Problem 67 - Calculus for Biology and Medicine

Question

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: Find the derivative

To find where the tangent is horizontal, we need to find the derivative of \( f(x) \) and set it to zero. The function is \( f(x) = 3x^3 - x^2 \). Using the power rule, the derivative is:\[ f'(x) = \frac{d}{dx}(3x^3) - \frac{d}{dx}(x^2) = 9x^2 - 2x \]

Step 2: Solve for zero derivative

Set the derivative equal to zero to find where the tangent line is horizontal:\[ 9x^2 - 2x = 0 \] Factor the equation:\[ x(9x - 2) = 0 \] Solve for \( x \):\[ x = 0 \quad \text{or} \quad 9x - 2 = 0 \rightarrow x = \frac{2}{9} \]

Step 3: Find the y-coordinates

Substitute the \( x \)-values back into the original function \( f(x) = 3x^3 - x^2 \) to find the corresponding \( y \)-coordinates.- For \( x = 0 \):\[ f(0) = 3(0)^3 - (0)^2 = 0 \]- For \( x = \frac{2}{9} \):\[ f\left(\frac{2}{9}\right) = 3\left(\frac{2}{9}\right)^3 - \left(\frac{2}{9}\right)^2 = \frac{24}{729} - \frac{4}{81} = \frac{24}{729} - \frac{36}{729} = -\frac{12}{729} = -\frac{4}{243} \]

Step 4: List the points

The points on the graph where the tangent is horizontal are the combined coordinates of \( x \) and \( f(x) \):- \( (0, 0) \)- \( \left(\frac{2}{9}, -\frac{4}{243}\right) \)

Key Concepts

Derivative

In calculus, derivatives play a crucial role. They help us understand how a function changes at any point. Simply put, the derivative of a function provides the slope of the tangent line to the graph at a particular point. This slope tells us how steep or flat the tangent is.

• When the slope is positive, the function is increasing at that point.
• When the slope is negative, the function is decreasing.
• And, when the slope is zero, it indicates a horizontal tangent.

To find the derivative, we must apply rules such as the power rule. For example, given a function like \( f(x) = 3x^3 - x^2 \), finding its derivative involves calculating the slopes for every point on this curve. Ultimately, this derivative, \( f'(x) = 9x^2 - 2x \), will inform us where the graph has certain features like peaks, valleys, or flat sections.

Horizontal Tangent

A tangent line that touches a curve but does not rise or fall is known as a horizontal tangent. These occur where the derivative of the function equals zero.
In practical terms, this means:

• Setting the derivative, like \( 9x^2 - 2x \), equal to zero.
• Solving this equation identifies the \( x \)-coordinates where the tangent is horizontal.
• By substituting these \( x \)-values back into the original function, we find the corresponding \( y \)-coordinates.

For instance, solving \( 9x^2 - 2x = 0 \) gives us \( x = 0 \) and \( x = \frac{2}{9} \). Substituting these back into \( f(x) \) gives the points \( (0, 0) \) and \( \left(\frac{2}{9}, -\frac{4}{243}\right) \) where the tangent is flat.

Power Rule

The power rule is one of the most straightforward techniques for finding derivatives. It's essential when dealing with polynomials, like \( f(x) = 3x^3 - x^2 \). The rule states: if \( f(x) = x^n \), then the derivative is \( f'(x) = nx^{n-1} \).
This means that:

• For \( 3x^3 \), apply the power rule to get \( 9x^2 \). Here, multiply the exponent 3 by the coefficient 3, then subtract one from the exponent.
• For \( x^2 \), the derivative is \( 2x \). Similarly, multiply the exponent 2 by the coefficient (which is 1 here) and reduce the exponent by one.

Using the power rule simplifies the process of finding derivatives, making it a vital tool in calculus. With it, you can easily determine where the horizontal tangents, peaks, troughs, and inflection points of a graph lie.