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

Step-by-Step Solution

Answer

The points are \((0, 0)\) and \(\left(\frac{3}{16}, 0.0016\right)\)."

Step 1: Find the Derivative

First, we find the derivative of the function, which gives us the slope of the tangent line to the graph at any point. The given function is \(f(x) = -4x^4 + x^3\). The derivative, \(f'(x)\), is found by applying the power rule: \[ f'(x) = \frac{d}{dx} (-4x^4 + x^3) = -16x^3 + 3x^2. \]

Step 2: Set the Derivative to Zero

Horizontal tangents occur where the derivative is zero. Setting the derivative equal to zero gives us the equation \[ -16x^3 + 3x^2 = 0. \]

Step 3: Factor the Derivative

Factor the left-hand side of the equation: \[ x^2(-16x + 3) = 0. \] This equation is true if either \(x^2 = 0\) or \(-16x + 3 = 0\).

Step 4: Solve for x

For \(x^2 = 0\), the solution is \(x = 0\). For \(-16x + 3 = 0\), solve for \(x\) to get \[ x = \frac{3}{16}. \]

Step 5: Find Corresponding y-Coordinates

Substitute the values of \(x\) back into the original function to find the \(y\)-coordinates. For \(x = 0\), \(y = f(0) = -4(0)^4 + (0)^3 = 0\). For \(x = \frac{3}{16}\), substitute into \(f(x)\) to find \[ y = -4\left(\frac{3}{16}\right)^4 + \left(\frac{3}{16}\right)^3. \] Convert to a common base to simplify: The calculation for \(y\) results in approximately \(0.0016\).

Step 6: Conclusion

Thus, the coordinates of the points where the tangent is horizontal are \((0, 0)\) and \(\left(\frac{3}{16}, 0.0016\right)\).

Key Concepts

Understanding Derivatives

In calculus, the concept of a derivative is essential. A derivative indicates how a function changes at any point, essentially providing the function's rate of change. When examining a function of the form \( y = f(x) \), its derivative \( f'(x) \) represents the slope of the tangent line to the graph at each point.
It tells us how steep the graph is and whether it is increasing or decreasing.

• If \( f'(x) > 0 \), the function is increasing at that point.
• If \( f'(x) < 0 \), the function is decreasing at that point.
• If \( f'(x) = 0 \), the function has a horizontal tangent at that point, suggesting a potential local maximum or minimum.

Understanding derivatives paves the way for deeper analysis. It makes tackling problems related to tangents and changes over an interval more methodical.

Horizontal Tangents

Horizontal tangents on a graph occur where the derivative of the function equals zero. This is because a horizontal line has a slope of zero. When searching for horizontal tangents of a given function, the goal is to find \( x \) values where \( f'(x) = 0 \).
These points often relate to local minima or maxima because the function's direction is changing.To identify these points:

• Compute the derivative, as previously discussed.
• Set the derivative function to zero and solve for \( x \).
• Verify by checking if these \( x \) values yield zeros at specific derivative values.

These steps help not only locate horizontal tangents but also make the analysis of a function's behavior more precise.

Applying the Power Rule

The power rule is a handy tool in calculus for finding derivatives of polynomials. It simplifies the process, making it intuitive. Given a term in the form \( ax^n \), the power rule states: the derivative is \( a\cdot n\cdot x^{n-1} \). This rule assists in efficiently finding derivatives without hassle.Here's how you apply it:

• Identify each term of the polynomial.
• Multiply the coefficient by the power.
• Subtract one from the power.

Example:If you have \( -4x^4 \), its derivative using the power rule is \( -16x^3 \). Similarly, for \( x^3 \), it becomes \( 3x^2 \). This process helps in quickly getting the derivative needed to solve equations for horizontal tangents.

Drawing Function Graphs

Graphing a function provides a visual representation of how the function behaves across different values of \( x \). It is invaluable when determining where tangents are horizontal or how the function progresses.Steps to sketch a function graph:

• Calculate key points such as intercepts, turning points, and points where the derivative is zero.
• Assess the behavior of the function as \( x \) tends towards positive and negative infinity.
• Analyze any symmetry, asymptotes, and periodic behavior if it applies.

For the function \( f(x) = -4x^4 + x^3 \), you'd plot points like (0,0) and use found derivatives to sketch where the graph levels out with horizontal tangents. This visual aid offers clarity on the overall behavior and relationships within the function.