91Ó°ÊÓ

Understanding differentiable functions is integral when studying the behavior of curves, especially in calculus. A function is said to be differentiable at a point if it has a derivative there, which means that there is a well-defined tangent to the curve at that point. Graphically, differentiability ensures that the curve has no sharp corners or breaks at the point in question.

For a function to be differentiable over an interval, it must be differentiable at every point within that interval. This means that the function is smooth and continuous, allowing for the computation of slopes at every point. In the context of the exercise provided, for functions like f(x) = x^2, f(x) = √x, and f(x) = 1/x, the differentiability enables the calculation of tangent lines and their point of intersection which is crucial for such problems. Differentiability also implies continuity, but the reverse is not always true; a function may be continuous but not differentiable at certain points. For instance, the absolute value function is continuous everywhere but not differentiable at zero.

The Mean Value Theorem (MVT) is a fundamental theorem in differential calculus that provides a connection between derivatives and differences of values of a function. It essentially states that for any differentiable function that is continuous on a closed interval [a, b], there exists at least one point c in the open interval (a, b) such that the derivative at c is equal to the average rate of change of the function over the interval from a to b.

The formula given by MVT is f'(c) = (f(b) - f(a)) / (b - a). This theorem is akin to saying that there is at least one point at which the instantaneous rate of change (slope of the tangent) matches the average rate of change (slope of the secant joining (a, f(a)) and (b, f(b))). In the exercise, the MVT could be used to verify the existence of a tangent line with a specific slope that validates the conditions of the tangent intersection points for different functions.

Derivatives represent the rate at which a function is changing at any given point and are fundamental in calculus for studying all kinds of changes. The derivative of a function at a point is visually represented by the slope of the tangent line to the function's graph at that point. Derivatives can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule, depending on the complexity of the function.

In the context of the exercise, the power rule is used for finding the derivatives of functions like f(x) = x^2 and f(x) = 1/x. This rule states that if f(x) = x^n, then f'(x) = nx^(n−1). For example, the derivative of f(x) = x^2 is f'(x) = 2x. Similarly, derivatives help us find the equations of the tangent lines at points P and Q and subsequently allow us to determine the intersection point of these tangents, leading us to the value of c in the various scenarios presented in the exercise.