6.2 Mean Value Theorem and Rolle's Theorem

The Mean Value Theorem and Rolle's Theorem are key tools in calculus. They help us understand how functions behave between two points, connecting average rates of change to instantaneous rates of change.

These theorems build on ideas of continuity and differentiability, showing there's always a point where a function's derivative equals its average rate of change. This concept is crucial for optimization and curve sketching in calculus.

Theorems

Mean Value Theorem and Rolle's Theorem

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• Mean Value Theorem states if a function $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c$ in $(a,b)$ such that [f'(c)](https://www.fiveableKeyTerm:f'(c)) = \frac{[f(b) - f(a)](https://www.fiveableKeyTerm:f(b)_-_f(a))}{b - a}
  • Geometrically, this means there is a point on the curve where the tangent line is parallel to the secant line through the endpoints $(a, f(a))$ and $(b, f(b))$
  • Helps to find the average rate of change of a function over an interval
• Rolle's Theorem is a special case of the Mean Value Theorem
  • If a function $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one point $c$ in $(a,b)$ such that $f'(c) = 0$
  • Geometrically, this means if a curve starts and ends at the same y-value, there must be at least one point where the tangent line is horizontal (slope = 0)
  • Can be used to find the critical points of a function

Function Properties

Continuity and Differentiability

• Continuity means a function has no breaks, gaps, or jumps in its graph
  • Formally, a function $f(x)$ is continuous at a point $a$ if $\lim_{x \to a} f(x) = f(a)$
  • Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper
  • Continuity is a necessary condition for differentiability and the application of the Mean Value Theorem and Rolle's Theorem
• Differentiability means a function has a well-defined derivative at each point in its domain
  • A function is differentiable at a point if its derivative exists at that point (i.e., the function has a non-vertical tangent line)
  • Differentiability implies continuity, but continuity does not imply differentiability (e.g., $f(x) = |x|$ is continuous but not differentiable at $x=0$)

Closed Intervals and Their Significance

• A closed interval includes its endpoints and is denoted by square brackets, e.g., $[a,b]$ means the interval from $a$ to $b$, including $a$ and $b$
  • In contrast, an open interval excludes its endpoints and is denoted by parentheses, e.g., $(a,b)$
• The Mean Value Theorem and Rolle's Theorem require the function to be continuous on a closed interval
  • This ensures the function is well-behaved and has no breaks or gaps within the interval of interest
  • Closed intervals also guarantee the existence of maximum and minimum values of a continuous function (Extreme Value Theorem)

Rates of Change

Intermediate Value Theorem and Its Applications

• The Intermediate Value Theorem states if a function $f(x)$ is continuous on the closed interval $[a,b]$ and $k$ is a value between $f(a)$ and $f(b)$, then there exists at least one point $c$ in $(a,b)$ such that $f(c) = k$
  • Intuitively, this means a continuous function takes on all values between $f(a)$ and $f(b)$ within the interval $[a,b]$
  • Can be used to prove the existence of roots or solutions to equations
  • Helps to understand the behavior of a function between two points
• Example: If $f(x)$ is continuous on $[0,2]$ with $f(0) = -1$ and $f(2) = 3$, then by the Intermediate Value Theorem, there must be a point $c$ in $(0,2)$ such that $f(c) = 0$ (i.e., the function crosses the x-axis)

Average Rate of Change and Its Relation to Secant Lines

• The average rate of change of a function $f(x)$ over the interval $[a,b]$ is given by $\frac{f(b) - f(a)}{b - a}$
  • Represents the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$
  • Provides information about how the function changes, on average, over the given interval
• The Mean Value Theorem guarantees the existence of a point $c$ in $(a,b)$ where the instantaneous rate of change (derivative) equals the average rate of change over $[a,b]$
  • This point corresponds to the tangent line parallel to the secant line through the endpoints
• Example: If the position of an object is given by $s(t) = t^2 + 2t$ for $0 \leq t \leq 3$, the average velocity (rate of change) over the interval $[0,3]$ is $\frac{s(3) - s(0)}{3 - 0} = \frac{15 - 0}{3} = 5$ units per second