91Ó°ÊÓ

In calculus, the derivative is a measure of how a function changes as its input changes. Derivatives are fundamental in finding the rate of change or slope of a curve at any given point.
In this problem, the deflection of the beam is represented by a polynomial function. To find where this deflection is maximized, we need to determine where the slope of the tangent to the polynomial is zero. This requires us to find the derivative of the polynomial function.To compute the derivative of a polynomial like \(f(x) = 2x^4 - 33x^3 + 1331x\), we apply the power rule, which states that the derivative of \(ax^n\) is \(nax^{n-1}\). Applying this rule, the derivative of our function is \(f'(x) = 8x^3 - 99x^2 + 1331\).
This new function, \(f'(x)\), gives us the rate of change of \(f(x)\) and is crucial in finding critical points where the function may reach maximum or minimum values.

Critical points are where the derivative of a function is zero or undefined. These points indicate potential locations for local maxima or minima, depending on the context.
For the function \(f(x) = 2x^4 - 33x^3 + 1331x\), we solve the equation \(f'(x) = 0\) to find critical points. Setting the derivative \(8x^3 - 99x^2 + 1331 = 0\) helps identify these points.
Critical points can be found by techniques such as factoring or using the Rational Root Theorem, which provides possible roots for polynomial equations with integer coefficients. Trying these possibilities, we find that one critical point is at \(x = 7\).
Determining the nature of critical points (whether they are maxima, minima, or points of inflection) usually involves further testing by evaluating the second derivative or comparing function values at critical points.

Polynomial functions are expressions that include terms in the form of coefficients and variables raised to whole number powers. Examples of polynomial functions involve basic algebraic equations and can range from simple, linear functions to complex, higher-degree equations incorporating various powers of a variable.
In this exercise, the polynomial function \(2x^4 - 33x^3 + 1331x\) represents the deflection of a beam. As a fourth-degree polynomial, it includes terms up to \(x^4\).
Such functions are continuous and smooth, making them ideal for determining maximum and minimum values using calculus. Specifically, they allow for derivatives to be easily calculated, aiding in the search for critical points and eventual optimization solutions.

The maximum value of a function refers to the highest point on its graph within a specified interval. In optimization problems, this concept is crucial for finding points of interest, such as where a structure's deflection might be the greatest.Having identified the critical point at \(x = 7\), the next step involves comparing the function values at this point and at the endpoints of the interval. In this scenario, evaluating the function \(f(x)\) at the endpoints \(x = 0\) and \(x = 11\)—both resulting in a deflection value of 0—confirms that neither endpoint provides a higher deflection than the critical point.
Calculating \(f(7)\) yields 13141, indicating that the maximum deflection occurs at \(x = 7\) meters. By comparing these values, we determine that the beam's deflection is greatest at this specific point, highlighting how calculus aids in optimizing real-world scenarios.