13-14 Find the number c that satisfies the conclusion of the...

$13-14$ Find the number $c$ that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at $(c, f(c)) .$ Are the secant line and the tangent line parallel?
$$f(x)=e^{-x}, \quad[0,2]$$

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus which guarantees that for any function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) where the instantaneous rate of change (i.e., the derivative) equals the average rate of change over [a, b]. This theorem forms a bridge between the overall behavior of a function and its local behavior at specific points.

Continuity and Differentiability

For the Mean Value Theorem to apply, a function must be continuous on the closed interval and differentiable on the open interval. Continuity ensures that the function has no breaks, jumps, or holes on the interval, while differentiability ensures that the function has a well-defined tangent (or instantaneous rate of change) at every point within the interval.

Secant Line

The secant line of a function over an interval [a, b] is the straight line connecting the endpoints of the function's graph at a and b. Its slope is calculated as (f(b) - f(a)) / (b - a) and represents the average rate of change of the function over that interval.

Tangent Line

The tangent line to a function at a point c is the line that best approximates the function at that specific point, having the same instantaneous rate of change as the function at c. According to the Mean Value Theorem, there exists a point where the slope of the tangent line equals the slope of the secant line across the entire interval.

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