Problem 15 - Calculus an Applied Approach

Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6-2 x ;(2,2) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the function \(f(x) = 6 - 2x\) at the point (2,2) is -2.

Step 1: Calculate the derivative of the function

Firstly, we need to calculate the derivative of the given function. For a function of the form f(x) = ax + b, the derivative f'(x) = a. So in this case, the derivative would be f'(x) = -2. This is the slope of the function and considering that the function is linear, it does not change over the range of the function.

Step 2: Application of the derivative at a given point

The next step is to find the slope of the tangent at the given point (2,2). However, we already determined that the slope of the function is a constant -2 because the function is linear. So, the slope of the tangent at any given point on this function, including the point (2,2), is also -2.

Step 3: Final Result

So, the slope of the tangent line to the graph of the given function at the point (2,2) is -2.

Key Concepts

Understanding The Limit Definition

The limit definition is synonymous with the study of calculus, laying the foundational concept for calculating derivatives. Essentially, the limit describes how a function behaves as it approaches a particular point from both directions.

For instance, consider the expression \(\lim_{x \to c} f(x)\), which signifies the value that the function \(f(x)\) gets closer to as \(x\) approaches \(c\) from both sides. Such a limit may be finite, infinite, or may not exist at all, depending on \(f\) and \(c\).

This concept becomes particularly crucial when assessing the behavior of a function at a point where direct substitution is impossible or undefined. It enables us to predict the continuous progression of a function without the need for actual values at specific points.

In applications, the limit definition leads us to compute the derivative, which ultimately gives the slope of the tangent line at a given point on a curve, such as linear functions. Both the understanding and application of this definition are pivotal for solving advanced calculus problems involving the instantaneous rate of change.

The Role of the Derivative

A derivative, in the most general sense, represents how a function changes as its input changes. In other words, it is a measure of the rate at which a function's output value is changing at any given point.

Mathematically, the derivative of a function \(f\) at a point \(x\) is defined as \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\), which is the foundation of the limit definition of the derivative. This expression provides us with the slope of the tangent line to the graph at that particular point, indicating the direction and steepness of the graph.

When the function is linear, as in the exercise provided, its derivative is constant because the rate of change is uniform across the entire graph. The derivative, thus, does not only give us a precise mathematical description of the instantaneous rate of change but also conveys geometric information about the gradient of the tangent to the graph at any given location.

Linear Functions

A linear function is a polynomial of degree one, typically represented in the slope-intercept form as \(f(x) = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. One of the defining features of linear functions is their graph – a straight line.

The simplicity of a linear function lies in its predictability. The rate of change remains constant, which means its graph is a straight line and the slope is the same at every point. Because of this, the derivative of a linear function \(f(x) = mx + b\) is simply the coefficient \(m\), which is constant for all values of \(x\).

Hence, the power of linear functions is in their clarity; identifying the slope is straightforward, and the y-intercept instantly gives us the starting point of the line on the graph. This direct relationship makes linear functions a crucial concept in algebra, calculus, and many real-world applications.

Tangent to a Graph

The idea of a tangent to a graph at a point encapsulates where a line touches the graph at precisely one point, exhibiting the direction of the curve at that instant – like a snapshot of the curve's motion.

In analytic geometry, when we say 'the tangent to the graph at a point,' we refer to a line with the same slope as the curve at that point. This slope is provided by the derivative of the curve's function at the given point. For curves, this tangent line can change its slope depending on where along the curve you are examining.

However, in the context of our exercise with a linear function, since the slope is constant anytime we talk about the tangent to the linear graph, we are referring to the line itself. As such, the tangent line to any point on a linear function will be the line itself, making it an interesting case where the tangent stays consistent along the entire length of the graph.