Differentiability and Continuity Worksheet :
Here we are going to see some practice questions on differentiability and continuity.
(1) Find the derivatives of the following functions using first principle.
(i) f(x) = 6 Solution
(ii) f(x) = -4x + 7 Solution
(iii) f(x) = -x2 + 2 Solution
(2) Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
(i) f(x) = |x - 1| Solution
(ii) f(x) = √(1 - x2) Solution
(3) Determine whether the following function is differentiable at the indicated values.
(i) f(x) = x | x | at x = 0 Solution
(ii) f(x) = |x2 - 1| at x = 1 Solution
(iii) f(x) = |x| + |x - 1| at x = 0, 1 Solution
(iv) f(x) = sin |x| at x = 0 Solution
(4) Show that the following functions are not differentiable at the indicated value of x.
(5) The graph of f is shown below. State with reasons that x values (the numbers), at which f is not differentiable.
(6) If f(x) = |x + 100| + x2, test whether f'(-100) exists.
(7) Examine the differentiability of functions in R by drawing the diagrams.
(i) | sin x | Solution
(ii) |cos x| Solution
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