Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the tangent line to \(y=\cos x\) at \(x=1\).

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = -\sin(1)(x - 1) + \cos(1) \).

Step 1: Determine the Slope (Derivative)

The first step in finding the equation of a tangent line is to determine the slope at the point of tangency. For the function \( y = \cos x \), the derivative \( y' \) with respect to \( x \) is \( y' = -\sin x \). Evaluate this derivative at \( x = 1 \) to find the slope of the tangent line. So, the slope is \( -\sin(1) \).

Step 2: Calculate the Function Value at x=1

Next, find the value of the function at \( x = 1 \). Substitute \( x = 1 \) into \( y = \cos x \) to find \( y \) at this point. Thus, \( y = \cos(1) \).

Step 3: Use Point-Slope Form

We use the point-slope form of a line, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency \((1, \cos(1))\). Substituting these values, we get \( y - \cos(1) = -\sin(1)(x - 1) \).

Step 4: Simplify the Equation

To express the tangent line equation clearly, simplify the equation from Step 3: \( y = -\sin(1)(x - 1) + \cos(1) \). This is the equation of the tangent line.

Key Concepts

Derivative

In calculus, the derivative is a crucial concept that measures how a function changes as its input changes. Think of it as a way to find the "rate of change" or slope of a function at any given point. When considering the curve of any function, the derivative helps us determine the steepness or incline at a specific point on the curve. In the context of our problem, we are interested in the derivative of the function \( y = \cos x \).

For the function \( y = \cos x \), the derivative is \( y' = -\sin x \). This tells us how \( y \) changes with respect to \( x \). To find the slope of the tangent line at a particular \( x \), in this case at \( x = 1 \), we compute \( y' = -\sin(1) \). This gives us the exact slope of the tangent line at \( x = 1 \), which is a crucial first step in finding the equation of the tangent line.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics, particularly in the fields of geometry and calculus. These functions relate the angles of a triangle to the lengths of its sides and are periodic, meaning they repeat at regular intervals.

In our problem, we're dealing with the cosine function \( y = \cos x \). This function produces a wave-like graph that oscillates between -1 and 1. When you differentiate \( \cos x \), you get \( -\sin x \), which indicates the rate of change of the cosine function with respect to \( x \).

At \( x = 1 \), by evaluating \( \cos(1) \), we find the y-coordinate of the point on the curve, and we also evaluate \( -\sin(1) \) to find the slope of the tangent line at this specific point. Understanding these trigonometric functions is key in calculus as they arise frequently in various problems.

Point-Slope Form of a Line

The point-slope form is a straightforward way to write the equation of a line when you know a point on the line and the slope. The formula is given by \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope, and \( (x_1, y_1) \) is the known point.

In our task of finding the tangent line to \( y = \cos x \) at \( x = 1 \), we determined the slope from the derivative, which is \( -\sin(1) \), and we calculated \( y_1 = \cos(1) \). The point on the curve at \( x = 1 \) is then \((1, \cos(1))\).

Substituting these values into the point-slope form, we get the equation of the tangent line \( y - \cos(1) = -\sin(1)(x - 1) \). This form is versatile and widely used in calculus for finding equations of lines through given points.

Calculus Problem-Solving

Calculus problem-solving involves methodically tackling problems like determining the equation of a tangent line, which might initially seem complex.

To solve such problems, you generally follow these steps:

• Identify the function and the point at which you need to find the tangent line.
• Compute the derivative to determine the slope of the tangent.


• Evaluate the function at the specific point to find the coordinates.
• Apply the point-slope form to write the equation of the tangent line.
• Simplify the equation for clarity.

In the example of finding a tangent line to \( y = \cos x \) at \( x = 1 \), we followed these steps to arrive at the final equation. These systematic procedures help break down the problem into smaller, manageable parts, making it easier to solve.