Problem 21 - Calculus for Biology and Medicine

Question

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=e^{x} \text { at } a=0 $$

Step-by-Step Solution

Answer

The linear approximation is \( L(x) = 1 + x \).

Step 1: Define the Function and Derivative

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

Step 2: Evaluate the Function at \( a = 0 \)

Since \( f(x) = e^x \), we substitute \( a = 0 \) to get \( f(0) = e^0 = 1 \).

Step 3: Evaluate the Derivative at \( a = 0 \)

Using the derivative \( f'(x) = e^x \), substitute \( a = 0 \) to get \( f'(0) = e^0 = 1 \).

Step 4: Write the Linear Approximation Formula

The linear approximation formula is \( L(x) = f(a) + f'(a)(x-a) \).

Step 5: Substitute into the Linear Approximation

Substitute \( f(0) = 1 \) and \( f'(0) = 1 \) into the approximation formula: \( L(x) = 1 + 1(x - 0) = 1 + x \).

Key Concepts

Understanding the Derivative

A derivative is a fundamental concept in calculus, representing the rate of change of a function. For any function, the derivative tells us how the function's output value changes as we slightly change the input value. When you imagine a curve on a graph, the derivative at a particular point gives the slope of the tangent line to that curve at that point. This slope is crucial because it provides insight into the behavior of the function.

The notation for a derivative is usually represented by \( f'(x) \) or \( \frac{df}{dx} \). In the context of our original exercise, understanding the derivative \( f'(x) = e^x \) helps us find how the exponential function \( e^x \) is increasing at any given x. When evaluating this at \( x = 0 \), we find \( f'(0) = e^0 = 1 \), indicating that the slope of the tangent line at this point is 1, meaning it rises at a consistent rate.

Function Evaluation - Finding Values at Specific Points

Function evaluation refers to the process of finding the value of a function at a certain input. To do this, you substitute the input value into the function's formula. This can help find specific outputs and understand the function's behavior better.

In our problem, we were tasked with evaluating the exponential function \( f(x) = e^x \) at the specific point \( a = 0 \). By substituting \( 0 \) for \( x \) in the function, we calculate \( f(0) = e^0 = 1 \). This tells us that the value of the function is 1 when the input is 0, which is a fundamental characteristic of the exponential function. Knowing this value is also key because it acts as the starting point for our linear approximation, acting as the y-intercept in the linear model formula.

Exploring the Exponential Function

The exponential function \( e^x \) is an important mathematical expression, widely used across different scientific fields. This function is unique because its rate of increase is proportional to its current value. In simpler terms, as \( x \) increases, \( e^x \) grows quickly, showing an exponentially increasing rate.

Some key properties of the exponential function include:

• Its value at \( x = 0 \) is always 1, as \( e^0 = 1 \).
• The function is never zero and never negative.
• The rate of growth is always positive, meaning the function is always increasing.

In our exercise, focusing on this function at \( x = 0 \) simplifies the linear approximation formula we derived, using the facts that \( e^0 = 1 \) both for the function and its derivative at that point. By understanding how \( e^x \) behaves, we can better approximate and predict values, especially close to specific points like \( a = 0 \).