91Ó°ÊÓ

In calculus, a continuous function is pivotal for understanding derivatives, integrals, and many theorems like the Mean Value Theorem (MVT) for Integrals. A function is continuous over an interval \( [a, b] \) if, intuitively, its graph can be drawn without lifting a pencil from the paper. There are no breaks, jumps, or holes.
For the Mean Value Theorem for Integrals to apply, the function must be continuous on the closed interval \( [a, b] \). This ensures that there will exist at least one value \( c \) in the interval where the function's average value equals the function value. In the given problem, \( f(x) = x(1-x) \), this quadratic function is continuous over the interval \( [0, 1] \), fulfilling the criteria for applying the theorem.

The definite integral is a fundamental concept in calculus that accumulates the values of a function over a specified interval. It represents the area under the curve of the function from one point to another. For a function \( f(x) \) with respect to \( x \), the definite integral over the interval \( [a, b] \) is denoted by \[ \int_{a}^{b} f(x) \, dx \].
To calculate a definite integral like \( \int_{0}^{1} x(1-x) \, dx \), you first find the antiderivative or indefinite integral. Then, you evaluate this antiderivative at the boundaries of the interval and find the difference.

• The process of calculating definite integrals often involves fundamental techniques of integration like substitution or integration by parts, but in this problem, direct polynomial integration suffices.
• In practice, definite integrals allow us to calculate physical properties like areas under curves, volumes, and other aggregated measures.

The quadratic formula is a powerful tool for finding solutions to quadratic equations, which are equations of the form \[ ax^2 + bx + c = 0 \]. These equations often arise in problems across various fields, including physics, engineering, and economics. The quadratic formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]- The term \( b^2 - 4ac \) is called the discriminant.- It determines the nature of the roots: * If positive, there are two distinct real solutions. * If zero, one real solution exists. * If negative, the roots are complex or imaginary.In the context of the original problem, finding \( c \) required solving a quadratic equation derived from equating the function value \( f(c) \) to the average value over the interval. The solutions provide the specific \( c \) values satisfying the Mean Value Theorem.

The indefinite integral, also known as an antiderivative, represents a family of functions whose derivative gives the original function. In symbol form, the indefinite integral of \( f(x) \) with respect to \( x \) is written as \[ \int f(x) \, dx \. \] An arbitrary constant \( C \) is always added to denote the constant of integration since differentiation of any constant is zero.- Finding the indefinite integral is akin to reversing the process of differentiation.- It provides the general form from which we can derive specific solutions by applying initial conditions or boundaries.- For example, the indefinite integral of \( x - x^2 \) would be \[ \frac{x^2}{2} - \frac{x^3}{3} + C \. \]In the given problem, calculating the indefinite integral is a step in determining the definite integral, which plays a critical role in solving the Mean Value Theorem for Integrals. The indefinite integral is evaluated over specific limits to find the definite integral.