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UTD / Art / AP 1301 / How to find the extreme values of f(x) = x2+2x-2 [-2,2]?

How to find the extreme values of f(x) = x2+2x-2 [-2,2]?

How to find the extreme values of f(x) = x2+2x-2 [-2,2]?

Description

School: University of Texas at Dallas
Department: Art
Course: Exploration of the Arts
Professor: Winston stone
Term: Fall 2015
Tags:
Cost: 50
Name: Exam 2 Fundamental Study Guide
Description: Okay guys I read through the book, analyzed, and gathered the most important information I could fine, including problem session questions. I had to upload hand-written notes because I couldn't find a text-editor that would let me upload equations so here it is and I really hope this helps you guys out on exam.
Uploaded: 11/13/2015
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How to find the extreme values of f(x) = x2+2x-2 [-2,2]?



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The chain Rule 1a (derivative of outside) (derivative of inside)

exemple fut (4x+3) 9 964x+3)*(47= 36(4x+3)

Chip = (x2+3x+7) hei=64x2+3x+77 - (2x+3) 25 ปี


What is fermat's theorem is all about?



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gux) = sin(x2-4*7 cos(x29xJ62x-47 double chain fey)= (It cos 2x)

f'(=261+ corax X-sinax) a sin 2x need its own the hor- fcx=201 tasar) (-sinax ) 2 4citas


Define derivative graphs.



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differentiate both sides if you differentiate ay term, multiply it @ sale for day We also discuss several other topics like Who are the three main gods of hindu?

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excerpledy of x2 + y2 =25

2x+2y=0 differen both sides H = 2x + Don't forget about the age old question of Which of the following is an example of an etic description of teen pregnancy in america?

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y = -2x solve 1 exemple 2 Sinx up ay ty=10x Don't forget about the age old question of How to determine the original mass or spring stiffness?

ex-by+Y'**]+ y = 10 If you want to learn more check out It is one of the biggest contributing factors to every organization, what is it?

Product Rule - 99% - 2 + x + y = 10

+dy = 10+ 2xy = Loss dy Cx2+1)= 10 + 2xy-cosx.

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Sinx-x +y= asХ Cosx-[

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- dyx+dys 20-208 x + 2xy. dy (-X2+1)= 20-cost axy = (20-cosx + 2xy.

(-x2+1) Trig Derivatives a sint cosx id tanx= secay a CSc x= -CSC Xcotx

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into caso I f(x) = in [(2x+372 7 2 In C2x+3)2 - InC 3-2) 6

[63x-206 = 2 In (2x+3) - 6 ln (3x-2)

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importante example find the deriv. of e2 - ex. (3x) =Bex

Experial Desistives.

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exemple (3*) 13. In 3

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use the linear approx to estimate 9402 plug in 244,02) = 2 + +64.02) -1 = 12.005

Absoluk min/max || Local min/max in

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Domain = absolute max Domain - absolute min

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example how to find the extreme values que tout

f(x) = x2+2x-2 [-2, 2] ayson 10 find critical numbers from (a,b).

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2 8 set to ot solve ax=-2 X- 2 3 evaluate fex) at critical numbers

f(-1) = -12 + 2(-1)-2=- 3 (3 evaluate fra) + f(b)

f(-2) = -22 +26-27-2 - - 2.

FCb) = 2ataca -2 - 6 14 conclude: biggest # = max, smallest =min

max= 6 or (2,6)

2 min=-3 (1-3) local min/max: point where a functions increased decrease changes exemple t x2+2x-2

P -tare derivative & find critical rumber at en

2x+2=0 crit = -1 - use number line test test a value on both sides of crit. #

- - f'(-2)=- (negative value)

f(0) = - (negative value) - the function stays – on sooth sides , so no

local min/max СХrpu KX) = х2

f'(x) = ax crito #io in low change in inasil decr. &

- f(-1)=

sono f(1) =+ ti so draw with the line

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Good conditions to know Ex Value Theocn (EVS).

if fet) is continues on Caib] then thace exists als minimax values if there is not a closed intertal. EXT Hors not apply

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Fermat's Theorem - it fex) has a local max of min at a t it fc exists

fcc7zo

+2 =6 dous n = o fona a Esc m4x4 a function may have a local minimax where it is not diffentiable

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Mein Value Theoren ( S cond Wons if, fois o cookies om C

b) differkuble on Calo) then there exists atleast I point cin [ab] fkas= f(b)-flad

exemple fex = 1 + 3x - 1 [2,97 note that 2 a +b=a a determine continuity Demain = (-09

19] = 62,92 Take the derivative

fox) = 1 + 25x-1 & it cx-1713 f'w=f(x-17213 f'( =

3473

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cannot differentiate at the 1 f is differentiable (-oo, DUCI ) which fits in interval Casa

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be 1 set derivative = to value found in step 2

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Concavity / hflection Points

f" x 70 - increasing f"(x240 - decreasing €" >o = concuje ce f ) 20 = Concave down exemple fcx) = x2 + 3x29x+4

f'c = 3+ CX-9

f"(x) = 6x+6 2 find critical numbers - 6x+6=0 (3 Test with number line a

Sture

TO

f"(-2): AFC) = + FH so concore up = (-1,0)

concave down= (-00-17 Inflection Point - change of

concavity

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The change of concavity in couve example is so to find the full point use

(-1, IL-11) (-lis)

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Derivative Graphs the information you need for a graph of a derivative

Course & sketch)

f(x)

Intercepts Asymptotes

use I find

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open intervals of incel decr. 7 local extreme values I

use f'(x) find

use suf Ex

Open intervals concow up down inflection points

L'harital's Rule & indeterminate Forms The 7 indeterminine forms 8. , 20-00, 0.00,1,0, so l'hopitals rule works directly on 8 and 2 example lim stanza o apply l'hopituds rule

Y-01 to 3% [l'hopitals rull

take decitative of top & bottan if the function is reduced to inteleoninale forms 8 to

keep taking the derivative cas long as its & until a value is reached 50+ lin 3ten Tx 2\sec?(78). - 31.c03 033)

tot tanax 3 sec 2(3x) 3 cos (7x) Plug in lim alcoscio = 21.1= 7

X>0+ 3.205470)

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Cuptum.zation doyeckie: momize or amize What to do?

come up with a fex Codepends on the prooem) @ Take the derivative of to 3 Find critical s. ex and the dimensions of a rectangle with an area of loouma whose perimeter is as small as possible

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P= x+14+Y | P=1x + y

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We are trying to minimize

the perinder so take the

derivative

of p=2x+2y

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first change P=2x+2y into a single variable problem. given by Alw 1000m2= xy werde proto change it into y so y 1000 then plug into P

P= 2x+2/1000) = ax+ 2000 take derivative -2x+ 2000x*-+ P = 2-2000 x 2 = 2 -2000 Superor B_2000) @ Find critical numbers

check critical 2x+-200050 Woo x2=0

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bring back 1000=xy. X=11000 so solve equation.

1000 - Cookp X=ALOL "Z11000

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