Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=2 x+7\)

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the function \(f(x) = 2x + 7\) at any point is always equal to 2.

Step 1: Find the slope of the secant line

Given the function \(f(x) = 2x + 7\), let's consider two points on the function: \(P(x_1, f(x_1))\) and \(Q(x_2, f(x_2))\). The slope of the secant line between these two points is given by the formula: \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\).

Step 2: Express the slope as a function of x

Now, substitute \(f(x_1) = 2x_1 + 7\) and \(f(x_2) = 2x_2 + 7\) into the slope formula from Step 1: \(\frac{(2x_2+7)-(2x_1+7)}{x_2-x_1} = \frac{2(x_2-x_1)}{x_2-x_1}\). We can simplify the slope formula and express the slope as the function of x: \(m(x) = 2\).

Step 3: Identify the point of tangency

Since the question asks for the tangent line at any point, we can choose any value of \(x_1\) for the point of tangency.

Step 4: Determine the slope of the tangent line

Finally, to find the slope of the tangent line, we must take the limit as the second point approaches the point of tangency, that is, the limit as \(x_2 \rightarrow x_1\). In more general terms, we want to find the limit \(m(x) = \lim_{x_2 \rightarrow x_1} 2 = 2\). Thus, the slope of the tangent line to the graph of the given function \(f(x) = 2x + 7\) at any point is always equal to 2.

Key Concepts

Tangent Line

In calculus, the tangent line to a curve at a given point is the line that touches the curve at that point without intersecting it. The tangent line provides a snapshot of the curve's behavior or the "direction" of the curve at that exact location. For a linear function like \(f(x) = 2x + 7\), the tangent line at any point on the graph is the same as the line itself. This is because linear functions have a constant slope everywhere, making every point a point of tangency.
If we were dealing with a curve, finding the tangent line would involve more steps, such as considering slight changes and approximations. In this case, however, since \(f(x)\) is linear, the tangent line is straightforward. So, the tangent at any point on this line is just the line itself.

Secant Line

A secant line cuts through a curve at two points, and it can be used to find an average rate of change between those points. In our example with the function \(f(x) = 2x + 7\), we initially find the slope of the secant line between two points, \(P(x_1, f(x_1))\) and \(Q(x_2, f(x_2))\).
The process involves calculating the change in \(y\) divided by the change in \(x\), using the formula:

• \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\)

By substituting into the formula, the calculation becomes \(\frac{2(x_2-x_1)}{x_2-x_1} = 2\). This shows that, for a linear function, the slope of the secant is equal to the slope at every point, reinforcing the consistency in linear functions.

Derivative

The derivative is a fundamental concept in calculus that describes how a function changes as its input changes, essentially measuring the "rate of change" or "slope" of the function at any point. It is symbolized by \(f'(x)\) or \(\frac{dy}{dx}\).
In the context of the exercise, we essentially calculated the derivative of \(f(x) = 2x + 7\) to find the slope of the tangent line. For linear functions, the derivative is constant because the rate of change doesn't vary as \(x\) changes. Hence, the derivative of \(f(x) = 2x + 7\) is simply 2.
This constant derivative signifies that the function is linear with a steady rate of change—a characteristic that greatly simplifies many calculus problems.

Slope

In mathematics, "slope" refers to the measure of the steepness or inclination of a line. It tells us how much \(y\) changes for a unit change in \(x\). For a straight line described by \(f(x) = mx + b\), \(m\) is the slope.
Regarding our function \(f(x) = 2x + 7\), the slope is 2. This means for every 1 unit increase in \(x\), \(f(x)\) increases by 2 units, showing a continuous upward inclination.
The importance of slope cannot be understated, as it is crucial in understanding rates of change. Calculating the slope of the tangent line also gives insights into how functions behave at any specific point, providing both a graphical and analytical tool to comprehend functional relationships.