L ectur e1 2| 1 Increasing/Decreasing Test
Def
A function is said to be increasing if .
A function is said to be decreasing if
L ectur e1 2| 2 Note that
and
Actually, the sign of tells us the property of increasing/decreasing.
L ectur e1 2| 3
Increasing/decreasing test (I/D test)
If on an interval then is increasing on that interval.
If on an interval then is decreasing on that interval.
L ectur e1 2| 4 EX Find where the function
is increasing and where it is decreasing.
L ectur e1 2| 5 EX For
find the interval of increasing and the intervals of decreasing.
L ectur e1 2| 6
First derivative test
Let be a critical number of that is d.n.e. or .
We are moving across from nearby to nearby .
(1) If changes sign from to – then is a local max.
(2) If changes sign from – to then is a local min.
(3) If does not change sign, then is no local max/min.
L ectur e1 2| 7
L ectur e1 2| 8 EX Given
find the local max/min.
L ectur e1 2| 9 EX Given
find the local max/min.
L ectur e1 2| 10 EX Given
find the local max/min.
L ectur e1 2| 11 Optimization Problems
1. Read problem carefully. Look for unknowns, quantities, conditions?
2. Draw diagrams.
3. Introduce notations and variables.
Says and for the desired quantity
4. Express in terms of of
5. Use the given condition to get rid of some variables, so finally get
6. Find the domain of ?
7. Use the closed interval method to find max/min if the domain is a closed interval.
L ectur e1 2| 12 EX (Maximize area) A farmer has 2400 ft of fencing and wants to fence off a
rectangular field that borders a straight river. He needs no fence along the river.
What are the dimensions of the field that has the largest area?
L ectur e1 2| 13 EX (Minimize cost) A cylindrical can is to be made to hold 1 L (= 1000 cm3) of oil.
Find the dimension that will minimize the cost of the metal to manufacture the can.
L ectur e1 2| 14 EX Find the point on the parabola that is closest to the point .
L ectur e1 2| 15 EX Find the area of the largest rectangle that can be inscribed in a semicircle of radius .
