Problem 62 - Precalculus with Limits

Question

In Exercises 59-62, find the derivative of \(f\). Use the derivative to determine any points on the graph of \(f\) at which the tangent line is horizontal. Use a graphing utility to verify your results. $$ f(x) = x^3 + 3x $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = x^3 + 3x\) is \(f'(x) = 3x^2 + 3\). The points on the graph of \(f\) at which the tangent line is horizontal are points at \(x = 0\).

Step 1: Find Derivative

To find the derivative of \(f(x) = x^3 + 3x\),follow the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\), where n is a constant. So, the derivative of \(x^3\) is \(3x^2\) and the derivative of \(3x\) is 3. Consequently, using the formula for the derivative of a sum of functions, which is the sum of the derivatives, the derivative of \(f(x)\), denoted \(f'(x)\), is given by \(f'(x) = 3x^2 + 3\)

Step 2: Find Points with Horizontal Tangent

The derivative of a function at some point is equal to the slope of the tangent line to the function at that point. So, to find the points on the graph of \(f\) where the tangent line is horizontal, it's necessary to solve the equation \(f'(x) = 0\) for \(x\). In other words, solve the equation \(3x^2 + 3 = 0\). Factoring the equation gives \(3(x^2 + 1) = 0\). Setting each factor equal to zero yields two equations: \(x^2 + 1 = 0\) and x = 0. Solving the first equation for \(x\) gives two complex roots, which are not points in the real plane, hence, discarded. Solving the second equation for \(x\) gives the root \(x = 0\).

Step 3: Verify Results using Graphing Utility

Plot the function \(f(x) = x^3 + 3x\) using a graphing utility and observe that the tangent line to \(f\) at \(x = 0\) is indeed horizontal, as the slope is 0. This verifies the results calculated algebraically.

Key Concepts

Derivative of a Function

Understanding the derivative of a function is essential in calculus. In simple terms, the derivative measures how a function's output value changes as its input value changes. It's like figuring out how quickly you're traveling at any given moment during a car trip.

For a function represented as f(x), the derivative at a certain point is denoted as f'(x). In the context of our exercise, where f(x) = x^3 + 3x, calculating the derivative f'(x) reveals the rate at which f(x) changes with respect to x.

• If f'(x) is positive, the function is increasing at that point.
• If f'(x) is negative, the function is decreasing at that point.
• If f'(x) is zero, the function may have a horizontal tangent, a peak, or a valley at that point.

When you calculate f'(x), you get the blueprint to the function’s slope everywhere along its curve. This step-by-step approach to solving problems is a powerful tool in understanding how functions behave and where they might have special features, like horizontal tangents.

Power Rule in Calculus

The power rule is a quick and handy method in calculus used to find the derivative of functions with powers of x. The formula is simple: for any function x^n, where n is a real number, the derivative is n*x^(n-1).

In our exercise, we have the function f(x) = x^3 + 3x, where we apply the power rule to each term separately.

• The term x^3 has an n of 3, so the derivative is 3*x^2.
• The term 3x is equivalent to 3x^1, which yields a derivative of 3 as x^0 equals 1.

We combine these derived pieces to find the complete derivative f'(x) = 3x^2 + 3. The power rule simplifies the process of finding derivatives and is vital in solving calculus problems efficiently. An important aspect to note for students is the significant time-savings when applying the power rule correctly, facilitating a better focus on more complex aspects of calculus.

Slope of Tangent Line

The slope of the tangent line to a curve at any given point is an important concept in calculus. It represents the instant rate of change of the function at that particular point, much like the slope of a straight line, which tells us how steep the line is.

In practical terms, finding the slope of the tangent line to the function f(x) at any point x is done by evaluating the derivative f'(x) at that point. A horizontal tangent line is a special case where the slope equals zero, indicating that at that point, the function is neither increasing nor decreasing.

In the exercise, we are particularly interested in points where f'(x) = 0, because these are the candidates for horizontal tangents. Through the calculation of the derivative, we set up the equation f'(x) = 3x^2 + 3 and solve for x where the derivative equals zero. In this case, the equation reveals that there’s a horizontal tangent at x = 0. By verifying with a graphing utility, one can see that at x = 0, the curve of f(x) indeed has a flat slope, confirming our result.

Understanding how to find and interpret the slope of the tangent line to a function is key not just for solving math problems, but also for applications in physics, economics, and various other fields where rates of change are integral.