test1

x^2=12 x=2+-(v--3) f''(x)=12x^2-48=0 12(x^2-4)=0 12(x+2)(x-2)=0 x=+-2 f'(x)=4x^3-48x =4x(x^2-12) f'(-4)=4(-4)((-4)^2-12) will be negative f'(-3)=4(-3)((-3)^2-12)positive f'(-1)=4(-1)((-1)^2-12)positive extrapolated application of number to variables rel max @ (0,f(0)) rel. min.@ (2V--3, f(2V--3)) and -2V--3, f(-2V--3)) (2root3, -66) and (-2root3, -66) f(x)=x^4-24x^2+80 f(0)=80 f(2V--3)=16*9-24(4*3)+80 146-288+80=0 =-146+80= -66 ^^same as f(-2V--3) Exam Prep Relative Extreme very similar to what we get on the test, parts a thru e just fill in all the blanks not too hard, just to remember what is going on any specific questions or walkthrough? some derivatives? probably one that involves more than one rule List out all the rules-2columns 1) first side: Derivatives we know f(x)=x^n =polynomial e^x or a^x ln^x sinx cosx 2) rules to use we can "break up functions" product rule u(x)-v(x) quotient rule: p(x)/q(x) chain rule: u(v(x)) ex) f(x) = sin^2(x+1/x) =(sin(x+1)/(x))^2 CHAIN RULE u(x)=x^2 u'(x)=2x v(x)=sin((x+1)/x) chain rule again g(x)=sinx g'(x)=cosx h(x)=(x+1)/x h(x)=(x/x)+1/x) h'(x)=x^-2 v'(x)=g'(h(x))*h'(x) =cos((x+1)/x)(-x^-2) =-cos((x+1)/x)/x^2 f'(x)=u'(v(x))*v'(x) 2(sin((x+1)/x)(-cos((x+1)/x)^2) ex) f(x)=(x^2-2)/e^x^2 Quotient rule: p(x)=x^2-2 p'(x)=2x q(x)=e^x^2 CHAIN RULE: q=u(v(x)) u(x)=e^x u'(x)=e^x) v(x)=x^2 v'(x)=2x q'(x)=e^x*(2x) f'=(e^x^2*2x-(x^2-2)(2xe^x^2))/(e^x^2)^2 =(2xe^x^2)-(2x^3e^x^2)+(4xe^x^2)/(e^2x^2) =(xe^x^2)(6-2x^2)/(e^x^2)^2 =x(6-2x^2)/(e^x^2) f(x)=3xe^x+2 find where ifs inc/dec, concave up/down PRODUCT RULE u(x)=3x u'(x)=3 v(x)=e^x v'(x)=e^x f'(x)=3e^x+3xe^x 3e^x+3xe^x=0<- to find CNS 3e^x(1+x)=0 e^ ANYTHING IS NEVER ZERO 1+x=0 x=-1 CN f'(-2)=3e^-2(1-2) f'(0)=3e^0(1-0) --> Positive! f is increasing on interval (-1,@@) and decreasing on (-@@,-1) would increase product rule but we are smart and we realize hey, this is the derivative of 3xe^x f''(x)=3e^x+(3e^x+3xe^x) 6e^x+3xe^x 3e^x(2+x)=0 x=-2 f''(-3)=3e^-3(2-3) gives NEGATIVE f''(0)=3e^0(2-0)>0 f is concave up on (-2,@@) down on (-@@,-2)|>