Problem 64 - Calculus

Question

Find the equation of the line tangent to the function at the given point.$$f(x)=x \text { at } x=20$$

Step-by-Step Solution

Answer

The tangent line is \(y = x\).

Step 1: Understanding the Objective

To find the equation of the tangent line to the function \(f(x) = x\) at \(x = 20\), we need two things: the slope of the tangent line and the point it passes through. Given that the function is linear, the slope is constant and equal to 1 (the coefficient of \(x\) in \(f(x) = x\)).

Step 2: Determine the Tangent Point

The point where the tangent line touches the curve is \((20, f(20))\). Since \(f(x) = x\), we have \(f(20) = 20\). Thus, the point is \((20, 20)\).

Step 3: Calculate the Slope of the Tangent

For linear functions, the derivative, which represents the slope of the tangent line, is constant and equal to the coefficient of \(x\). So, for \(f(x) = x\), the derivative \(f'(x) = 1\). Therefore, the slope of the tangent line is 1.

Step 4: Formulate the Equation of the Tangent Line

A tangent line at a point \((x_0, y_0)\) with slope \(m\) is given by: \[ y - y_0 = m(x - x_0) \] Plugging in \((x_0, y_0) = (20, 20)\) and \(m = 1\), the equation becomes: \[ y - 20 = 1(x - 20) \]

Step 5: Simplify the Tangent Line Equation

Simplify the equation from Step 4: \[ y - 20 = x - 20 \] Thus, the equation simplifies to \( y = x \).

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. Specifically, it represents the rate of change of the function with respect to one of its variables. Think of it as the function's speed. When dealing with a linear function like \( f(x) = x \), finding the derivative is straightforward. The derivative of \( f(x) = x \) is \( f'(x) = 1 \). This means the rate of change of the function is constant, and that is precisely why the graph of \( f(x) = x \) is a straight line. As such, the slope of the tangent line to the function at any point is equal to 1. In more complex functions, the derivative is not always a constant and the process of finding it may involve more intricate rules such as the product rule or chain rule.
To compute the derivative:

• You differentiate each term of the function based on its power of \(x\).
• For \( f(x) = x \), the derivative is determined by the coefficient of \( x \), which is 1.
• Thus, the tangent line at any point on this curve will maintain the same slope, 1.

Linear Function

A linear function is any function that can be graphically represented in a straight line. The general form of a linear function is \( f(x) = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. In the context of the exercise, \( f(x) = x \) is particularly simple with a slope \( m = 1 \), and y-intercept \( b = 0 \). This simplicity indicates that the function passes through the origin and has a uniform rate of change.
Some properties of linear functions include:

• They are continuous and defined for all real numbers.
• The graph is always a straight line regardless of the values of \( m \) and \( b \).
• The slope \( m \) determines the line's steepness and direction.

Linear functions like \( f(x) = x \) make it easier to visualize tangents since the tangent line overlaps with the function itself, due to the constant slope.

Slope

Slope is the measure of the steepness or incline of a line and is an integral part of linear functions. It is often represented by the letter \( m \). The slope can be thought of as the rate at which \( y \) changes for a given change in \( x \). Thus, for each increase of 1 unit in \( x \), \( y \) changes by the amount of the slope \( m \).
There are important characteristics of slope:

• Positive slope: The line rises as it moves from left to right.
• Negative slope: The line falls as it goes from left to right.
• A zero slope implies a horizontal line, and an undefined slope indicates a vertical line.

In our exercise, the slope of the function \( f(x) = x \) is 1, which means for every unit increase in \( x \), \( y \) also increases by 1. This consistent slope is why the function appears linear, making computations involving tangents relatively straightforward compared to non-linear functions.

Equation of a Line

An equation of a line is a mathematical expression that describes all the points along a line in Cartesian coordinates. It is usually expressed in the form \( y = mx + b \) or in point-slope form \( y - y_0 = m(x - x_0) \), where \( m \) is the slope and \( (x_0, y_0) \) is a specific point on the line.
To form the equation of a tangent line, you need:

• Its slope \( m \).
• A point through which it passes \((x_0, y_0)\).

In our particular problem, using the point-slope form helps derive the tangent equation easily. By substituting \( m = 1 \) and the point \( (20, 20) \), the tangent line becomes \( y - 20 = 1(x - 20) \). Simplified, this returns to the formula of the function itself, \( y = x \), reinforcing the idea that the tangent line shares the same properties as our linear function \( f(x) = x \). In cases involving non-linear functions, this method ensures precision when determining tangents.