Using critical points and to identify extreme values
Using critical points and to identify extreme values 4.3-ish Extrema Are local and global maximum and minimum values of a graph. There is a global or absolute maximum at f(c) if f(c) f(x) for every x value in the domain of the function. There is a global (absolute) minimum at f(c) if
f(c) f(x) for every x value in the domain of the function. More Extrema You can still have fundamental changes in the behavior of a graph even if it not a global max or min. There is a local maximum when f(c) f(x) for every x value near c. There is a local minimum when f(c) f(x) for every x value near c.
Any maximum or minimum can be called an extreme value (plural is extrema). Critical Values If a function has a relative extreme value at (c, f(c)) the function changes its behavior at c regarding whether it is increasing or decreasing before or after the point. This can only happen if f(c) = 0 or f(c) is undefined. These c values are called Critical values.
Critical values are possible locations for extrema. Not all critical values will have extreme values. Increasing and Decreasing Functions A function is increasing on an interval if as x moves to the right, its graph move up. A function is decreasing on an interval if as x moves to the right the graph moves down. A function is constant if as x moves to the right the graph is moving neither up nor down.
Theorem: Increasing/Decreasing Function Test If f is a function that is continuous on a closed interval [a,b] and differentiable on the open interval (a,b). 1) If f(x) > 0 for all x in (a,b), then f is increasing on [a,b]. 2) If f(x) < 0 for all x in (a,b), then f is decreasing on [a,b] 3) If f(x) = 0 for all x in (a,b) then f is constant on [a,b]
Intervals on Which f is increasing or decreasing Find the open intervals on which: Interval Test Value Sign of f(x) Direction of f(x) Steps to identify increasing and
decreasing intervals 1) Locate the critical numbers of f in (a,b), use these to determine your test intervals. 2) Determine the sign of f(x) at one test value in each interval. 3) Determine whether f(x) is increasing or decreasing on the interval Determine all intervals where the following function is increasing or decreasing.
Interval Test Value Sign of f(x) Direction of f(x) Try to sketch the graph by hand. Graph your critical points and use the information about the direction of f(x). Use your calculator to check your answer afterwards. Homework
Pg 226 17, 21, 27, 33, 35, 37, 41, 47 Or Khan Academy 1) Critical numbers and 2) extreme values from a graph. Theorem: First Derivative Test c is a critical number of the function f that is continuous on an interval which contains c. If f differentiable on the interval (except possibly at c) then f(c) can be classified as follows: 1) If f(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)).
2) If f(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)). 3) If f(x) is positive on both sides or negative on both sides of c, then f(c) is neither a relative minimum nor a relative maximum. Find the relative extrema - in the interval (0, 2) over the real numbers = over the real numbers Shape of a Graph Part 2:
Concavity 4.4-ish Concavity If a graph is curving upwards we call it concave up (smile ) If a graph is curving downwards we call it concave down (frown )
More precise definition Let f be differentiable on an open interval. The graph is concave upward if f is increasing on the interval. The graph will lie above all of its tangent lines The graph is concave downward if f is decreasing on the interval The graph will lie below all of its tangent lines Test for Concavity We use the second derivative
If the second derivative of f exists on the open interval: If f(x) > 0 for all x in the interval, the graph of f(x) is concave upward on the interval. If f(x) < 0 for all x in the interval, the graph of f is concave downward in the interval Determine the open intervals over which is concave up and concave down
( )= Interval Test Value Sign of f(x) Concavity 2
2 2+ 1 ( )= 2 4 Interval Test Value
Sign of f(x) Concavity Points of inflection A point of inflection is when a graph changes concavity (from concave up to concave down or visa-versa) f is a function that is continuous (on some interval) and c be a point in the interval. If the graph of f has a tangent line at (c, f(c)), then this point is a point of inflection of the graph if
the concavity of f changes at this point. More inflection If (c, f(c)) is a point of inflection of the graph of f, then either f(c)=0 or f does not exist at x=c. 4 ( )= 4
3 Determine the points of inflection and discuss the concavity of the graph Interval Test Value Sign of f(x) Concavity Hw Pg. 235 2,4,5, 9, 11, 13, 17, 21, 27, 31, 35, 45,
49 Second derivative test Let f be a function such that f(c)= 0 and the second derivative of f exists on an open interval containing c. 1) If f(c) > 0, then f(c) is a relative minimum 2) If f(c) < 0, then f(c) is a relative maximum. If f(c)= 0, the test FAILS. (It could be anything but we can use the first derivative test to verify for sure what it is.)
Wait What? Ok find the critical points (the zeroes of the first derivative) Substitute the critical points themselves into the second derivative. Positive -> Max Negative -> Min Zero -> we have no idea so wed better try something else
Find the relative extrema of Here we substitute in critical values rather than use test points on intervals Point Sign of f(x) Type of extrema How to identify key features of functions without really graphing
Things we might want to know about a function
Intercepts Symmetry Domain and range Continuity Differentiablitiy Asymptotes Relative Extrema Concavity Points of inflection
Guidelines for analyzing functions 1) Find Domain and range of the function (Algebra 2 ish stuff) 2) Determine intercepts, asymptotes and symmetry of the graph. (Algebra 1,2 and Pre-Calc ish stuff) 3) Locate the x-value for which f(x) and f(x) are zero or do not exist. Use this to determine relative extrema, points of inflection. 4) Use these intervals to note where the function is increasing/decreasing and concave up vs. concave down.
( )= f(x) -< x <-2 x = -2 x = -2 -2< x < 0 -2< x < 0 x=0 x=0 0
2 f(x) 4 Characteristics Of Graph ( )= f(x) -< x <-2
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