CK12 Calculus - Single Variable

Fromwhatwe havelearnedalreadyaboutdifferentiability, it will not be difficult to showthat continuityis an
importantconditionfor differentiability. The followingtheoremis one of the mostimportanttheoremsin cal-
culus:

Differentiabilityand Continuity

Iffis differentiableatx 0 , thenfis also continuousatx 0.

The logicallyequivalentstatementis quiteuseful:Iffisnotcontinuousatx 0 , then

fis not differentiableatx 0.

(The converseis not necessarilytrue.)

We havealreadyseenthat the converseis not true in somecases.The functioncan havea cusp,a corner,
or a verticaltangentand still be continuous,but it is not differentiable.
ReviewQuestions
In problems1–6, use the definitionof the derivativeto findf(x) and thenfind the equationof the tangent
line at x = x 0.

1.

2.

3. ;

4.

5. (whereaandbare constants);x 0 = b
   6.f(x) =x1/3;x 0 = 1.


6. Finddy/dx|x= 1giventhat


7. Showthat is continuousatx= 0 but it is not differentiableatx= 0. Sketchthe graph.


8. Showthat

is continuousand differentiableatx= 1. Sketchthe graphoff.

10. Supposethatfis a differentiablefunctionand has the propertythat
    f(x+y) =f(x) +f(y) + 3xy