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?

{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200e^−t/10, for t≥0.

a. Graph the velocity function, for t≥0.

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.

$s\left(t\right)=\frac{30}{t+5}$, $-4\le t\le0$

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

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.

$s\left(t\right)=9t-t^2$, $0\le t\le3$

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?

Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

a. Graph the position function.

$f\left(t\right)=6t^3+36t^2-54t;0\le t\le 4$

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y=10e^{-\frac{t}{2}}\cos \left(\frac{\pi t}{8}\right)$.

a. Graph the displacement function. 

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the velocity and acceleration of the object at t = 1. 

f(t) = t² − 4t; 0 ≤ t ≤ 5

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the acceleration of the object when its velocity is zero.

f(t) = t² - 4t; 0 ≤ t ≤ 5

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

On what intervals is the speed increasing?

f(t) = t² - 4t; 0 ≤ t ≤ 5

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the acceleration of the object when its velocity is zero.

f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

On what intervals is the speed increasing?

f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3