4. Applications of Derivatives

{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.

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

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

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) = 18t - 3t²; 0 ≤ t ≤ 8

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) = 2t³ - 21t² + 60t; 0 ≤ t ≤ 6

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) = 6t³ + 36t² - 54t; 0 ≤ t ≤ 4