Question

suppose an object moves along the y axis so that its location is y = f(x)=x² + 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. (c) the instantaneous velocity at x = 4 seconds. (a) the average velocity is m/sec.

Explanation:

Step1: Recall average - velocity formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is given by $\frac{f(b)-f(a)}{b - a}$.

Step2: Calculate $f(4)$ and $f(8)$

Given $f(x)=x^{2}+x$. Then $f(4)=4^{2}+4=16 + 4=20$, and $f(8)=8^{2}+8=64 + 8=72$.

Step3: Calculate average velocity for part (A)

Using the average - velocity formula $\frac{f(8)-f(4)}{8 - 4}=\frac{72-20}{4}=\frac{52}{4}=13$.

Answer:

13