Problem 55 - Calculus Single Variable

Question

In Exercises 55-58, use the given information to estimate \(f^{\prime}(c)\) at the given point \(c\) $$ f(4)=5.7 \text { and } f(4.1)=6.2, c=4 $$

Step-by-Step Solution

Answer

The estimated derivative \(f'(4)\) is 5.

Step 1: Understanding the Problem

We need to estimate the derivative of the function \(f\) at the point \(c=4\). The derivative at a point \(c\), denoted by \(f'(c)\), is the slope of the tangent line at that point. We can approximate this using the average rate of change of the function over the interval that includes \(c\).

Step 2: Identify Given Values

The problem provides the function values at two points: \(f(4) = 5.7\) and \(f(4.1) = 6.2\). These values will help us calculate the average rate of change around \(c = 4\).

Step 3: Apply the Difference Quotient

The difference quotient for estimating the derivative \(f'(c)\) is given by \[\frac{f(4.1) - f(4)}{4.1 - 4}\]. We use the given function values to find this quotient.

Step 4: Calculate the Difference Quotient

Substitute the given values into the difference quotient: \[\frac{6.2 - 5.7}{4.1 - 4} = \frac{0.5}{0.1}\].

Step 5: Simplify the Expression

The calculation from the previous step yields \(\frac{0.5}{0.1} = 5\). This is the estimated value of \(f'(4)\).

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus used to estimate the derivative of a function at a specific point. It is essentially a formula that helps us find the rate of change of the function over a small interval. The difference quotient is given by \[\frac{f(x+h) - f(x)}{h}\]where \(h\) is a small value, and \(x\) is the point at which you're calculating the derivative. In our example, \(x = 4\) and \(h = 0.1\).To estimate derivatives, we choose \(h\) to be a small increment to make the calculation accurate. The smaller the \(h\), the closer the difference quotient comes to the actual derivative.In the case of the provided exercise, using the difference quotient leads us to the approximate slope of the tangent at the point \(c = 4\). This gives us a very close estimation of \(f'(c)\).

Average Rate of Change

The average rate of change of a function over an interval gives us a sense of how the function behaves between two points. Calculating this is similar to finding the slope of a line that connects these two points.Think of it as knowing how far you've traveled within a period of time. It's a straightforward calculation, using the formula:\[\frac{f(b) - f(a)}{b - a}\]where \(a\) and \(b\) are the endpoints of the interval.In our problem:

• We have \(a = 4\) and \(b = 4.1\)
• The function values are \(f(4) = 5.7\) and \(f(4.1) = 6.2\)

This results in an average rate of change of \[\frac{6.2 - 5.7}{4.1 - 4} = 5\]which indicates how quickly the function's value is increasing or decreasing between the points \(x = 4\) and \(x = 4.1\).

Calculus Problems

Calculus problems often involve finding derivatives and rates of change. These problems can at first seem daunting, but with practice, they become approachable.When tackling a calculus problem:

• First, understand what needs to be found. Here, that's the derivative \(f'(c)\). This tells you how the function's value is changing at a specific point.
• Identify the information given. This could be values of the function at specific points or an equation of the function.
• Apply relevant concepts and formulas, such as the difference quotient or average rate of change, to find a solution.

Through consistent practice, you can develop intuition for solving calculus problems, making them simpler as you progress. A key is breaking down each problem into understandable steps.

Slope of Tangent Line

The slope of the tangent line at a point on a curve represents the instant rate of change of the function at that exact location. It is essentially the value of the derivative at that point. Calculating this slope provides an insight into how the function behaves at that specific point.In simpler terms, the slope of a tangent line tells us:

• How steep the curve is at a particular point
• Whether the function is increasing or decreasing at that moment
• How sensitive the function is to changes at that exact position

From the exercise, the slope of the tangent line at \(c = 4\) is estimated to be \(5\). This suggests the function is increasing at this point, with every small advancement in \(x\) leading to a relatively large increase in \(f(x)\). By understanding the slope of a tangent line, you gain valuable insights into the nature of a function's behavior at any given point.