Differentiable Functions

[johnny101]

Find the values of a and b for which the function h below is differentiable

h(x)= {ax +b x(less than or equal to) -1}
{ax^3+x+2b x(greater than) -1}


Can someone show how to do this example? Can't find anything like this in the text. Lost. Thanks

 

[Romsek]

Find the values of a and b for which the function h below is differentiable

h(x)= {ax +b x(less than or equal to) -1}
{ax^3+x+2b x(greater than) -1}


Can someone show how to do this example? Can't find anything like this in the text. Lost. Thanks

start with the definition of a derivative at a point x.

h'(x) = lim_(dx->0) (h(x+dx) - h(x))/dx

You need to ensure this limit exists at x = -1 by judicious choice of a and b.

 

[HallsofIvy]

Find the values of a and b for which the function h below is differentiable

h(x)= {ax +b x(less than or equal to) -1}
{ax^3+x+2b x(greater than) -1}


Can someone show how to do this example? Can't find anything like this in the text. Lost. Thanks

Unless you are at the very beginninng of a Calculus course and have just learned the limit definition of the derivative, I would NOT recomend using that definition.

Do you know what the derivative of ax+ b is? Do you know what the derivative of \(\displaystyle ax^3+ x+ 2b\)? You should see immediately that h is differentiable at all x except, possibly x=-1.

Further, although the derivative of a function is not necessarily continuous it must satisfy the "intermediate value theorem". Here, that means that the two derivatives, on either side of x= -1, must have the same value at x= -1.