Continuity tests, differentiability, chain rule, implicit and parametric differentiation, Rolle's and MVT.
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Q1 · Continuity and Differentiability (Class 12) · EASY
Let f(x) = (x^2 - 4)/(x - 2) for x ≠ 2 and f(2) = k. For what value of k is f(x) continuous at x = 2?
Q2 · Continuity and Differentiability (Class 12) · MEDIUM
If y = x^3 + 3x^2y + y^3 = 0 defines y implicitly as a function of x, find dy/dx at (0, 0).
Q3 · Continuity and Differentiability (Class 12) · HARD
Verify Rolle's theorem for the function f(x) = x^2 - 4x + 3 on the interval [1, 3]. What is the value of c in (1, 3) where f'(c) = 0?
Q4 · Continuity and Differentiability (Class 12) · MEDIUM
The function f(x) = |x − 3| is not differentiable at