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 (h + 9) m/sec. (c) the instantaneous velocity is m/sec.

Explanation:

Step1: Recall the formula for instantaneous velocity

Instantaneous velocity is the derivative of the position - function. Given $y = f(x)=x^{2}+x$, use the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$.

Step2: Differentiate the function

The derivative $y^\prime=f^\prime(x)=\frac{d}{dx}(x^{2}+x)=\frac{d}{dx}(x^{2})+\frac{d}{dx}(x)$. By the power rule, $\frac{d}{dx}(x^{2}) = 2x$ and $\frac{d}{dx}(x)=1$. So $f^\prime(x)=2x + 1$.

Step3: Evaluate the derivative at $x = 4$

Substitute $x = 4$ into $f^\prime(x)$. $f^\prime(4)=2\times4+1$.
$f^\prime(4)=8 + 1=9$.

Answer:

9