Exercise 1.1.1-4.

In Exercises 1-4, \(y\) represents the distance a bus has travelled after \(x\) seconds. Find \(\Delta y\) and the average velocity during the interval \(\Delta x\) for the following situations.

\(x_0 = 2, \; \Delta x = 0.5 \)
\(x_0 = 2, \Delta x = 0.01 \)
\(x_0 = 4, \; \Delta x = 0.1 \)
\(x_0 = 4, \Delta x = 0.01 \)

Formulas:

\(\Delta y = f(x_0 + \Delta x) - f(x_0)\)

Average Velocity: \( \frac{\Delta y}{\Delta x} \)

Exercise 1. \(y = x^2 + 3x\)

Solution:

\( \Delta y = f(2 + 0.5) - f(2) = 3.75\). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{3.75}{0.5} = 7.5 \)
\( \Delta y = f(2 + 0.01) - f(2) = 0.0701 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.0701}{0.01} = 7.01 \)
\( \Delta y = f(4 + 0.1) - f(4) = 1.11 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{1.11}{0.1} = 11.1 \)
\( \Delta y = f(4 + 0.01) - f(4) = 0.1101 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.1101}{0.01} = 11.01 \)

Exercise 2. \(y = 3x^2 + x\)

Solution:

\( \Delta y = f(2 + 0.5) - f(2) = 7.25\). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{7.25}{0.5} = 14.5 \)
\( \Delta y = f(2 + 0.01) - f(2) = 0.1303 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.1303}{0.01} = 13.01 \)
\( \Delta y = f(4 + 0.1) - f(4) = 2.53 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{2.53}{0.1} = 25.3 \)
\( \Delta y = f(4 + 0.01) - f(4) = 0.2503 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.2503}{0.01} = 25.03 \)

Exercise 3. \(y = x^2 + 10x\)

Solution:

\( \Delta y = f(2 + 0.5) - f(2) = 7.25\). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{7.25}{0.5} = 14.5 \)
\( \Delta y = f(2 + 0.01) - f(2) = 0.1401 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.1401}{0.01} = 14.01 \)
\( \Delta y = f(4 + 0.1) - f(4) = 1.81 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{1.81}{0.1} = 18.1 \)
\( \Delta y = f(4 + 0.01) - f(4) = 0.1801 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.2503}{0.01} = 18.01 \)

Exercise 4. \(y = 2x\)

Solution:

\( \Delta y = f(2 + 0.5) - f(2) = 1\). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{1}{0.5} = 0.5 \)
\( \Delta y = f(2 + 0.01) - f(2) = 0.02 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.02}{0.01} = 2 \)
\( \Delta y = f(4 + 0.1) - f(4) = .2 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{.2}{0.1} = 2 \)/p>
\( \Delta y = f(4 + 0.01) - f(4) = 0.02 \). Avg. Velocity = \( \frac{\Delta y}{\Delta x} = \frac{0.02}{0.01} = 2 \)

arrow_forward