Of coursewe couldzoomin on the graphto see that the lowestpointon the graphlies withinthe fourth
quadrant,but let’s use theCALCVALUEfunctionof the calculatorto verifythat thereis a zero in the interval
In orderto applythe IntermediateValue Theorem,we needto find a pair ofx-valuesthat have
functionvalueswith differentsigns.Let’s try somein the tablebelow.
We see that the sign of the functionvalueschangesfrom negativeto positivesomewherebetween1.2 and
1.3. Hence,by the IntermediateValue theorem,thereis somevaluecin the interval(1.2,1.3)suchthatf(c)
= 0.
LessonSummary
- We learnedto examinecontinuityof functions.
- We learnedto find one-sidedlimits.
- We observedpropertiesof continuousfunctions.
- We solvedproblemsusingthe Min-Maxtheorem.
- We solvedproblemsusingthe IntermediateValue Theorem.
ReviewQuestions - Generatethe graphoff(x)= (|x+ 1|)/(x+ 1) usingyour calculatorand discussthe continuityof the function.
- Generatethe graphof usingyour calculatorand discussthe continuityof
the function.
Computethe limitsin #3–6.
3.
4.
5.
6.
In problems7 and 8, explainhow you knowthat the functionhas a root in the giveninterval.(Hint:Use the
IntermediateValue Functionto showthat thereis at leastone zero of the functionin the indicatedinterval.):
- , in the interval