Problem 9 - Brief Applied Calculus

Question

Find the average rate of change of the given function between the following pairs of \(x\) -values. [Hint: See pages 95-96.] a. \(x=1\) and \(x=3\) b. \(x=1\) and \(x=2\) c. \(x=1\) and \(x=1.5\) d. \(x=1\) and \(x=1.1\) e. \(x=1\) and \(x=1.01\) f. What number do your answers seem to be approaching? $$ \text {} f(x)=x^{2}+x $$

Step-by-Step Solution

Answer

The average rates are approaching 3 as \( x \) values get closer to 1.

Step 1: Recall the Formula

The average rate of change of a function \( f(x) \) between two values \( x=a \) and \( x=b \) is given by \( \frac{f(b) - f(a)}{b-a} \). We'll use this formula to calculate the average rate of change for each pair of \( x \) values.

Step 2: Calculate \( f(x) \) Values for (a)

We need to find \( f(1) \) and \( f(3) \):\[ f(1) = 1^2 + 1 = 2 \]\[ f(3) = 3^2 + 3 = 12 \].

Step 3: Compute Average Rate for (a)

Using the formula from Step 1, compute \( \frac{f(3) - f(1)}{3-1} \):\[ \frac{12 - 2}{3-1} = \frac{10}{2} = 5 \].

Step 4: Calculate \( f(x) \) Values for (b)

Find \( f(2) \):\[ f(2) = 2^2 + 2 = 6 \].

Step 5: Compute Average Rate for (b)

Calculate \( \frac{f(2) - f(1)}{2-1} \):\[ \frac{6 - 2}{2-1} = 4 \].

Step 6: Calculate \( f(x) \) Values for (c)

Find \( f(1.5) \):\[ f(1.5) = (1.5)^2 + 1.5 = 2.25 + 1.5 = 3.75 \].

Step 7: Compute Average Rate for (c)

Calculate \( \frac{f(1.5) - f(1)}{1.5-1} \):\[ \frac{3.75 - 2}{1.5-1} = \frac{1.75}{0.5} = 3.5 \].

Step 8: Calculate \( f(x) \) Values for (d)

Find \( f(1.1) \):\[ f(1.1) = (1.1)^2 + 1.1 = 1.21 + 1.1 = 2.31 \].

Step 9: Compute Average Rate for (d)

Calculate \( \frac{f(1.1) - f(1)}{1.1-1} \):\[ \frac{2.31 - 2}{0.1} = \frac{0.31}{0.1} = 3.1 \].

Step 10: Calculate \( f(x) \) Values for (e)

Find \( f(1.01) \):\[ f(1.01) = (1.01)^2 + 1.01 = 1.0201 + 1.01 = 2.0301 \].

Step 11: Compute Average Rate for (e)

Calculate \( \frac{f(1.01) - f(1)}{1.01-1} \):\[ \frac{2.0301 - 2}{0.01} = \frac{0.0301}{0.01} = 3.01 \].

Step 12: Analyze Converging Values

Observe the average rates: 5, 4, 3.5, 3.1, 3.01. These values are approaching 3, which suggests that the instantaneous rate of change of the function at \( x=1 \) might be 3.

Key Concepts

Average Rate of Change

The concept of 'average rate of change' is central in calculus. It helps us understand how a function behaves over a given interval. Think of it as how much the function's output changes on average as you move from one point to another along the curve of the function. For a function like our example, \( f(x) = x^2 + x \), the average rate of change gives insight into how quickly the values of the function are rising or falling between two specific \(x\) values.

To compute the average rate of change between two points \(x = a\) and \(x = b\), you apply the formula:

• \( \frac{f(b) - f(a)}{b-a} \)

This formula is akin to finding the slope of a straight line connecting the two points on the function graph. By applying this to our given \(x\) values, we gradually see how the rate of change decreases as the interval becomes smaller, edging closer to a specific value.

Understanding this helps lay the groundwork for grasping more advanced concepts like instantaneous rate of change. It's like measuring your average speed over a trip when monitoring your car's odometer and the clock.

Instantaneous Rate of Change

As the name suggests, the 'instantaneous rate of change' takes the idea of average rate to a more precise level. Imagine wanting to find the rate at which a function is changing at exactly one point, not over an interval. This idea is crucial in calculus and is tied closely with the concept of derivatives.

In our exercise, as the \(x\) intervals\(\ (x = 1.5, 1.1, 1.01)\ \) become smaller, the average rate of change approaches a limit. This gives a hint at what the instantaneous rate of change might be at \(x = 1\). Mathematically, this is essentially the derivative of the function \(f(x)\) at that point.

The closer inspection of the rates approaching 3 as \(\Delta x\) (change in \(x\)) gets smaller, suggests that the instantaneous rate of change or the derivative at \(x = 1\) is 3. This means that at \(x = 1\), the function is increasing at the rate of 3 units for every unit increase in \(x\). It’s like checking your speedometer at a particular moment rather than calculating speed over a journey.

Function Analysis

Function analysis revolves around understanding the behavior of functions, which includes examining rates of change, among other aspects. With functions like \(f(x) = x^2 + x\), it's essential to dissect various aspects to comprehend their characteristics.

By analyzing rates of change, both average and instantaneous, we gain insight into how a function behaves at every point or interval. It's a bit like profiling a character in a story: understanding not only who they are at a specific moment but also how they change over the course of the story.

• The average rate shows us the pace of change over intervals.
• The instantaneous rate reveals immediate behavior at single points.

These analyses help us not just with solving immediate problems, but also lay a solid foundation for more complex calculus applications.

Mastering these concepts allows for exploratory studies in physics, economics, biology, and any other field where change is a constant factor. As we become adept at predicting and understanding function behavior, we better equip ourselves to solve real-world problems.