Problem 115 - Calculus and its applications

Question

Graph fand \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) $$ f(x)=x^{4}-x^{3} $$

Step-by-Step Solution

Answer

\(f'(1) = 1\)

Step 1: Differentiate the Function

We start by finding the derivative of the function \( f(x) = x^4 - x^3 \). The derivative is calculated using standard differentiation rules: \( f'(x) = 4x^3 - 3x^2 \). This new function \( f'(x) \) represents the slope of \( f(x) \) at any point \( x \).

Step 2: Evaluate the Derivative at x = 1

Next, we substitute \( x = 1 \) into the derivative \( f'(x) = 4x^3 - 3x^2 \) to find \( f'(1) \). Thus, \( f'(1) = 4(1)^3 - 3(1)^2 = 4 - 3 = 1 \).

Step 3: Graph the Function f(x)

When graphing \( f(x) = x^4 - x^3 \), note that it is a polynomial of degree 4. This kind of function typically has a general 'w'-shaped curve due to its degree. Identify critical points through its derivative, which may help sketch the graph.

Step 4: Graph the Derivative f'(x)

Graph \( f'(x) = 4x^3 - 3x^2 \). This cubic function helps visualize where the slope of \( f(x) \) is positive, negative or zero. Critical points in \( f(x) \) will correspond to where \( f'(x) \) crosses the x-axis.

Key Concepts

Polynomial functions

Polynomial functions like \( f(x) = x^4 - x^3 \) belong to a family of functions defined by expressions made up of terms composed of variables raised to powers, usually whole numbers. The degree of a polynomial is determined by the highest power of the variable within the expression, and in this case, the degree is 4, characterized as a quartic polynomial.
Addition and subtraction connect these terms, giving polynomials a distinctive algebraic structure.
Understanding polynomial functions is crucial for analyzing behaviors such as growth rate and end behavior. In this function:

• The term \( x^4 \) dominates the behavior due to its highest degree, influencing the growth of \( f(x) \) as \( x \) becomes large.
• The function will have a general 'W' shape, typical for quartic polynomials, with possibly complex intersections and turning points.

Mastering polynomial functions sets the groundwork for exploring more intricate mathematical models that model naturally occurring patterns and phenomena.

Graphical interpretation

The graphical interpretation of polynomial functions can reveal numerous important characteristics of the function itself, such as its curve, turning points, and intersections with axes.
For the function \( f(x) = x^4 - x^3 \), plotting its graph can help us visualize how it behaves:

• The graph will show a fourth-degree curve, featuring 3 potential turning points where the direction of change of the function alters.
• The endpoints extend towards positive infinity due to the positive leading coefficient of \( x^4 \).
• The function crosses the x-axis at the points where \( f(x) = 0 \), which are critical for determining root values.

Utilizing a graph to interpret the function provides visual insights into the rate of change and can help identify regions of increase and decrease, offering a complete picture of its geometry and algebraic properties. This enhances comprehension and is beneficial for visual learners.

Critical points

In the study of calculus, critical points are key as they indicate positions where the derivative of the function equals zero or is undefined, lending insights into local minima, maxima, or potential inflection points.
For \( f(x) = x^4 - x^3 \), the critical points are identified by setting the derivative \( f'(x) = 4x^3 - 3x^2 \) to zero:
- \( 4x^3 - 3x^2 = 0 \)
This simplifies to:
- \( x^2 (4x - 3) = 0 \)
Resulting in the solutions:

• \( x = 0 \)
• \( x = \frac{3}{4} \)

These points suggest positions where the function may transition from increasing to decreasing, or vice versa. Evaluating the sign change of \( f'(x) \) around these critical points further aids in determining their nature—either as peaks (maxima), troughs (minima), or neither.
Through critical points, we gain valuable insights into the structure and behavior of the polynomial function, which is pivotal in both theoretical and applied contexts.