4. Applications of Derivatives

Suppose the position of a particle moving along a straight line is given by the graph below. At time $t=2$ seconds, estimate the value of the velocity and acceleration of the particle using the graph.

Given the vector-valued function $r\left(t\right)=(t^2,\frac{2}{3}{t}^3,t)$, find the unit tangent vector $T$, the unit normal vector $N$, and the unit binormal vector $B$ at the point $(4,\frac{-16}{3},-2)$. Which of the following correctly gives the unit tangent vector $T$ at that point?

Given the position equation $s\(\left\)(t\(\right\))$ , calculate the average velocity (in meters per second) based on the given time interval, and the instantaneous velocity (in meters per second) at the end of the time interval.

$s\(\left\)(t\(\right\))=\(\frac{30}{t+5}\)$, $-4\(\le\) t\(\le\)0$

Which of the following best describes the gradient vector field of the function $f(x,y)=x^2+y^2$?

Given the position of an object $s\(\left\)(t\(\right\))=12t-t^2$ (in meters), find the acceleration of the object at $t=5$ seconds.

Given below is the graph of velocity with respect to time. At which time(s) would acceleration be 0?

Given the position equation $s\(\left\)(t\(\right\))$, calculate the average velocity (in meters per second) based on the given interval, and the instantaneous velocity (in meters per second) at the end of the time interval.

$s\(\left\)(t\(\right\))=9t-t^2$, $0\(\le\) t\(\le\)3$