A function is not differentiable at x = a under any of the following conditions.
1) The function has discontinuity.
2) The graph of the function has a sharp corner or cusp.
3) The tangent line at x = a has a vertical slope.
Determine whether the following function is continuous, differentiable, neither or both at the point.
Problem 1 :
Solution :
The function is not continuous, it is not differentiable. It has jump discontinuity. So, it is neither.
Problem 2 :
Solution :
The function has removable discontinuity. The function f(x) is discontinuous, then it is not differentiable.
So, it is neither.
Problem 3 :
Solution :
The function is continuous, it is differentiable. So, it is both.
Problem 4 :
Solution :
The function is continuous, at x = 0, we see the vertical tangent. So, the function is continuous but not differentiable at the point x = 0.
Problem 5 :
Solution :
The function is discontinuous at x = 2, it has infinite discontinuity. So, the function is discontinuous and it is not differentiable. Then it is both.
Problem 6 :
Solution :
The function is continuous. Since we have sharp point at x = 0. The function is not differentiable. So, the function is continuous but not differentiable.
Problem 7 :
Solution :
The function is continuous. At x = 2, we can draw the vertical tangent line. Then the function is continuous but not differentiable at the point x = 2.
Problem 8 :
Solution :
The function is continuous. At x = -1, we can draw the vertical tangent line. Then the function is continuous but not differentiable at the point x = -1.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM