Problem 13 - 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. \(f(x)=3 x^{2}\)

Step-by-Step Solution

Answer

In short, to find the slope of the tangent line to the graph of the given function \(f(x) = 3x^2\) at any point, we first find the derivative \(f'(x) = 6x\). Then, the slope of the tangent line at any point is given by the derivative evaluated at that point, which is simply \(m = 6x\).

Step 1: Find the derivative of the function

The function is given by \(f(x) = 3x^2\). To find the derivative of this function, we will use the power rule, which states that the derivative of a function in the form of \(x^n\) is equal to \(n*x^{n-1}\). So, the derivative of the function, denoted as \(f'(x)\), is given by: \[f'(x) = 3*(2*x^{2-1}) = 6x\]

Step 2: Evaluate the derivative at the given point

To find the slope of the tangent line at any point, we simply need to evaluate the derivative of the function at that point. Let the point be represented by coordinates (x, f(x)). Then, the slope, m, of the tangent line will be equal to the derivative at that point: \[m = f'(x) = 6x\]

Step 3: Write the equation of the tangent line in slope-intercept form

Given the slope, m, and the coordinates of the point (x, f(x)), we can write the equation of the tangent line to the graph in slope-intercept form, which is given by: \[y = mx + b\] First, we can substitute the slope (m) and the coordinates of the point (x, f(x)) into the equation: \[f(x) = 6x(x) + b\]

Step 4: Simplify and find the slope of the tangent line

Now, let's simplify the equation obtained in Step 3: \[3x^2 = 6x^2 + b\] We don't actually need to solve for the y-intercept (b) here, as our goal is to find the slope of the tangent line at any point, which we already have in the form of the derivative evaluated at the given point: \[m = f'(x) = 6x\] So, the slope of the tangent line to the graph of the function at any point is given by \(m = 6x\).

Key Concepts

Derivative

In calculus, a derivative represents a rate of change. When we find a derivative, we are essentially calculating how a function changes as its input, or x-value, changes. This is a fundamental concept because it provides a way to compute the slope of a curve at any given point.
A practical way to imagine this is by thinking about how fast you're going in a car. If you're tracking your change in distance over time, the derivative would be your speed at any given moment.

• This speed—your instant rate of change—is what the derivative gives us for a function and its changes with respect to x.
• The notation used for a derivative of a function is often expressed as either \(f'(x)\) or \(\frac{df}{dx}\).

Tangent Line

The tangent line to a curve at a given point is a straight line that best approximates the curve near that point. It's like the line trying to "touch" the curve just enough without cutting through it. The slope of this tangent line is precisely the value of the derivative at that point.
Why is this important? Well, the tangent line gives a linear approximation of the function at a specific point, reflecting where the curve is headed nearby.

• If you imagine running your fingers along a track, the tangent line tells you which way your fingers will move if they continue beyond the track.
• Finding the equation of a tangent line is crucial for understanding how functions behave locally at any point.

To write the equation of the tangent line, we typically use the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Power Rule

The power rule is one of the most essential rules in calculus for finding derivatives. When you have a function of the form \(x^n\), the power rule allows you to quickly determine its derivative. This rule states that the derivative of \(x^n\) is \(n*x^{n-1}\).
Let's break it down:

• The exponent "n" comes down to multiply the coefficient.
• The new exponent is the original exponent minus one.
• For example, the derivative of \(3x^2\) is \(6x\), as we first multiply 3 by 2 (the original exponent), and then reduce the exponent by one.

The power rule is like a shortcut for derivatives of power functions, saving time and effort, especially compared to calculating limits.

Slope of a Function

The slope of a function at a particular point is synonymous with the derivative value at that point. It represents how steep the function is at any given spot on the graph.
This is where derivatives really shine, providing not just an average rate of change, but the exact "steepness" or "flatness" at a specific point.

• Where the slope is positive, the function is increasing.
• If the slope is negative, the function is decreasing.
• A zero slope indicates a flat line at that point, possibly a max, min, or a point of inflection.

So, mastering how to find and interpret the slope of a function using derivatives can unveil important insights about the behavior and shape of the graph. In contexts like physics, economics, or any field involving change, knowing the slope at a point can reveal valuable predictions about future movement and trends.