s ection 3.3 product and quotient rules & high-order derivatives
TRANSCRIPT
THE PRODUCT RULE
Due to the fact that addition and multiplication are commutative, we can move the above around to be
EXAMPLE 8The radius of a right circular cylinder is given by and its height is , where is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.
HIGHER ORDER DERIVATIVES
Do you think we can continue on in this process of taking derivatives?
What would that mean in the contexts of “rates of change”?
What about with regard to the position function, for example?
THINK BACK . . .
What is a rate of change that we use to measure the speed of our vehicle?
The function that gives the position (relative to the origin) of an object as a function of time is called the position function.
Average Velocity
POSITION, VELOCITY & ACCELERATION
VelocityGiven a position function, , for an object moving along a straight line, the velocity of the object at time is
the instantaneous rate of change of the position function.
Thus, “velocity is the derivative of position”
“acceleration is the derivative of velocity.”Hence,
.