Mean Value Theorem and Inequalities

These flashcards cover the key vocabulary and concepts related to the Mean Value Theorem and Inequalities as discussed in Lecture 14.

Mean Value Theorem (MVT)

States that if f is differentiable on (a, b) and continuous on [a, b], then f(b) - f(a) = f'(c)(b - a) for some c in (a, b).

Secant Line Slope

The average rate of change of the function f over the interval, defined as (f(b) - f(a)) / (b - a).

Tangent Line Slope

The instantaneous rate of change of the function at a point, represented as f'(c).

Differentiability

A function is differentiable at a point if it has a derivative at that point, implying it has a defined slope of the tangent line.

Increasing Function

A function f is increasing on an interval if for any a < b, f(a) < f(b).

Decreasing Function

A function f is decreasing on an interval if for any a < b, f(a) > f(b).

Constant Function

A function f is constant if f'(x) = 0 for all x in its domain.

Inequalities from MVT

The conclusions drawn from the MVT indicate the behavior of a function based on the sign of its derivative, specifically that positive derivative implies increase.

Example of MVT

If you travel 1,000 miles in 3 hours, at some point you must have traveled at exactly 333.33 miles per hour.

Linear Approximation

Estimates the value of a function f(x) using the derivative at a point a, expressed as f(x) ≈ f(a) + f'(a)(x - a).