Problem 23 - Calculus for Biology and Medicine

Question

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{-x}\) at \(a=0\)

Step-by-Step Solution

Answer

The linear approximation of \( f(x) = e^{-x} \) at \( a = 0 \) is \( 1 - x \).

Step 1: Identify the function and the point of approximation

We have the function \( f(x) = e^{-x} \) and we want to compute its linear approximation around \( a = 0 \). This means we need to find \( f(a) \), \( f'(a) \), and substitute these into the linear approximation formula.

Step 2: Calculate f(a)

Substitute \( a = 0 \) into the function to find \( f(a) \). Thus, \( f(0) = e^{0} = 1 \).

Step 3: Differentiate the function

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

Step 4: Calculate f'(a)

Substitute \( a = 0 \) into the derivative to find \( f'(a) \). Thus, \( f'(0) = -e^{0} = -1 \).

Step 5: Use the linear approximation formula

Substitute \( f(a) = 1 \), \( f'(a) = -1 \), and \( a = 0 \) into the approximation formula: \[f(x) \approx f(a) + f'(a)(x - a)\]\[f(x) \approx 1 - 1(x - 0)\]\[f(x) \approx 1 - x\]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, focusing on how a function changes as its input varies. It involves finding the derivative, which is an expression that gives the rate of change of a function with respect to one of its variables. The process of differentiation enables us to understand how the function behaves locally, by examining its slope or gradient.

When you differentiate a function, such as \( f(x) = e^{-x} \), you apply rules like the chain rule, product rule, or quotient rule to find its derivative. In this particular example, differentiating \( e^{-x} \) involves using the basic rule for the derivative of an exponential function and multiplying it by the derivative of the exponent (-1 in this case).

• Derivatives provide insights into the function's behavior and are essential for linear approximations.
• Understanding differentiation helps in predicting trends and making approximations.

Exponential Function

An exponential function is characterized by its constant base raised to a variable exponent. These functions take the form \( f(x) = a^x \), where "a" is a positive constant. In the case of \( f(x) = e^{-x} \), the base "e" is Euler's number, approximately 2.71828, which is a special mathematical constant.

Exponential functions are known for their rapid growth or decay. The function \( e^x \) continuously increases as \( x \) increases, whereas \( e^{-x} \) decreases. This is crucial when computing linear approximations because it involves understanding how the function behaves around a specific point, like \( a = 0 \) in this exercise.

• Exponential decay functions, like \( e^{-x} \), are used in modeling processes that decrease quickly.
• These functions are crucial in various fields, including physics, biology, and finance.

Derivative

The derivative of a function represents the rate at which the function’s value changes as the input changes. It's a key tool in calculus for analyzing and predicting the behavior of functions. For the function \( f(x) = e^{-x} \), the derivative \( f'(x) = -e^{-x} \) provides us with a direct measure of how rapidly the function decreases.

To find this derivative, you apply the rule that the derivative of \( e^u \) with respect to \( x \) is \( e^u \cdot u' \). Applying this to \( -x \), we get \( f'(x) = -e^{-x} \). The negative sign indicates that for every increase in \( x \), the function decreases.

• Derivatives are essential for finding linear approximations, as they define the slope of the tangent at any point.
• Understanding the derivative function helps in making predictions about future behavior of the function.