Problem 36 - Calculus for Biology and Medicine

Question

. Suppose \(f(x)=e^{x}, x \in[0,1]\). (a) Find the slope of the secant line connecting the points \((x, y)=(0,1)\) and \((1, e) .\) (b) Find a number \(c \in(0,1)\) such that \(f^{\prime}(c)\) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in \((0,1)\).

Step-by-Step Solution

Answer

(a) Slope = \(e - 1\). (b) \(c = \log(e-1)\), exists by Mean Value Theorem.

Step 1: Identify Points

Identify the points on the curve based on given values. From \((x, y)=(0, 1)\) and \((1, e)\), note that the coordinates are \((0, 1)\) and \((1, e)\).

Step 2: Calculate Slope of Secant

Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find the slope between \((0, 1)\) and \((1, e)\). This results in \(\frac{e - 1}{1 - 0} = e - 1\).

Step 3: Derivative of Function

Find the derivative of the function \(f(x) = e^x\). The derivative is \(f'(x) = e^x\).

Step 4: Equating Derivative to Secant Slope

Equate the derivative \(f'(c) = e^c\) to the slope of the secant line \(e - 1\). Set \(e^c = e - 1\).

Step 5: Solving for c

Take the natural logarithm on both sides to solve for \(c\). Thus, \(c = \log(e-1)\).

Step 6: Verify c in Interval

Since \(0 < e-1 < e\), \(\log(e-1)\) must lie in the interval \((0,1)\), thus verifying \(c \in (0, 1)\).

Step 7: Application of Mean Value Theorem

The Mean Value Theorem guarantees that there exists a \(c \in (0, 1)\) such that \(f'(c) = \frac{f(1) - f(0)}{1 - 0}\), confirming that a \(c\) exists within the interval.

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus, particularly useful for understanding the behavior of differentiable functions over an interval. Suppose you have a continuous function on a closed interval \([a, b]\), which is also differentiable on the open interval \( (a, b) \). The Mean Value Theorem states that there exists at least one point \((c)\) in the interval where the instantaneous rate of growth of the function—its derivative—is equal to the average rate of change over that interval.
This theorem can be visualized as guaranteeing that somewhere between \(a\) and \(b\), the tangent to the curve at that point is parallel to the secant line connecting \( (a, f(a)) \) and \((b, f(b))\). In simpler terms:

• The instantaneous rate of change (derivative) equals the average rate of change (slope of secant line).
• The existence of such a point \(c\) is guaranteed provided the conditions of the function being continuous and differentiable are met.

In the given problem for \(f(x) = e^{x}\) over \[0,1\], the function is both continuous and differentiable, which fulfils the conditions of the theorem.

Derivative

The derivative is a central concept in calculus and represents the rate at which a function is changing at any given point. More precisely, the derivative of a function \(f(x)\) at a point \((x_0)\) provides the slope of the tangent to the function's curve at that specific point. For exponential functions like \(f(x) = e^{x}\), this smooth curve is characterized by its derivatives.The process of finding a derivative is called differentiation. Here:

• For \(f(x) = e^x\), the derivative \(f'(x)\) is also \(e^x\), which means the slope of the tangent at any point \(x\) is \(e^x\) itself.
  
• This feature of exponential functions makes them unique as their rate of change is proportional to their value at any point.

Understanding derivatives helps in analyzing various phenomena such as growth rates modeled by exponential functions.

Exponential Functions

Exponential functions, such as \(f(x) = e^{x}\), are crucial in modeling diverse natural and biological processes, such as population growth, radioactive decay, and the spread of diseases.
These functions are defined by the constant base 'e', approximately equal to 2.71828, raised to a variable exponent. This constant is significant because it allows exponential functions to properly model continuous growth or decay.

• The unique property of \(f(x) = e^x\) is that its derivative is identical to the function itself, making calculations involving rates of change straightforward.
• This allows for easy interpretation in real-world contexts—for instance, if a population grows at a rate proportional to its current size, \(e^x\) function models it perfectly.

In summary, understanding exponential functions enables more profound insights into processes that exhibit exponential growth or decay, which is essential in fields like biology and medicine.