Calculus I for Bus
and Econ
By Daria Vyachkileva
Optimization
Source: Quickmeme
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Application of first derivative
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Application of first derivative
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Application of first derivative
Warning:
The theorem makes a statement over intervals !, # and not just at a single point $.
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Application of first derivative
Method:
1. Find all values of ! for which " ! ! = 0 or " ! is discontinuous and identify intervals determined by these numbers.
2. Select a test number % in each interval found in step 1 and determine the sign of " ! % in that interval
1. If " ! % > 0, " is increasing on that interval
2. If " ! % < 0, " is decreasing on that interval
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Application of first derivative
Example: Determine the intervals where the function ! " = " " is increasing and where it is decreasing.
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Application of first derivative
!
Example: Determine the intervals where the function ! " = " " + 6" " is increasing and where it is decreasing.
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Application of first derivative
!
Example: Determine the intervals where the function ! " = " " + 6" " is increasing and where it is decreasing.
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Application of first derivative
!
Example: Determine the intervals where the function ! " = " " + 6" " is increasing and where it is decreasing.
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Application of first derivative
Example: Determine the intervals on which ! " = $ #$ is increasing or decreasing.
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Application of first derivative
Example: Determine the intervals on which ! " = ln " is increasing or decreasing.
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Application of first derivative
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Application of first derivative
Method: First derivative test on finding relative extrema.
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Application of first derivative
First derivative test on finding relative extrema cont.
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Application of first derivative
Explanation.
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Application of first derivative
Example. Find the points at which the function ! " = " " has relative maxima and minima.
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Application of first derivative
!
Example. Find the points at which the function ! " = " " + 6" " has relative maxima and minima.
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Application of first derivative
!
Example. Find the points at which the function ! " = " " + 6" " has relative maxima and minima cont.
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Application of first derivative
Example. Sketch the graph and find the local extrema of
! " = sin " '( 0, +
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Application of first derivative
Example. Let ! be a positive integer and let " be a polynomial of degree !. How many critical points can " have?
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Application of first derivative
Example. Find the points at which the function ! " = " "$ #$ has relative maxima and minima.
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Application of first derivative
Example. Find the points at which the function ! " = " "$ #$ has relative maxima and minima.
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