Problem 24 - Calculus for Biology and Medicine

Question

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{-3 x}\) at \(a=0\)

Step-by-Step Solution

Answer

The linear approximation of \(f(x) = e^{-3x}\) at \(a = 0\) is \(1 - 3x\).

Step 1: Identify Function and Point

The given function is \(f(x) = e^{-3x}\). We need to find the linear approximation at the point \(a = 0\).

Step 2: Evaluate Function at Given Point

Calculate \(f(a)\). Since \(a = 0\), we have: \[f(0) = e^{-3\times0} = e^0 = 1\].

Step 3: Find the Derivative

Differentiate the function \(f(x) = e^{-3x}\). Using the chain rule, the derivative is: \[f'(x) = -3e^{-3x}\].

Step 4: Evaluate Derivative at Given Point

Calculate \(f'(a)\). Since \(a = 0\), we have: \[f'(0) = -3e^{-3\times0} = -3e^0 = -3\].

Step 5: Use Linear Approximation Formula

Substitute \(f(a)\), \(f'(a)\), and \(a\) into the linear approximation formula: \[f(x) \approx f(a) + f'(a)(x - a) = 1 + (-3)(x - 0)\].

Step 6: Simplify the Expression

Simplify the linear approximation expression: \[f(x) \approx 1 - 3x\].

Key Concepts

Understanding the Chain Rule

The chain rule is a critical technique in calculus for finding the derivative of composite functions. Composite functions are when one function is nested inside another, such as in our exercise where we have the function \(f(x) = e^{-3x}\). Here, the outer function is the exponential \(e^u\) and the inner function is \(-3x\).
The chain rule helps us differentiate complex functions by expressing the derivative of the outer function in terms of the inner function and multiplying it by the derivative of the inner function.

• Step 1: Differentiate the outer function, \(e^u\), with respect to \(u\), which is \(e^u\).
• Step 2: Multiply by the derivative of the inner function, \(-3\), since \(u = -3x\).

Applying this process, we find that the derivative of \(f(x)\) is \(-3e^{-3x}\), as seen in the exercise. This step is crucial for further calculations in finding the linear approximation.

Exploring Derivatives

Derivatives measure how a function changes as its input changes. They are fundamental in understanding rates of change and are used extensively in physics, engineering, and economics.
In our exercise, the function \(f(x) = e^{-3x}\) represents exponential decay, and the derivative \(f'(x) = -3e^{-3x}\) measures the rate of this decay. The negative sign indicates that the function decreases as \(x\) increases.

• Calculate the derivative using appropriate rules, such as the chain rule.
• Evaluate the derivative at specific points to understand the behavior of the function at those points.

In step 3 of our solution, we found the derivative and in step 4, we evaluated it at \(x=0\), yielding \(-3\). This tells us that at \(x=0\), the function is decreasing at a rate of 3 units per increase of 1 in \(x\).

The Role of Function Evaluation

Function evaluation is the process of finding the value of a function at a specific point. It allows us to gain insights into the behavior of the function at that point.
For the function \(f(x) = e^{-3x}\), evaluating it at \(a=0\) gives \(f(0) = 1\). This result tells us the exact value of the function at that point, which is critical for calculating linear approximations.

• Determine \(f(a)\) by substituting the point \(a\) into the function.
• Use this value to better understand the function's output at specific points.

The function evaluation at \(x=0\) is used in the linear approximation process to provide an initial value from which the approximation extends.

Understanding the Linear Approximation Formula

Linear approximation is a method used to approximate complex functions using linear functions, which are easier to analyze and compute. It is akin to zooming in on the function close to a particular point \(a\), so the curve of the function resembles a straight line.
The linear approximation formula is \(f(x) \approx f(a) + f'(a)(x-a)\). It uses:

• The value of the function at \(a\), \(f(a)\), which gives the starting point of the linear approximation.
• The derivative of the function at \(a\), \(f'(a)\), which provides the slope of the tangent line that closely resembles the function.
• \(x-a\), which indicates how far \(x\) is from \(a\).

In our example, substituting the values found in the exercise gives \(f(x) \approx 1 - 3x\). This approximation provides a simple linear expression that estimates the function \(f(x)\) near \(x=0\), making calculations easier and faster for small changes around this point.