SECTION 3.3 Product and Quotient Rules & High-Order Derivatives
S ECTION 3.3 Product and Quotient Rules & High-Order Derivatives.
Dec 14, 2015
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SECTION 3.3Product and Quotient Rules & High-Order Derivatives
QUESTION TO PONDER So how would we take the derivative of the
following:
THE PRODUCT RULE
Due to the fact that addition and multiplication are commutative, we can move the above around to be
EXAMPLE 1Use the product rule to differentiate.
a.
b.
THE QUOTIENT RULE
Think: “Low-DeeHi minus Hi-DeeLow over Low squared.”
EXAMPLE 2Use the quotient rule to differentiate.
a.
b.
EXAMPLE 3Find the derivative by rewriting the function and NOT using the quotient rule.
a.
b.
DERIVATIVES OF TRIG FNC.’S
How can we prove these?
EXAMPLE 4Find the derivative of each function.
a.
b.
c.
EXAMPLE 5Evaluate the derivative at the given point and check with a calculator.
EXAMPLE 6Find an equation of the line tangent to the graph of at the given point.
EXAMPLE 7Determine the points at which the graph has a horizontal tangent line.
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,
.
NOTATION FOR HIGHER-ORDER DERIVATIVES
EXAMPLE 10Find given that
.
QUESTIONS???
Don’t forget to be working the practice problems.
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