Problem 9 - Calculus for Biology and Medicine

Question

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(\ln (1.01)\)

Step-by-Step Solution

Answer

Using linear approximation, \( \ln(1.01) \approx 0.01 \), versus calculator: \( \approx 0.00995 \).

Step 1: Understand the Formula

The formula provided is the linear approximation or tangent line approximation, which is generally expressed as \( f(x) \approx f(a) + f'(a)(x-a) \). This formula uses the first derivative to approximate the value of a function near a point \( a \).

Step 2: Select a Suitable a Value

For the function \( f(x) = \ln(x) \) and the value \( x = 1.01 \), we need to select \( a \) close to \( x \). Choosing \( a = 1 \) is convenient because \( \ln(1) = 0 \) and the derivative is simple to calculate.

Step 3: Calculate f(a) and f'(a)

For \( f(x) = \ln(x) \), \( f(a) = \ln(1) = 0 \). The derivative \( f'(x) = \frac{1}{x} \), so \( f'(a) = f'(1) = 1 \).

Step 4: Apply the Approximation Formula

Substitute \( a = 1 \), \( f(a) = 0 \), \( f'(a) = 1 \), and \( x = 1.01 \) into the formula: \[ f(1.01) \approx 0 + 1 \times (1.01 - 1) = 0.01 \].

Step 5: Use a Calculator to Find the Actual Value

Calculate \( \ln(1.01) \) using a calculator. This typically equals approximately 0.00995.

Step 6: Compare the Results

The linear approximation gives us 0.01, while the calculator gives us approximately 0.00995. The approximation is quite close to the actual value.

Key Concepts

Derivative

A derivative represents the rate of change of a function with respect to a variable. Imagine you're tracking a car's speedometer as you drive; the speed shown is essentially a derivative, reflecting the rate of change of distance over time. In mathematics, derivatives give us a similar measure of how a function changes as its input varies.
For example, the derivative of the natural logarithm function, denoted as \( f(x) = \ln(x) \), is \( f'(x) = \frac{1}{x} \). This tells us how rapidly \( \ln(x) \) changes for small changes in \( x \).

• The derivative \( \frac{1}{x} \) reaches a higher value when \( x \) is small, meaning \( \ln(x) \) changes faster when \( x \) is smaller.
• For \( x = 1 \), \( f'(x) = 1 \), meaning a very small increase in \( x \) translates directly into an approximately identical change in the logarithmic value at that point.

Natural Logarithm

The natural logarithm, symbolized as \( \ln(x) \), is a logarithmic function that provides the inverse of the exponential function \( e^x \). It's a tool for transforming multiplicative processes into additive ones, simplifying complex calculations.

• \( \ln(x) \) is defined only for positive values of \( x \) since you cannot take the logarithm of a negative number or zero.
• The natural logarithmic curve rises slowly; it starts at \( \ln(1) = 0 \) because \( e^0 = 1 \), showing us every product of \( e \) results in a smaller growth of \( \ln(x) \).

The logarithm is particularly helpful when dealing with exponential growth patterns or when you need to log-transform data for statistical analysis. It smooths out exponential trends, making them easier to work with.

Tangent Line Approximation

Tangent line approximation, also known simply as linear approximation or linearization, helps estimate the value of a function at a given point using the function's derivative. Imagine standing at a specific point on a curve and examining it with a locally straight (linear) perspective. This local straight view is what the tangent line offers.
The approximation formula is:\[ f(x) \approx f(a) + f'(a)(x-a) \]This formula uses the slope of the tangent line, which is the derivative \( f'(a) \), to project the function's value near \( a \).

• By substituting values and calculating using the formula \( f(x) \approx f(a) + f'(a)(x-a) \), you can get an approximate value without needing a calculator.
• Although it simplifies problems, the approximation is less precise further from point \( a \). So always compare your results with exact values when necessary.

This method is beneficial in various real-world applications where minute changes require quick approximations without needing complex calculations.