Problem 26 - Calculus for Biology and Medicine

Question

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ . \(f(x)=e^{2 x+1}\) at \(a=-1 / 2\)

Step-by-Step Solution

Answer

The linear approximation is \( f(x) \approx 2x + 2 \).

Step 1: Identify the Function and Calculate the Derivative

We start with the function given, \( f(x) = e^{2x+1} \). To find the linear approximation, we first need to compute its derivative. The derivative of \( e^{u} \) with respect to \( x \), where \( u = 2x+1 \), is \( f'(x) = e^{u} \cdot \frac{d}{dx}(u) = e^{2x+1} \cdot 2 \). So, \( f'(x) = 2e^{2x+1} \).

Step 2: Evaluate the Function and Its Derivative at \( x = a \)

We need to evaluate both \( f(x) \) and \( f'(x) \) at \( a = -\frac{1}{2} \). First, calculate \( f(-\frac{1}{2}) = e^{2(-\frac{1}{2})+1} = e^{0} = 1 \). Then, calculate the derivative at this point: \( f'(-\frac{1}{2}) = 2e^{2(-\frac{1}{2})+1} = 2e^{0} = 2 \).

Step 3: Write the Linear Approximation Formula

The linear approximation formula is given by \( f(x) \approx f(a) + f'(a)(x-a) \). Substitute \( a = -\frac{1}{2} \), \( f(a) = 1 \), and \( f'(a) = 2 \) into the formula: \[ f(x) \approx 1 + 2(x + \frac{1}{2}) \].

Step 4: Simplify the Linear Approximation Formula

Simplify the expression: \( 1 + 2(x + \frac{1}{2}) = 1 + 2x + 1 = 2x + 2 \). Thus, the linear approximation is \( f(x) \approx 2x + 2 \).

Key Concepts

Derivative Calculation

Derivatives play a crucial role in understanding how functions behave and change. Here, we are working with the function

• \( f(x) = e^{2x+1} \)
• To find the derivative, we need to know the chain rule.

This rule helps us differentiate composite functions. For instance, for a function like \( e^u \), where \( u \) is a function of \( x \), the derivative with respect to \( x \) is \( e^u \cdot \frac{du}{dx} \).
In our case:

• \( u = 2x+1 \)
• The derivative \( \frac{du}{dx} = 2 \).

Multiplying these together, the derivative of \( f(x) = e^{2x+1} \) becomes \( f'(x) = 2e^{2x+1} \). This tells us how fast our function's value changes for small changes in \( x \).

Exponential Function

Exponential functions like \( e^{2x+1} \) are powerful tools in mathematics, known for their distinctive growth properties. By understanding these functions, you can predict how rapidly a quantity will increase. The base of the natural exponential function is \( e \), an irrational number approximately equal to 2.718.
The function \( e^{2x+1} \) specifically represents transformations and shifts of the basic exponential function:

• The term \( 2x \) indicates the stretching of the graph of \( e^x \) horizontally by a factor of 2.
• The additional constant \(+1\) shifts the entire graph vertically.

Such transformations are crucial in forming accurate models in scientific and financial contexts. They allow us to understand and predict behaviors like population growth and interest accrual.

Function Evaluation

Function evaluation helps us find specific values of the function and its derivatives at given points. In this exercise, the point is \( a = -\frac{1}{2} \).
To evaluate \( f(-\frac{1}{2}) \):

• Substitute \( -\frac{1}{2} \) into \( f(x) = e^{2x+1} \).
• Calculate \( f(-\frac{1}{2}) = e^{2(-\frac{1}{2})+1} = e^{0} = 1 \).

Similarly, for its derivative at \( a \):

• Evaluate \( f'(-\frac{1}{2}) = 2e^{0} = 2 \).

This evaluation tells us both the current value of the function at a particular point and the rate at which it changes there. By using these evaluated values, we construct the linear approximation, \( f(x) \approx 1 + 2(x + \frac{1}{2}) \), showing how the function behaves near a small neighborhood around the point \( a = -\frac{1}{2} \).