Problem 68 - 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. $$ f(x)=-4 x^{4}+x^{3} $$

Step-by-Step Solution

Answer

The graph has horizontal tangents at \((0, 0)\) and \(\left(\frac{3}{16}, \frac{27}{65536}\right)\)."

Step 1: Find the derivative of the function

To determine where the tangent to the graph is horizontal, we need to find its derivative. The derivative represents the slope of the tangent line. Let's find the derivative of the function, \( f(x) = -4x^4 + x^3 \). Use the power rule: \( f'(x) = \frac{d}{dx}(-4x^4) + \frac{d}{dx}(x^3) = -16x^3 + 3x^2 \).

Step 2: Set the derivative equal to zero

The tangent to the curve is horizontal where the slope is zero. This means that we set the derivative equal to zero: \(-16x^3 + 3x^2 = 0\).

Step 3: Factor the derivative

To make it easier to solve, factor the derivative: \(-16x^3 + 3x^2 = x^2(-16x + 3) = 0\).

Step 4: Solve for x

The factored equation \(x^2(-16x + 3) = 0\) gives us two possible solutions: \(x^2 = 0\) and \(-16x + 3 = 0\). Solving these, we find that \(x = 0\) and \(x = \frac{3}{16}\).

Step 5: Find corresponding y-coordinates

Substitute the values of \(x\) back into the original function to find the corresponding \(y\)-coordinates. For \(x = 0\), \(f(0) = -4(0)^4 + (0)^3 = 0\). For \(x = \frac{3}{16}\), substitute and compute \(f\left(\frac{3}{16}\right) = -4\left(\frac{3}{16}\right)^4 + \left(\frac{3}{16}\right)^3\) to find the \(y\)-coordinate.

Step 6: Simplify and calculate y for \(x = \frac{3}{16}\)

Calculate the value for \(f\left(\frac{3}{16}\right)\):\[-4\left(\frac{3}{16}\right)^4 + \left(\frac{3}{16}\right)^3 = -4\left(\frac{81}{65536}\right) + \left(\frac{27}{4096}\right)\]. Simplifying gives \(-\frac{324}{65536} + \frac{27}{4096} = \frac{27}{65536}\). The points are \((0, 0)\) and \(\left(\frac{3}{16}, \frac{27}{65536}\right)\).

Key Concepts

Derivative

In calculus, the derivative of a function is a key concept that represents how the function's output changes as its input changes. In simpler terms, it tells us the rate at which the function is changing at any given point. For the function given in the exercise, which is \[ f(x) = -4x^4 + x^3\]we find its derivative to see how the slope of the tangent line behaves at different points on the graph.

• The derivative of \(f(x)\) using the power rule results in\(f'(x) = -16x^3 + 3x^2\).
• This derivative is a polynomial, indicating that the slope of the tangent line will change with different values of \(x\).

The derivative is fundamental because it allows us to determine where a function increases or decreases, and specifically, in this problem, where the tangent to the graph is horizontal.

Horizontal Tangent

A horizontal tangent occurs when the slope of the tangent line is zero. This condition can reveal important information about the behavior of a function, such as identifying maximum or minimum points.

• To find where the tangent is horizontal, we set the derivative equal to zero: \[-16x^3 + 3x^2 = 0\].
• Solving this equation gives us the x-values where the slope is zero.

In the given exercise, finding these points is crucial to pinpoint where the function flattens out, helping in understanding the function's overall shape and behavior.

Power Rule

The power rule is a basic technique in calculus used to find the derivative of polynomial functions. It simplifies the process of differentiation by following a straightforward formula: If \(f(x) = ax^n\), then the derivative is \(f'(x) = nax^{n-1}\).

• This rule was applied to each term of \(f(x) = -4x^4 + x^3\), leading to the derivative \(f'(x) = -16x^3 + 3x^2\).
• The power rule provides a quick and efficient way to find derivatives of power functions.

By using the power rule, the derivative computation becomes a predictable process, conducive to handling more complex functions easily.

Slope of Tangent

The slope of a tangent line to a graph at a given point is the value of the derivative at that point. It tells us whether the function is increasing, decreasing, or flat at that specific \(x\)-value.

• In the exercise, finding the x-values where the slope is zero allows us to determine where the tangent is horizontal.
• The slope can be positive, negative, or zero, indicating different behaviors of the function.

Understanding the slope of the tangent is important as it helps visualize the function's growth or decline and can indicate critical points of interest.

Graph of a Function

The graph of a function visually represents the relationship between the input \(x\) and the function's output \(y\). It helps in understanding the behavior of the function, such as increases, decreases, and flat or turning points.

• For \(f(x) = -4x^4 + x^3\), the graph can show where tangents are horizontal, maxima, or minima.
• Plotting both \(f(x)\) and its derivative \(f'(x)\) gives a comprehensive view of how the function behaves and changes.

Utilizing graphing tools to visualize the function often makes it easier to comprehend complex behavior and outcomes than purely algebraic methods. It provides a detailed perspective on how changes in the input reflect in the output curve of the function.