Theorem 2(B): Concave DOWNward

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Edgar Henderson
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1 Montana State University M161: Survey of Calculus 61 Section Applications of the Second Derivative Honeybees This is a population graph for Cyprian honeybees raised in an apiary. The population is increasing, but how does the rate of increase change over time? When is the rate of increase highest? Theorem 2(A): Concave UPward If! (!) 0 for every x in an interval (a,b), Then!(!) is CONCAVE UPWARD on the interval (a, b) Theorem 2(B): Concave DOWNward If! (!) 0 for every x in an interval (a,b), Then!(!) is CONCAVE DOWNWARD on the interval (a, b)

2 Montana State University M161: Survey of Calculus 62 Concave Up/Down Intervals of a Function Umbrellas and Bowls Step 1: Find! (!) Step 2: Set!!!! = 0, find all values of x for which!!!! = 0 OR is NOT continuous Step 3: Make intervals using the numbers you found in Step 2 Step 4: Pick a test number, c, in each of the intervals Step 5: Compute f (c) or at least figure out the sign of f (c) If!!!! > 0, then f(x) is Concave Upward on that interval If!!!! < 0, then f(x) is Concave Downward on that interval If!!!! = 0, then you either made an algebra error, or you missed something in Step 2 Step 6: Find INFLECTION POINTS, places where the function changes concavity If! (!) changes sign as we move across a critical number c, ie, changes from concave up to concave down, then! is an inflection point

3 Montana State University M161: Survey of Calculus 63 Example Find the intervals the function is concave upward and concave downward, and inflection points. #!!! =!!!"!"#$!:!"#$%&'$%"(!"#$!:!"#$%!!!!! = 0!"!"#!!!! = 0!"!"# x = INTERVALS Test Value a Find! (!) OR sign of! (!) C- Up / C- Down Inflection Point(s)? at x = Conclusions: Intervals CONCAVE UP: Intervals CONCAVE DOWN: INFLECTION POINT(S):

4 Montana State University M161: Survey of Calculus 64 #!!! =!!!!!!"#$!:!"#$%&'$%"(!"#$!:!"#$%&!!!! = 0!"!"#!!!! = 0!"!"# x = x = INTERVALS Test Value a Find! (!) OR sign of! (!) C- Up / C- Down Inflection Point(s)? at x = at x = Conclusions: Intervals CONCAVE UP: Intervals CONCAVE DOWN: INFLECTION POINT(S):

5 Montana State University M161: Survey of Calculus 65 #!!! =!!!!!! +!!"#$!:!"#$%&'$%"(!"#$!:!"#$%&!!!! = 0!"!"#!!!! = 0!"!"# x = x = INTERVALS Test Value a Find! (!) OR sign of! (!) C- Up / C- Down Inflection Point(s)? at x = at x = Conclusions: Intervals CONCAVE UP: Intervals CONCAVE DOWN: INFLECTION POINT(S):

Montana State University M161: Survey of Calculus 66 Second Derivative Test (Relative Max/Min) Mountains and Valleys of a Function Step 1: Find! (!) Step 2: Find the critical numbers of f(x), at which!

ie f(x) is Concave- Down then!(!) has a relative maximum at x = c If!! >! ie f(x) is Concave- Up then!(!) has a relative minimum at x = c If!! =! then the test fails and is inconclusive YOU WILL NEED TO USE THE FIRST DERVIATIVE TEST!

6 Montana State University M161: Survey of Calculus 66 Second Derivative Test (Relative Max/Min) Mountains and Valleys of a Function Step 1: Find! (!) Step 2: Find the critical numbers of f(x), at which!! = Step 3: Find! (!) Step 4: Compute! (!), for each critical number found in Step 2 If! (!) <! ie f(x) is Concave- Down then!(!) has a relative maximum at x = c If!! >! ie f(x) is Concave- Up then!(!) has a relative minimum at x = c If!! =! then the test fails and is inconclusive YOU WILL NEED TO USE THE FIRST DERVIATIVE TEST! (!) <! Thus x = - 3 is a Relative Max!! >! Thus x = 1 is a Relative Min

7 Montana State University M161: Survey of Calculus 67 Example Use the Second Derivative Test to find the Relative Maximum(s) and Minimum(s). #!!! =!!!"!! +!"#!"#$!:! (!)!"#$!:!"#$#%&'!"#$%&!!! = 0!"#$!:!"#$%&'!!!!!"#$!:!"#"$%&'"!"#$%&'"!"#$%&' #!!! =!!!!" +!!"#$!:! (!)!"#$!:!"#$#%&'!"#$%&!!! = 0!"#$!:!"#$%&'!!!!!"#$!:!"#"$%&'"!"#$%&'"!"#$%&'

8 Montana State University M161: Survey of Calculus 68 Graph of a Function and Its Derivatives The Function!! =!!!"!"#!"#$%&'$%":!! =!!!!!"#$#%&'!"#$%&:! = ± 2 3, , , INC DCR INC By 1 st Derivative Test: x = 2 3!"#$ x = 2 3!"#$!"#!"#$%&'$%":!! =!", 0 C- DOWN 0, C- UP By 2 nd Derivative Test!!! 2 3 < 0!!! 23 > 0 x = 2 3!"#$ x = 2 3!"#$

9 Montana State University M161: Survey of Calculus 69 Comparing the First and Second Derivative Tests Ways to find the Relative Extrema Pros First Derivative Test Pros It ALWAYS works to find RMax/RMin Cons Takes more work to see if RMax/RMin Requires two calculations Second Derivative Test Pros Can be faster/easier to compute RMax/RMin IF! (!) is easy to compute Cons IF!! = 0, THEN the test fails Have to do 1 st Derivative Test Cons ONLY works when! (!) exists! (!) can be difficult to compute General Shape of the Graph of f(x)! (!) > 0 ie:!(!) is Increasing First Derivative!! < 0 ie:!(!) is decreasing Derivative! (!) > 0!(!) is Concave Up f(x) is Increasing Concave Up f(x) is Decreasing Concave Up Second!! < 0!(!) is Concave Down f(x) is Increasing Concave Down f(x) is Decreasing Concave Down

10 Montana State University M161: Survey of Calculus 70 EXAMPLE Sketch the graph of a function having the properties listed below. 1 st Derivative: Shape of Graph: 2 nd Derivative: Function Values First Derivative Info Second Derivative Info o!( 1) = 4 o!(0) = 2 o!(1) = 0 o! ( 1) = 0 o! (1) = 0 o!!! > 0!", 1 o!!! > 0!" (1, ) o!!! < 0!" 1, 1 o!!! < 0!", 0 o!!! > 0!" 0,

4.3, Math 1410 Name: And now for something completely different... Well, not really.

4.3, Math 1410 Name: And now for something completely different... Well, not really. How derivatives affect the shape of a graph. Please allow me to offer some explanation as to why the first couple parts