Problem 56 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The function \(y = x^{2} + 2x\) has a horizontal tangent at the point (-1, -1).

Step 1: Find the derivative of the function

The derivative of \(y = x^{2} + 2x\) is given by applying the power rule of derivatives, i.e., if we have \(y = ax^{n}\), then its derivative \(\(y'= n*ax^{n-1}\) \). So, \(y' = 2x + 2\)

Step 2: Solve for x where the derivative equals zero

Setting the derivative equal to zero, we solve for x to find the x-coordinate(s) of the point(s) where the function has a horizontal tangent line. 2x + 2 = 0. Solving this equation for x gives x = -1.

Step 3: Substitute x into the original function

To get the y-coordinate of the point where the slope is zero, we substitute the obtained x-value into the original function. Substituting x = -1 into the original function \(y = x^{2} + 2x\), we get \(y = (-1)^{2} + 2(-1) = -1\). Therefore, the function \(y = x^{2} + 2x\) has a horizontal tangent at the point (-1, -1).

Key Concepts

Derivative

When working with derivatives, we are essentially finding the rate at which a function's value changes at any given point. For a function like \(y = x^{2} + 2x\), the derivative helps us understand how the slope of the function graph changes. To find the derivative, we use the power rule, which simplifies the process of differentiation.

Power Rule

The power rule is a quick way to differentiate functions that have variables raised to a power. If you have a term like \(ax^n\), where \(a\) is a constant and \(n\) is the exponent, the derivative is \(n \cdot ax^{n-1}\). For the function \(y = x^{2} + 2x\), applying the power rule means:

• The derivative of \(x^2\) is \(2x^{2-1} = 2x\).
• The derivative of \(2x\) is simply \(2\) because the first power drops out, as \(x^1\) becomes \(x^0 = 1\).

By combining these, we find that the derivative \(y' = 2x + 2\). Often, knowing shortcuts like the power rule makes solving derivatives much easier and efficient, especially when dealing with polynomials.

Function Graph

A function graph is a visual representation of all the possible points \((x, y)\) that satisfy a function equation. For \(y = x^{2} + 2x\), the graph is a parabola. Understanding how this graph looks helps us find features like intercepts and tangent lines.
The slope of the tangent line changes at different points on the graph. By analyzing these, we pinpoint where the slope is zero, which indicates horizontal tangent lines, providing us information about key points and changes in direction.
Graphs allow us to visually interpret the derivative's results. They show us what happens when we solve for slopes, especially as they approach zero, connecting concepts to real geometric interpretations on the graph.

Slope

Slope is a measure of steepness or the inclination of a line. For functions, the slope tells us how fast they are changing at any point. Calculating the slope of tangent lines involves finding the derivative at specific \(x\)-values.
To determine where a function has a horizontal tangent line, we set its slope (or derivative) to zero and solve for \(x\). In our equations, a horizontal tangent means a zero slope. This equates to setting \(y' = 2x + 2 = 0\). Here the solution, \(x = -1\), informs us that the slope is zero at this point on the function.
By plugging \(x = -1\) back into the original equation \(y = x^{2} + 2x\), we find \(y = -1\), confirming that at the point \((-1, -1)\), the slope is indeed zero, thus confirming a horizontal tangent line.