1.10. Practice Exercises (for Students)

1. Find the average rate of change of 𝑓(𝑥) = 3𝑥² − 2𝑥 from 𝑥 = 1 to 𝑥 = 4.

To find the average rate of change (AROC) for the function 𝑓(𝑥) = 3𝑥² − 2𝑥 between the points where 𝑥 = 1 and 𝑥 = 4, you first need to calculate the value of the function at both points.

1. Calculate 𝑓(1): 𝑓(1) = 3(1)² − 2(1) = 3(1) - 2 = 1.
2. Calculate 𝑓(4): 𝑓(4) = 3(4)² − 2(4) = 3(16) - 8 = 48 - 8 = 40.
3. Now, plug these values into the AROC formula:
   AROC = (𝑓(4) - 𝑓(1)) / (4 - 1) = (40 - 1) / (3) = 39 / 3 = 13. This means that, on average, the function increases by 13 units for every 1 unit increase in 𝑥 over the interval from 1 to 4.

Imagine you're tracking how much water a tank fills over time. If you check the tank's water level at two specific times and notice it increased from 1 liter to 40 liters over a 3-hour period, the average rate of change tells you that, on average, about 13 liters of water filled the tank every hour.

2. Given 𝑠(𝑡) = 4𝑡² − 𝑡 + 1, find the instantaneous rate of change at 𝑡 = 2.

To determine the instantaneous rate of change (IROC) for 𝑠(𝑡) = 4𝑡² − 𝑡 + 1 at 𝑡 = 2, apply the concept of derivatives.

1. First, calculate the derivative of the function:
   𝑠'(𝑡) = 8𝑡 - 1.
2. Now, substitute 𝑡 = 2 into the derivative:
   𝑠'(2) = 8(2) - 1 = 16 - 1 = 15. This indicates the function is changing at a rate of 15 units per 1 unit change in 𝑡 at that specific moment.

Think about a racecar on a track. If you want to find out how fast it's going at exactly the 2nd second of a race, you would look at its speedometer — that's an example of the instantaneous rate of change, giving you a specific speed at that very moment just like finding the IROC for our function.

3. Sketch a graph of 𝑓(𝑥) = 𝑥³ and draw:
   • a secant line from 𝑥 = 0 to 𝑥 = 2
   • a tangent line at 𝑥 = 1.

For this exercise, the task is to sketch the graph of the cubic function 𝑓(𝑥) = 𝑥³.

1. Start by plotting key points based on the function:
   𝑓(0) = 0, 𝑓(1) = 1, 𝑓(2) = 8.
2. Connect these points smoothly, noting that the curve will have a characteristic shape bending upward as 𝑥 increases.
3. To draw the secant line, find the slope between the points (0, 0) and (2, 8):
   Slope = (8 - 0) / (2 - 0) = 4. Draw the line between these two points on the graph reflecting this average rate of change.
4. For the tangent line at 𝑥 = 1, calculate the derivative: 𝑓'(𝑥) = 3𝑥², so at 𝑥 = 1, 𝑓'(1) = 3. The tangent line at the point (1,1) has a slope of 3, meaning it rises 3 units for each 1 unit it moves right. Plot this line touching only at the point (1,1).

Imagine a roller coaster ride: the secant line is like the average steepness of the track between the start and the second hill, while the tangent line represents the steepness at the point where you're at the peak of the first hill, indicating how quickly you could roll down from that point.

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