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2 4 5 Lecture topics 10 - 12 - 98 Worked example Page 1 Worked example Page 2 Worked example and Graph
Lecture topics 10 - 12 - 98
1) Worked an example: f (x) = x2 (x 1)3 . Dierentiate and simplify, nd all critical points, and classify whether they are a local maximum, minimum, or nothing at all. The points were x = 0, LM AX ; x = 2 , LM IN ; x = 1, N OT HIN G. Here 5 NOTHING has to be understood as meaning not an extreme. In fact, it is a point of an inection, tho not an extreme. Located these points on a graph. 2) Stated a test to nd absolute extremes on an open interval; showed how it works for y = x2 . 3) Found the extremes for y = LMIN as well as an AMIN,
x2 x2 +1
; again showed that x = 0 is an
4) Remarked that if f is dened on a closed interval [a, b], then the extremes could be at the endpoints. Also, theres something similar to the local extremes LMAX and LMIN, called endpoint extremes: EMAX and EMIN. 5) Stated the rst derivative test for endpoint extremes. 6) Said Id do a problem on Wednesday: Let f (x) = x + sin x, on 2 the closed interval [0, 2 ]. a) Dierentiate and simplify, b) Find all critical points, c) Use the rst derivative to nd local and endpoint extremes, d) Find absolute extremes.
Worked example Page 1
EXAMPLE Let f (x) = x2 (x 3 1) . a) Dierentiate and simplify. b) Find all critical points, c) Classify whether they are a local maximum, minimum, or nothing at all.
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f (x) = (x2 ) (x 1) 3 + x2 [(x 1) 3 ] 2 1 2 = (2x) (x 1) 3 + x2 [ (x 1) 3 ] 3 2 1 2 2x2 (2x) (x 1) 3 3(x 1) 3 + 2x2 = (2x) (x 1) 3 + = 1 1 3 3(x 1) 3(x 1) 3 (2x) (x 1)(3) + 2x2 (2x) [3(x 1) + x] (2x) [4x 3] = = = 1 1 1 3(x 1) 3 3(x 1) 3 3(x 1) 3 b) Find the critical points: f (x) = 0 where the numerator is zero: (2x) [4x 3] = 0 or x = 0, x = 3 . f (x) dne where the denominator 4 1 3 = 0 or x = 1. The critical points are: is zero: 3(x 1) x = 0, x = 3 , x=1 4
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c) Classify the critical points: Heres the number line:
Worked example Page 2
Now I need to make an f chart, and plug in numbers in each pla...