Problem 171 - Calculus Volume 1

Question

Determine whether the Mean Value Theorem applies for the functions over the given interval \([a, b] .\) Justify your answer. $$ y=e^{x} \text { over }[0,1] $$

Step-by-Step Solution

Answer

The Mean Value Theorem applies because \(y = e^x\) is continuous and differentiable on \([0, 1]\).

Step 1: Confirm Continuity on [a, b]

The Mean Value Theorem requires that the function is continuous on the closed interval \([a, b]\). The function \(y = e^x\) is continuous everywhere because the exponential function does not have any points of discontinuity. Hence, \(y = e^x\) is continuous on \([0, 1]\).

Step 2: Confirm Differentiability on (a, b)

The Mean Value Theorem also requires that the function is differentiable on the open interval \((a, b)\). The derivative of \(y = e^x\) is \(\frac{dy}{dx} = e^x\), which exists for all \(x\). Hence, \(y = e^x\) is differentiable on \((0, 1)\).

Step 3: Conclusion about Mean Value Theorem

Since \(y = e^x\) satisfies both conditions of being continuous on \([0, 1]\) and differentiable on \((0, 1)\), the Mean Value Theorem applies for this function over the interval \([0, 1]\).

Key Concepts

Continuity of Functions

To determine if a function like the exponential function is subject to the Mean Value Theorem, we first check if it is continuous within the specified interval. Continuity means that there are no breaks, jumps, or vertical asymptotes in the graph of the function over that interval. For the function \( y = e^x \), which represents an exponential curve, it is continuous everywhere, without exceptions. This property holds because exponential functions smoothly extend over all real numbers without interruptions. Therefore, for any interval, including \([0, 1]\), \( y = e^x \) does not have any points of discontinuity. It moves seamlessly across the interval, meeting the continuity requirement needed for the Mean Value Theorem.

Differentiability of Functions

The next step in applying the Mean Value Theorem is to ensure the function is differentiable on a specific interval. Differentiability in mathematics refers to a function having a derivative at every point within an interval. Essentially, you should be able to create a tangent line at every point on the function's graph within the selected interval.For \( y = e^x \), the function is not only continuous but also differentiable across all real numbers. The derivative of \( y = e^x \) is \( \frac{dy}{dx} = e^x \). This derivative exists and is continuous everywhere, including over the open interval \((0, 1)\). Thus, \( y = e^x \) meets the differentiability requirement for applying the Mean Value Theorem.

Understanding Exponential Functions

Exponential functions are a cornerstone of mathematics due to their unique properties. The function \( y = e^x \) exemplifies exponential growth, where the rate of increase at any point is proportional to its current value. This characteristic makes it powerful for modeling scenarios such as population growth and compound interest.Consider some fundamentals about exponential functions:

• They have the form \( y = a \, e^{b\,x} \) where \( e \approx 2.71828 \), a mathematical constant.
• Exponential functions are always positive and never touch the x-axis.
• They exhibit consistent growth or decay, depending on the context, over every interval.

In applying the Mean Value Theorem to \( y = e^x \), its continuous and differentiable nature over the interval \([0, 1]\) ensures that there must exist at least one point \( c \in (0,1) \) where the derivative equals the average rate of change over the interval, confirming the theorem applies effortlessly to this function.