Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose \(f\) is a function satisfying \(f(3)=8\) and \(f^{\prime}(3.05)=\frac{1}{4} .\) Use this information to approximate \(f(3.05) .\)

Step-by-Step Solution

Answer

The approximate value of \( f(3.05) \) is 8.0125.

Step 1: Identify the Given Information

We know that \( f(3) = 8 \) and the derivative at a nearby point is \( f'(3.05) = \frac{1}{4} \). We need to find an approximate value for \( f(3.05) \).

Step 2: Understand the Role of the Derivative

The derivative \( f'(3.05) = \frac{1}{4} \) represents the rate of change of the function. This implies how much \( f(x) \) changes as \( x \) changes from 3 to 3.05, approximately in a linear manner in a small neighborhood.

Step 3: Use Linear Approximation Formula

The linear approximation formula is given by:\[ f(x) \approx f(a) + f'(a) \cdot (x - a) \]where \( a = 3 \) and \( x = 3.05 \).

Step 4: Substitute the Values into the Formula

Substitute \( a = 3 \), \( x = 3.05 \), \( f(3) = 8 \), and \( f'(3) = \frac{1}{4} \) into the linear approximation formula:\[ f(3.05) \approx f(3) + f'(3) \cdot (3.05 - 3) \]

Step 5: Calculate the Approximation

Calculate each component of the formula:- Calculate the difference: \( 3.05 - 3 = 0.05 \)- Multiply the derivative by the difference: \( \frac{1}{4} \times 0.05 = 0.0125 \)- Add this result to \( f(3) = 8 \):\[ f(3.05) \approx 8 + 0.0125 = 8.0125 \]

Step 6: Write the Final Answer

The approximate value of \( f(3.05) \) is 8.0125.

Key Concepts

Derivative

The derivative of a function at a certain point tells us how the function's value changes as we make tiny changes in the input value, typically known as "rate of change." When we say the derivative of a function at a specific point, such as \( f'(3.05) = \frac{1}{4} \), it implies the slope of the tangent line at that point on the curve of the function. In simpler terms, it's like the function's speed at that instant.
This numerical value of \( \frac{1}{4} \) means every slight increase in the input results in a quarter of that increase in the output.

• If the input increases slightly by 1 unit, the output increases by 0.25 units.
• A negative derivative indicates the function's value decreases with an increase in input.

This concept of derivative is essential in understanding how functions behave locally, and is used heavily in calculus and real-world applications.

Rate of Change

"Rate of Change" is a valuable concept that gives us an understanding of how much one quantity changes with respect to another. In functions, it explains how quickly the function's output is increasing or decreasing as its input changes.
In the context of linear approximation, we use the rate of change at a known point to predict how the function behaves near that point. Hence, the rate of change (or the derivative) at \( x = 3.05 \) helps us approximate \( f(3.05) \). By knowing that the rate of change is \( \frac{1}{4} \), we predict that a small shift from \( x = 3 \) to \( x = 3.05 \) results only in a modest 0.0125 increase in the function's value (as calculated using linear approximation).
Understanding rate of change is crucial for interpreting graphical behavior and practical applications like motion, economy, and physics.

Approximation Formula

The linear approximation formula is a powerful tool in calculus that allows us to estimate the value of a function near a given point using its derivative. When we do not have explicit data on a function at a particular point, we can use this formula to make an educated guess.
The approximation formula is:\[ f(x) \approx f(a) + f'(a) \cdot (x - a)\]

• \( f(a) \) represents the function's value at a known point \( a \).
• \( f'(a) \) is the rate of change of the function at point \( a \).
• \( (x - a) \) represents the small difference of \( x \) from \( a \).

For our problem, \( a \) was 3, \( x \) was 3.05, and using the formula, we derived \( f(3.05) \approx 8 + 0.0125 = 8.0125 \). This formula simplifies the process of function estimation without needing comprehensive calculations across each interval.

Function Analysis

Function analysis involves understanding the behavior, continuity, and differentiability of functions. Analyzing a function means looking at its various attributes, how it shifts, transforms, and the impact of changes on its graph.
When working to approximate \( f(3.05) \) for a function \( f \), we were able to analyze a segment of this larger function. By using function values and derivatives at a point, we assess a segment's behavior and provide trustworthy estimations. This boils down to studying vital characteristics such as:

• Behaviour within intervals and at particular points
• Symmetry and asymmetry within broader analysis
• Local and global extrema, inflection points

Through systematic analysis, we solve complex problems and better predict how small changes affect our outputs. These skills are crucial for comprehending the broad and local dynamics of functions.