4. Applications of Derivatives

The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.

b. Find the velocity function for the marble.

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

Terminal velocity Refer to Exercises 95 and 96.

a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).

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.

b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left?

f(t) = 18t-3t²; 0 ≤ t ≤ 8

Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.

c. Find the total distance traveled by the body from t = 0 to t = 2.

106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?

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$

Understanding Motion from Graphs

The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.

b. When is the particle’s acceleration positive? Negative? Zero?