineering disciplines.
