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 13 m/sec. (b) the average velocity is m/sec.

Explanation:

Step1: Recall average - velocity formula

The average velocity of an object moving along the $y$-axis with position function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$.

Step2: Find $f(4)$ and $f(4 + h)$

Given $f(x)=x^{2}+x$. Then $f(4)=4^{2}+4=16 + 4=20$, and $f(4 + h)=(4 + h)^{2}+(4 + h)=16+8h+h^{2}+4 + h=h^{2}+9h + 20$.

Step3: Calculate the average velocity for $x$ changing from $4$ to $4 + h$

The average velocity $v_{avg}=\frac{f(4 + h)-f(4)}{(4 + h)-4}=\frac{(h^{2}+9h + 20)-20}{h}=\frac{h^{2}+9h}{h}=h + 9$.

Answer:

$h + 9$