MATHEMATICS  CASE STUDY SOLUTION



Although there are situations when a function cannot be differentiated,
the discussion in this sectionfocuses on the function's differentiability. 





Differentiable

y = 3x^{2 }+ 2x + 1
dy/dx = 6x + 2


Consider for example, the function y = 3x^{2 }+ 2x 
1, then dy/dx = 6x + 2, which is reasonable.
When x < 1/3, dy/dx or tanθ is negative
and when x > 1/3, dy/dx or tanα is positive.
When x= 1/3, the slope of the tangent line is horizontal and thus equal
to 0. These correspond to its derivative function figure and it is obvious
that the function is differentable at all points. 



3x ^{2 }+ 2x  1 is differentiable


A function is differentiable at a point if the derivative
of the function exists at that point. A function is differentiable on
an interval if it is differentiable
at every point in the interval, as already concluded.
According to the above definition, y = 3x^{2 }+ 2x  1 function
is differentiable. 



y = (x^{2})^{0.5}
(x^{2})^{0.5} is not
Differentiable at all Points


Now, consider another example, y = (x^{2})^{0.5}.
Is y differentiable in this case at all points?
The definition of derivative gives
In order to applying the definition to the y = (x^{2})^{0.5} function, Δy/Δx
need to be find first.
At x = 0,
The left hand limit is:
The right hand limit is:
The left hand limit is not equal to the right hand limit. Thus does
not exist, in other words, the function y = (x^{2})^{0.5} is
not differentiable when x = 0. 





Differentiability and Continuity



If a function is differentiable at point b, then this
function is continuous at this specific point.
proof:
When x approaches b, the difference between f(x) and f(b)
is y_{x}  y_{b}. Since x ≠ b, y_{x}  y_{b} can
be written as:
Thus




Differentiability and Continuity I
Differentiability and Continuity II


Therefore,
This proves that when y is differentiable at point b, it is continuous
at that point.
Function y = 3x^{2 }+ 2x  1 is differentiable in its variable
range, and so it is continuous.
If a function is continuous, is it differentiable? The answer is may not
be. For example y = (x^{2})^{0.5} is continuous, but it
was shown that it is not differentiable. 





Differentiability and Continuity

Not Differentiable at a  A Corner
Not Differentiable at a  Kink or Loop


Function y = (x^{2})^{0.5} is not differentiable at 0
because its left hand limit is not equal to its right hand limit. Normally,
if a function's graphic has a corner or kink (loop), then the function
is not differentiable.




Not Differentiable at a  Function Discontinuous 

If a function's graphic is discontinuous, then this function is not differentiable.
since the function's left and right hand limits are different. 



Not Differentiable at a  Vertical
Tangent Line 

When the curve of a continuous function has a vertical tangent line
at point a, the slope of the tangent line is infinity. This means
Thus the function is not differentiable. 


