Problem :
Find the derivative of f (x) = 3x^{4} -2x^{2} +5x^{-1} and evaluate it at x = 2.
Using the basic calculus rules established in this section, we find that
f'(x) = 12x^{3} -4x - 5x^{-2} andf'(2) = 96 - 8 - 5/4 = 86 + 3/4
Problem :
Find the velocity and acceleration functions corresponding to the position
function x(t) = 3t^{2} - 8t + 458.
v(t) = x'(t) and
a(t) = v'(t) = x''(t), so using our basic calculus rules again we
find that
v(t) = 6t - 8 and a(t) = 6
Notice that the acceleration in this case is constant, and that its value is
equal to twice the coefficient of
t^{2} in
x(t).
Problem :
What happens when a car which is traveling along at constant velocity
screeches to a halt?
The velocity of the car decreases rapidly, corresponding to a large negative
acceleration (or
deceleration) of the vehicle (courtesy of good
brakes). While the car was traveling at constant velocity, on the other
hand, the acceleration was zero.