Problem 12 - Calculus - AP Edition

Question

Determine whether the statement is true or false. Explain your answer. If \(f\) is continuous on a closed interval \([a, b]\) and differentiable on \((a, b),\) then there is a point between \(a\) and \(b\) at which the instantaneous rate of change of \(f\) matches the average rate of change of \(f\) over \([a, b]\)

Step-by-Step Solution

Answer

True. It satisfies the Mean Value Theorem conditions.

Step 1: Identify the Given Scenario

The problem provides a function \( f \) that is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). We need to determine if there is a point where the instantaneous rate of change equals the average rate over this interval.

Step 2: Recall the Mean Value Theorem

The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \(c\) in \((a, b)\) where \(f'(c) = \frac{f(b) - f(a)}{b - a}\).

Step 3: Apply the Theorem to the Problem

Since \(f\) meets all conditions of the Mean Value Theorem, it guarantees the existence of at least one point \(c\) in \((a, b)\). At this point, the instantaneous rate of change \(f'(c)\) will be equal to the average rate of change \(\frac{f(b) - f(a)}{b - a}\). Therefore, the statement holds true.

Key Concepts

continuous function

A continuous function is one that does not have any breaks, jumps, or holes in its graph. This means you can draw it without lifting your pencil from the paper.

A function being continuous on a closed interval \([a, b]\) signifies that there are no abrupt changes from point \(a\) to point \(b\). Continuous functions are smooth and without interruptions within the given interval.

Being continuous is a crucial condition for many important mathematical theorems, like the Mean Value Theorem, because it ensures the stability and predictability of the function's behavior over an interval.

differentiable function

A differentiable function is one where a derivative exists at each point in its domain. Having a derivative means the function has a defined slope or rate of change at every point.

When we say a function is differentiable on an open interval \(a, b\), it means that for every point between \(a\) and \(b\), the function has a tangent line with a defined slope. This excludes the endpoints, focusing on the behavior inside the interval.

Differentiability implies continuity, but not the other way around. Therefore, a function must be continuous to be differentiable, but just being continuous doesn't ensure it is differentiable.

• Continuous function: No breaks, continuous line.
• Differentiable function: Continuous with a specific slope at each point.

average rate of change

The average rate of change of a function over an interval gives a simplified view of how a function behaves between two points. It is calculated by taking the difference in the function's values at two points and dividing by the difference in the points themselves, like so:

\[\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}\]

This formula provides a single slope value that connects the endpoints \(a\) and \(b\), akin to finding the slope of a straight line that passes through the two points on the function.

Understanding the average rate of change helps us get a general idea of the function's behavior over an interval. It does not, however, provide specific details or variations within the interval itself.