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Unformatted text preview: ejeétige: Determine values of
constants to create a continuous
function. Deﬁne the function Make the limit exist Check to see that they are equal a Review solving systems of equations " skewsu, V t A anlues of the constant that Fin
cause the functions to be H) XI“: l
. 31.11.
continuous ,, L3 / it h(x)={ax2+bx,if,x—2<l I 4 X P.”
L 2x+3,zfix—2[21 ‘ 8/29/2011 8/29/2011 Objective: Use the Intermediate
Value Theorem If f is continuous on
[a, b] and k is any
number between f(a)
and f(b), then there is
at least one number 0
in [a, b] such that
f(c) = k Rat) 4L L Existence theorems u 5) “ Guarantee that some c exists We may not always be able to ﬁnd that "c The Mean Value Theorem is another
important existence theorem that we will study later. 8/29/2011 A function is continuous on an open interval (a,b) if
it is continuous at each point in the interval. m A function is continuous on the closed interval [a,b] if it is continuous on the open interval (ab) and
limﬂx) = fla) and lim f(x) = f(b) x—>a+ X—W' illéﬁ‘a'9ﬁ1s  Verify that the NT applies and ﬁnd
the value of if possible. \) f(x)=x2 +x__1 on [0,5] Is f(x) coniinuous‘? \ﬁQSL {C’L‘T’km’m .
“6):” f(a)=f(0)=U"U" 7 ” I f(b)=f(5)= Z‘it/S 477/” IS f(a)<f(0)<f(b)?/ _ I In lives!
I L.(l L‘L‘l V f(c)=c2+c—1 11=c2+c—1 Solve to ﬁnd c ;( ’5)(?(Hf) 8/29/2011 If f(x)is continuous on
[a, b] and if f(a) and
f(b) are nonzero and
have opposite signs,
then f(x) has a zero in
(a. b) a a .
Explain why f(x) has a zero in the
interval (a, b) J f(a)=f(l)= in 3 : a f(b)=f(2)= fag/3:
negative < c < p_0sitive
1% c is zero somewhere in (1, 2) V6 Is f (x) continuous? 21 ) ...
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This note was uploaded on 09/19/2011 for the course CALCULUS 1431 taught by Professor All during the Fall '11 term at University of Houston.
 Fall '11
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