Problem 54 - Calculus an Applied Approach

Question

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=x^{3}+3 x^{2} $$

Step-by-Step Solution

Answer

The points at which the function \(y = x^{3} + 3x^{2}\) has a horizontal tangent line are (0,0) and (-2,4).

Step 1: Differentiate the Function

The derivative of the function \(y = x^{3} + 3x^{2}\) is \(y' = 3x^{2} + 6x\).

Step 2: Solve the Derivative Equals Zero

To determine the x-values where the function g has a horizontal tangent, solve the equation \(3x^{2} + 6x = 0\). Factorise the equation to obtain \(3x(x + 2) = 0\). So, the x-values are \(x = 0\) and \(x = -2\).

Step 3: Find the Corresponding y-values

Substitute these x-values into the original function to find the corresponding y-values. For \(x = 0\), \(y = 0^{3} + 3*0^{2} = 0\). For \(x = -2\), \(y = (-2)^{3} + 3*(-2)^{2} = -8 + 12 = 4\).

Key Concepts

Derivative of a Function

Let's delve into the heart of calculus: the derivative. Often perceived as simply a formula to memorize and apply, the derivative is actually a powerful tool that measures how a function's output changes as its input changes. It's the rate of change or the slope of the curve at any point. In the textbook problem, we discovered the derivative of the function y = x^{3} + 3x^{2} to be y' = 3x^{2} + 6x.

Every time you calculate a derivative, you're finding a new function that reveals the slope at each point along the original function's graph. In practical terms, this allows us to predict 'how fast' and 'in which direction' the function's graph is changing at any given point. The power rule, which was used here to obtain the derivative—multiplying the exponent by the coefficient and reducing the exponent by one—is one of the basic rules that makes finding derivatives straightforward in many cases.

Solving Equations

Solving equations is fundamental in all of mathematics, and calculus is no exception. We solve equations to find unknown values, often representing points where certain conditions are met. In this problem, we focus on 'when is the slope of the tangent line to the curve horizontal?' To answer this, we set the function's derivative equal to zero and solve for the x-values.

Why zero? Because the slope of a horizontal line is zero, and the derivative tells us the slope at a point on the curve. In our example, we got the equation 3x^{2} + 6x = 0. Factoring, a versatile and often-used technique in algebra, came to the rescue to simplify and solve for the values of x, which are the solutions to the equation. Identifying and implementing the correct method to solve equations is crucial, and for many students, practicing these techniques helps solidify a strong mathematical foundation.

Tangent Lines in Calculus

Understanding tangent lines in calculus opens a door to visualizing functions and their rates of change. A tangent line 'touches' the curve at a single point and has the same slope as the curve at that point. The magic of the derivative is that it gives us that exact slope.

When we're referring to horizontal tangent lines, we're seeking the places where the slope of the function is zero. This is particularly meaningful in the study of the function's behavior—it often corresponds to local maxima, minima, or points of inflection, depending on the function's broader behavior. In our textbook problem, we figured out the points of tangency by solving for when the derivative equals zero, indicating horizontal tangents at those points. The exercise nicely demonstrates how the abstract notion of derivatives translates into something concrete—a recognizable line on the graph of a function.