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