piecewise-defined function: f(x) = ( − sin(a x) if x ≤ 0 x if x > 0 (a) For what values of a, if any, is the function f(x) continuous at x = 0? what parameter a that makes f differentiable at every x?

Question

Consider the following piecewise-defined function:

f(x) = ( − sin(ax) if x ≤ 0

( x if x > 0

(a) For what values of a, if any, is the function f(x) continuous at x = 0?

(b) Is there a value of the parameter a that makes f differentiable at every x?

The outer brackets after the = sign represents the squiggly bracket for piecewise functions.

Answer 1

f(x) = -sin(ax) if x</=0

f(x) = x if x>0

for what value of a is f(x) continuous at x=0

-sin(ax) = x

sinax = -x

ax = sin^-1(-x)

a(0) = sin^-1(-0)

0=sin^-1(0)=0

for any value of a

graph the two pieces

quadrant 1 is a straight line from the origin, diagonally upward at 45 degrees

quadrants 2&3 have the usual sine curve reflected across the y access and horizontally stretched or compressed by a factor of "a"

the two sections meet at the origin

they're continuous at the origin regardless of the value of a

that's part a) of the problem

then part b)

for f(x) to be differentiable at the origin the slopes of the two sections have to be the same at the origin

f'(x) = (x)'= 1 for all x>0

f'(x)= [-sin(ax)]'= -acos(ax)

f'(0) = -acos(0) = -a(1) =-a for x</=0

set -a = 1

solve for a

a =-1 makes the piece wise function differentiable at x=0