Question

Given \(f(7)=26\) and \(f(8)=22,\) approximate \(f^{\prime}(7)\).

Short answer

The approximate derivative \( f'(7) \) is \(-4\).

Step-by-step solution

Understand the Problem

We are given the values of a function at two points, specifically \( f(7) = 26 \) and \( f(8) = 22 \). We need to approximate the derivative of the function at \( x = 7 \), which is \( f'(7) \). The derivative represents the rate of change of the function at that point.

Apply the Formula for the Derivative Approximation

To approximate \( f'(7) \), we can use the formula for the average rate of change between two points: \[f'(x) \approx \frac{f(x+h) - f(x)}{h}\] where \( h \) is the difference in the \( x \)-values. In this case, \( x = 7 \) and \( x + h = 8 \), so \( h = 1 \).

Substitute Values into the Formula

Substitute the given values into the formula: \[f'(7) \approx \frac{f(8) - f(7)}{8 - 7} = \frac{22 - 26}{1}\]Calculate the result.

Calculate the Approximation

Perform the calculation to find:\[f'(7) \approx \frac{22 - 26}{1} = \frac{-4}{1} = -4\]Thus, the approximate value of \( f'(7) \) is \(-4\).

Key concepts

Rate of Change

The concept of "rate of change" is an essential idea in calculus and applied mathematics, describing how one quantity changes in relation to another. In mathematical terms, it measures the change in one variable, usually called "dependent," concerning the change in another, usually called "independent." This is widely represented by the derivative in calculus.
This concept is crucial for understanding how systems evolve over time, such as the speed of a moving car (the rate of change of distance with respect to time) or the growth of a population.

• When considering a straight line, the slope of the line represents the rate of change.
• The derivative, denoted as \( f'(x) \), is the rate of change of the function \( f \) at a specific point \( x \).

In the context of our exercise, the rate of change tells us how quickly the function \( f \) is increasing or decreasing as we move from \( x = 7 \) to \( x = 8 \).

Average Rate of Change

The average rate of change provides a simple method to calculate how a function changes over an interval. Unlike the instantaneous rate of change, which considers a single point, the average rate of change looks at the overall change between two points.
Calculated using \[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]where \( f(x_1) \) and \( f(x_2) \) are the function values at two points \( x_1 \) and \( x_2 \). This formula essentially gives the slope of the secant line connecting the two points on the graph of the function.

• This method is particularly useful when finding derivatives analytically is challenging or impossible.
• It offers an estimated rate of change over an interval in which the function operates.

In our example, evaluating between the points at \( x = 7 \) and \( x = 8 \) means the average rate of change is computed as \(\frac{f(8) - f(7)}{8 - 7} = \frac{22 - 26}{1} = -4.\)

Finite Difference Method

The finite difference method is a numerical tool used to estimate derivatives, particularly useful in scenarios where analytical derivatives are difficult to ascertain. This concept is foundational in fields such as engineering and computational mathematics where approximations are needed for complex equations.

• This method works by taking discrete intervals and points to approximate derivatives.
• The most basic version is the forward difference, calculated as \( \frac{f(x+h) - f(x)}{h} \).

The idea is fairly straightforward: we pick two points close to each other on the function and determine the difference in their \( y \)-values, divided by the difference in their \( x \)-values. By using very small intervals, the finite difference method delivers precise approximations of derivatives which reflect the function's behavior. In our problem, using \( h = 1 \) simplifies the calculation to provide an approximation for the rate of change at \( x = 7 \).
This method is extremely useful in approximating derivatives like in our example, where one finds \( f'(7) \approx -4 \) by applying the finite difference formula over a linear distance of one unit.