Questions: Question 10, 2.4.47 Part 1 of 3 Suppose an object moves along the y axis so that its location is y=f(x)=x^2+x at time x (y is in meters, x is in seconds). Find (A) The average velocity (the average rate of change of y with respect to x) for x changing from 4 to 8. (B) The average velocity for x changing from 4 to 4+h.

Solution Steps

To solve these problems, we need to calculate the average velocity, which is the average rate of change of the function \( y = f(x) = x^2 + x \) over a given interval.

(A) For the interval from \( x = 4 \) to \( x = 8 \), the average velocity is calculated using the formula for the average rate of change: \(\frac{f(b) - f(a)}{b - a}\), where \( a = 4 \) and \( b = 8 \).

(B) For the interval from \( x = 4 \) to \( x = 4 + h \), the average velocity is calculated similarly using the formula: \(\frac{f(4 + h) - f(4)}{h}\).

To find the average velocity over the interval from \( x = 4 \) to \( x = 8 \), we use the formula for the average rate of change of the function \( y = f(x) = x^2 + x \):

\[ \text{Average velocity} = \frac{f(8) - f(4)}{8 - 4} \]

First, calculate \( f(8) \) and \( f(4) \):

\[ f(8) = 8^2 + 8 = 64 + 8 = 72 \]

\[ f(4) = 4^2 + 4 = 16 + 4 = 20 \]

Substitute these values into the formula:

\[ \text{Average velocity} = \frac{72 - 20}{8 - 4} = \frac{52}{4} = 13.0 \]

For the interval from \( x = 4 \) to \( x = 4 + h \), the average velocity is given by:

\[ \text{Average velocity} = \frac{f(4 + h) - f(4)}{h} \]

Assuming \( h = 0.01 \), calculate \( f(4 + h) \):

\[ f(4 + h) = (4 + 0.01)^2 + (4 + 0.01) = 16.0801 + 4.01 = 20.0901 \]

Substitute these values into the formula:

\[ \text{Average velocity} = \frac{20.0901 - 20}{0.01} = \frac{0.0901}{0.01} = 9.01 \]

Final Answer