Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Rearrange the formula, Equation 12.14, \(f(b)=f(a)+f^{\prime}\left(c_{0}\right)(b-a)\) and show that it is the a statement of the Mean Value Theorem.

Step-by-Step Solution

Answer

The equation is a statement of the Mean Value Theorem by expressing the derivative as the average rate of change.

Step 1: Identify Expression

The given equation is \( f(b) = f(a) + f'(c_0)(b-a) \). This equation represents an approximation of \( f(b) \) using the value of the function and its derivative at some point \( c_0 \) between \( a \) and \( b \).

Step 2: State the Mean Value Theorem

The Mean Value Theorem (MVT) states that 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 the interval \((a, b)\) such that \( f'(c) = \frac{f(b) - f(a)}{b-a} \).

Step 3: Match the Equation to MVT

Rearrange the given equation as follows: \( f'(c_0) = \frac{f(b) - f(a)}{b-a} \). This is the formulation in the MVT, showing that \( f'(c_0) \) is the average rate of change over \([a, b]\), which matches the MVT formula: \( f'(c) = \frac{f(b) - f(a)}{b-a} \).

Step 4: Conclusion

The rearranged equation \( f'(c_0) = \frac{f(b) - f(a)}{b-a} \) directly corresponds to the Mean Value Theorem, where \( c_0 \) is some point within the interval \((a, b)\). Thus, the given equation is a statement of the Mean Value Theorem.

Key Concepts

Differentiability

When we talk about differentiability in mathematics, we are referring to a function's ability to have a derivative at any given point on its curve. A function is differentiable at a point if it’s smooth enough and does not have any sharp corners or discontinuities at that point. Differentiability implies that there is a tangent line to the curve, which represents the instantaneous rate of change of the function at that specific point. Essentially, if you zoom in closely on a differentiable function at any point, it practically looks like a straight line.

• A function that's differentiable over an interval is easy to predict and analyze because it behaves nicely without jumps or abrupt bends.
• If a function is differentiable throughout an interval, then it is also continuous in that interval, though the reverse is not always true.

Therefore, for a function to satisfy the Mean Value Theorem, it needs to be differentiable on the open interval mentioned.

Average Rate of Change

The concept of the average rate of change is significant, especially when dealing with the Mean Value Theorem. It essentially measures how much the value of a function changes on average between two points. The average rate of change between two points, say \(a\) and \(b\), is calculated with the formula: \[\text{Average Rate of Change} = \frac{f(b) - f(a)}{b-a}.\]This formula is incredibly useful as it gives a simple expression for assessing overall change in a function across a closed interval.

• This is different from instantaneous rate of change, which is what differentiability deals with at specific points.
• The Mean Value Theorem uses this concept to assert that there's at least one point where this average is exactly equal to the instantaneous rate of change.

So in essence, the average rate of change over a closed interval helps connect points over larger spans, showcasing the overall behavior of a function.

Closed Interval

A closed interval in mathematics refers to an interval that includes its endpoints. For example, the closed interval \[ [a, b] \] includes all values from \(a\) to \(b\), inclusive of \(a\) and \(b\) themselves. Closed intervals are crucial in the context of the Mean Value Theorem because it ensures boundaries are well-defined and included. The Mean Value Theorem requires the function to be continuous over this closed interval, \[ [a, b] \]. This means there should be no breaks, holes, or jumps within these boundaries.

• Continuity on a closed interval is an essential prerequisite for ensuring that the function behaves nicely within these endpoints.
• Even though the function must be differentiable on the open interval \(a, b\), being continuous on the closed interval \[ [a, b] \] helps in establishing the existence of the average rate of change within those endpoints.

Simply put, closed intervals help encapsulate our defined area in function analysis, particularly under the Mean Value Theorem.