Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find all points on the graph of \(y=x^{3}-x^{2}\) where the tangent line is horizontal.

Step-by-Step Solution

Answer

The points are \((0, 0)\) and \(\left(\frac{2}{3}, -\frac{4}{27}\right)\)."

Step 1: Understand the Problem

To find where the tangent line is horizontal on the graph of \(y = x^3 - x^2\), we need to find the points where the slope of the tangent is zero. The slope of the tangent line is given by the derivative of the function.

Step 2: Find the First Derivative

Find the derivative of the function \(y = x^3 - x^2\) with respect to \(x\). The derivative, \(y'\), represents the slope of the tangent line. So, we calculate:\[y' = \frac{d}{dx}(x^{3} - x^{2}) = 3x^{2} - 2x\].

Step 3: Set Derivative Equal to Zero

Set the derivative equal to zero to find the points where the tangent is horizontal:\[3x^{2} - 2x = 0\].

Step 4: Solve for x

Factor the equation from Step 3:\[x(3x - 2) = 0\].Set each factor equal to zero and solve for \(x\):\[x = 0\] or \[3x - 2 = 0\].Solving \(3x - 2 = 0\) gives \(x = \frac{2}{3}\).

Step 5: Find Corresponding y-values

Substitute the \(x\)-values back into the original function to find the \(y\)-coordinates:- For \(x = 0\), \(y = 0^3 - 0^2 = 0\).- For \(x = \frac{2}{3}\), \(y = \left(\frac{2}{3}\right)^3 - \left(\frac{2}{3}\right)^2\).Calculate:\(\frac{2^3}{3^3} - \frac{2^2}{3^2} = \frac{8}{27} - \frac{4}{9} = \frac{8}{27} - \frac{12}{27} = -\frac{4}{27}\).

Step 6: Write the Solution

The points where the tangent line is horizontal are \((0, 0)\) and \(\left(\frac{2}{3}, -\frac{4}{27}\right)\).

Key Concepts

Understanding Derivatives

In mathematics, a derivative represents the rate at which a function is changing at any given point. It's like the speedometer of a function, telling you how fast or slow it's moving at a specific moment.
Derivatives give us the slope of the tangent line to the function's graph at any point. The slope of this tangent line is crucial when trying to find points where certain conditions, such as horizontality, occur. If a tangent line is horizontal, it means that the slope is zero, and this is indicated by a derivative that equals zero.

• The process of differentiation allows us to find the derivative of a function.
• For a polynomial function, like the one in our example, we use the power rule to find derivatives. This involves multiplying the exponent by the coefficient and reducing the exponent by one.
• The resulting expression tells us how the slope changes with each point on the graph.

In the problem, by differentiating the function, we found that its derivative is \(3x^{2} - 2x\). Setting this to zero helps us find where the tangent line is horizontal.

Basic Algebra Techniques

Algebra is like the toolkit we use to manipulate and understand mathematical expressions and equations. When working with derivatives, algebra helps us solve for unknowns and analyze the behavior of functions.
In the step-by-step solution, we used algebra to set the derivative equal to zero, finding points on the function where the tangent line is horizontal.

• By factoring out the equation \(3x^2 - 2x = 0\), we found the expression \(x(3x - 2) = 0\).
• Here, the Zero Product Property tells us that if a product of factors is zero, at least one of the factors must be zero.
• This means we solve two simple equations: \(x = 0\) and \(3x - 2 = 0\).

Algebra gives us the x-values of interest, in this case, \(x = 0\) and \(x = \frac{2}{3}\). These values tell us where on the x-axis the slope of our function will be zero.

Graphing Functions and Tangent Lines

Analyzing the graph of a function can visually demonstrate important characteristics, like where tangent lines are horizontal, just as in our example. The original function \(y = x^3 - x^2\) is a type of cubic polynomial with an intriguing shape.

• The function graph changes direction where the derivative is zero; these are often called critical points.
• From these critical points, we can determine if the function is at a maximum, minimum, or point of inflection.
• In our example, \(x = 0\) and \(x = \frac{2}{3}\) were found to have horizontal tangents, helping us to understand the function's shape better.

By using graph analysis and knowing what the derivative tells us about the slope, we gain insight into not just where these critical points are, but also what the function is doing at these points. Drawing a graph allows one to visualize a function's behavior over an interval and see where the slope changes, turns, or becomes zero.