Problem 58 - Calculus Single Variable

Question

Use the given information to estimate \(f^{\prime}(c)\) at the given point \(c\) $$ f(3.47)=2.61 \text { and } f(3.49)=2.67, c=3.48 $$

Step-by-Step Solution

Answer

The estimated derivative \( f'(3.48) \) is approximately 3.0.

Step 1: Understand the Problem

We are given two values of a function, namely \( f(3.47) = 2.61 \) and \( f(3.49) = 2.67 \), and we need to estimate the derivative of the function \( f \) at the point \( c = 3.48 \). The derivative \( f'(c) \) represents the rate of change or the slope of the tangent line to the curve at the point \( c \).

Step 2: Approximation Using Difference Quotient

To estimate the derivative at \( c = 3.48 \), we can use the symmetric difference quotient, which is defined as:\[f'(c) \approx \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]where \( x_1 = 3.47 \) and \( x_2 = 3.49 \). This method uses both points on either side of \( c \) to get a better approximation.

Step 3: Calculate the Difference

Compute \( f(3.49) - f(3.47) \):\[f(3.49) - f(3.47) = 2.67 - 2.61 = 0.06\]

Step 4: Calculate the Difference in Points

Compute \( 3.49 - 3.47 \):\[3.49 - 3.47 = 0.02\]

Step 5: Calculate the Derivative Approximation

Now divide the difference in function values by the difference in \( x \) values:\[f'(3.48) \approx \frac{0.06}{0.02} = 3.0\]This gives us the estimated rate of change of \( f \) at \( c = 3.48 \).

Key Concepts

Difference Quotient

The difference quotient is a valuable tool in calculus for estimating the derivative of a function at a particular point. It helps us understand how a function behaves as we move slightly from one point to another. In essence, it gives us the slope of the secant line, which is a rough approximation of the instantaneous rate of change, or the derivative.

Mathematically, the difference quotient is expressed as:

• For a function \( f \) at points \( x_1 \) and \( x_2 \), the difference quotient is \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).
• This formula takes two values of the function, \( f(x_2) \) and \( f(x_1) \), and divides the change in function values by the change in the \( x \)-values.

Using the difference quotient helps us approximate the derivative at a point close to these \( x \)-values, especially when the points are close to the desired point of estimation. The smaller the distance between \( x_1 \) and \( x_2 \), the more accurate the approximation of the derivative.

Rate of Change

The concept of the rate of change is central to understanding calculus and functions. It describes how one quantity changes in relation to another. In the context of derivatives, it specifically refers to how a function's output values change as its input values change.

In practical terms, the rate of change is the same as the slope we find using the difference quotient. When approximating a derivative using two points:

• The rate of change tells us how steep the function is around those points.
• A positive value indicates the function is increasing, while a negative value shows it's decreasing.

For instance, in our given exercise, we found that \( f'(3.48) \approx 3.0 \), meaning the function is increasing at a rate of 3 units of \( f(x) \) for every 1 unit increase in \( x \) at \( c = 3.48 \). This gives a clear picture of how the function behaves near that point.

Slope of Tangent Line

Finding the slope of the tangent line at a specific point on a curve is a classic problem in calculus, representing the derivative at that point. The tangent line is a straight line that just "touches" the curve at a particular point without crossing it.

Here's why it matters:

• The slope of this tangent provides the best linear approximation of the function around that point.
• If the slope is positive, the function is rising at that point; if negative, it is falling.
• In our case, the slope of the tangent line at \( c = 3.48 \) is estimated to be 3.0.

This slope tells us the function's immediate tendency at that point, capturing the essence of how quickly and in what direction the function changes precisely at that point.