Problem 67 - Calculus for Biology and Medicine

Question

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x^{3}-x^{2} $$

Step-by-Step Solution

Answer

The points with horizontal tangents are (0, 0) and \(\left(\frac{2}{9}, \frac{-4}{243}\right)\).

Step 1: Understand Horizontal Tangents

Horizontal tangents occur at points where the derivative of the function is equal to zero. This means we need to find where the slope of the tangent to the curve is horizontal.

Step 2: Calculate the Derivative

Find the derivative of the function \(f(x) = 3x^3 - x^2\).The derivative is calculated as \(f'(x) = \frac{d}{dx}(3x^3) - \frac{d}{dx}(x^2)\).This gives us \(f'(x) = 9x^2 - 2x.\)

Step 3: Set the Derivative to Zero

To find where the tangent is horizontal, set the derivative equal to zero:\(9x^2 - 2x = 0.\)

Step 4: Solve the Equation

Solve the equation \(9x^2 - 2x = 0\).Factor out an \(x\):\(x(9x - 2) = 0\).This gives the solutions \(x = 0\) and \(x = \frac{2}{9}\).

Step 5: Find the Corresponding y-values

Determine the y-coordinate for each x-value found. For \(x = 0\), substitute into the original function:\(f(0) = 3(0)^3 - (0)^2 = 0\).For \(x = \frac{2}{9}\), substitute into the original function:\(f\left(\frac{2}{9}\right) = 3\left(\frac{2}{9}\right)^3 - \left(\frac{2}{9}\right)^2 = \frac{24}{729} - \frac{4}{81}\).Simplify the fraction to \(f\left(\frac{2}{9}\right) = \frac{-4}{243}.\)

Step 6: Conclusion

The points on the graph where the tangents are horizontal are \((0, 0)\) and \(\left(\frac{2}{9}, \frac{-4}{243}\right)\).

Key Concepts

Derivative

The derivative of a function is a core concept of calculus. It represents the rate at which a function is changing at any given point. Imagine you are riding a bike down a hill, the steeper the hill, the faster the bike speeds up. Similarly, the derivative tells us how steep a function is at a specific point.When it comes to finding the derivative of a polynomial function such as \(f(x) = 3x^3 - x^2\), we apply the power rule. The power rule states that for any term of the form \(ax^n\), the derivative is \(n \times ax^{n-1}\). Using this rule:

• The derivative of \(3x^3\) is \(9x^2\).
• The derivative of \(-x^2\) is \(-2x\).

By combining these, we find the derivative \(f'(x) = 9x^2 - 2x\).Derivatives are fundamental because they help us understand how to find slopes of tangents, locate critical points, and analyze the behavior of functions.

Horizontal Tangent

A horizontal tangent line touches the graph of a function without sloping upwards or downwards. This occurs when the slope, or the derivative, of the function is zero at a point. Essentially, horizontal tangents are flat lines occurring at specific points on a curve.To locate such points for a function like \(f(x) = 3x^3 - x^2\), we must set the derivative equal to zero. For our function, the derivative is \(f'(x) = 9x^2 - 2x\). Solving \(9x^2 - 2x = 0\) involves:

• Factoring the equation: \(x(9x - 2) = 0\).
• Leading to solutions \(x = 0\) and \(x = \frac{2}{9}\).

These solutions are where the slope is zero, indicating horizontal tangents. Hence, at these \(x\)-values, the function has zero slope and thus horizontal tangents.

Function Analysis

Function analysis involves understanding how a function behaves across its domain, including identifying key characteristics like where it increases, decreases, peaks, and valleys.For the function \(f(x) = 3x^3 - x^2\), function analysis begins with calculating derivatives, which helps us understand critical points. We determined horizontal tangents occur at \(x = 0\) and \(x = \frac{2}{9}\). It is beneficial to determine the \(y\)-coordinates associated with these \(x\)-values:

• At \(x = 0\), \(f(0) = 0\) leading to the coordinate point \((0, 0)\).
• At \(x = \frac{2}{9}\), calculation shows \(f\left(\frac{2}{9}\right) = \frac{-4}{243}\), leading to the point \(\left(\frac{2}{9}, \frac{-4}{243}\right)\).

These points indicate where the graph flattens and potentially changes behavior from increasing to decreasing, or vice versa. By knowing these characteristics, we gain a deeper insight into the nature of the function's graph. This understanding is crucial for sketching the graph accurately and predicting its general shape.