Continuity and Differentiability

34 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Continuity and Differentiability' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Continuity and Differentiability'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Continuity and Differentiability'? medium MCQ Problem Solving & Optimization
• 4 Evaluate displaystyle lim_(x→ 0)(sin(5x)-5x)/x^3 using L'Hôpital's rule. medium 3 marks L'Hôpital's rule for higher-order limits
• 5 Let f(x)=(x^3-8)/(x-2), x≠ 2 12, x=2. (i) Show that f is continuous at x=2. (ii) Find f'(2). hard 4 marks Continuity and derivative of piecewise function via simplification
• 6 Find ddx(x^2 e^x) and evaluate it at x=3. medium 2 marks Product rule differentiation
• 7 Using the definition of derivative, find f'(1) for f(x)=√(x+3). medium 2 marks Derivative from first principles
• 8 Evaluate lim_(x→0)(e^2x-1-2x)/x^2. medium 2 marks Limit using L'Hôpital's rule
• 9 Let g(x)=x^3-6x^2+9x+2 on [0,3]. Show that g satisfies Rolle's theorem and find all c∈(0,3) such that g'(c)=0. hard 4 marks Rolle's theorem and finding stationary points
• 10 Let f(x)=3x+1, x≤ 2 x^2-4, x>2. Which statement about the continuity of f at x=2 is correct? easy MCQ 1 mark Continuity of piecewise functions
• 11 Using the definition of derivative, find f'(3) for f(x)=√(2x+1). medium 2 marks Derivative from first principles
• 12 Let h(x)=x^4-4x^3+6x^2-4x be defined on the closed interval [0,2]. (i) Verify that h satisfies the hypotheses of Rolle’s theorem on [0,2]. (ii) Find all numbers c in (0,2) such that h'(c)=0. medium 3 marks Rolle’s theorem and finding critical points
• 13 The curve defined implicitly by x^3+y^3=6xy passes through the point (3,3). (i) Find dy/dx at (3,3) using implicit differentiation. (ii) State whether the tangent at this point is horizontal, vertical, or neither. hard 4 marks Implicit differentiation and tangent classification
• 14 Consider the function f(x)=(sin x)/x, x≠ 0 1, x=0. Which of the following statements about the continuity of f at x=0 is correct? easy MCQ 1 mark Continuity of piecewise function at a point
• 15 Using the definition of derivative, find f'(2) for f(x)=√(5x-3). easy MCQ 1 mark Derivative by definition
• 16 The curve defined by x^2y+y^3=4 passes through the point (1,1). (i) Find dydx at (1,1) using implicit differentiation. (ii) State whether the tangent at this point is horizontal, vertical, or neither. hard 3 marks Implicit differentiation and tangent nature
• 17 Let f(x)=x^2-4x+5, x<3 2x-1, x≥ 3. Determine whether f is continuous at x=3. medium 2 marks Continuity of piecewise functions
• 18 Let p(x)=x^3-3x^2+2x be defined on [0,2]. (i) Verify that p satisfies the hypotheses of Rolle’s theorem on [0,2]. (ii) Find all numbers c in (0,2) such that p'(c)=0. hard 3 marks Rolle’s theorem application
• 19 Let f(x)=(x^2-1)/(x-1), x≠ 1 3, x=1. Which of the following is true about the continuity of f at x=1? easy MCQ 2 marks Continuity of piecewise functions
• 20 Verify that p(x)=x^3-3x satisfies the hypotheses of Rolle’s theorem on [0,√3] and find all numbers c in (0,√3) such that p'(c)=0. medium 3 marks Rolle’s theorem application
• 21 The curve defined implicitly by x^2+xy+y^2=3 passes through the point (1,1). (i) Find dy/dx at (1,1) using implicit differentiation. (ii) State whether the tangent at this point is horizontal, vertical, or neither. hard 4 marks Implicit differentiation and tangent nature
• 22 Find f'(2) for f(x)=√(2x+5) using the definition of derivative. medium 3 marks Derivative by definition
• 23 Evaluate displaystylelim_(x→0)(sin(7x)-7x)/x^3 medium 3 marks Limit using series
• 24 Let h(x)=x^4-8x^3+18x^2-8x+1 on the closed interval [1,3]. hard 4 marks Rolle’s theorem
• 25 Using the definition of derivative, find f'(5) for f(x)=sqrt[3],2x+1, medium 3 marks Derivative via definition
• 26 Let g(x)=x^3-6x^2+9x on the closed interval [0,3]. hard 4 marks Rolle’s theorem and critical points
• 27 Consider the function f(x)=(sin x)/x, x≠0 1, x=0. Which of the following statements about the continuity of f at x=0 is correct? medium MCQ 2 marks Continuity of piecewise trigonometric function
• 28 Using the definition of derivative, find f'(2) for f(x)=√(3x+4). easy 2 marks Derivative by definition
• 29 Evaluate lim_(x→0)(e^4x-1-4x-8x^2)/x^3. hard 3 marks Limit using series expansion
• 30 Let h(x)=x^4-6x^3+9x^2+4 be defined on [0,3].\n(i) Verify that h satisfies the hypotheses of Rolle’s theorem on [0,3].\n(ii) Find all numbers c∈(0,3) such that h'(c)=0.\n(iii) Compute h''(c) for each such c. hard 5 marks Rolle’s theorem, differentiation of polynomials
• 31 Using the definition of derivative, find f'(3) for f(x)=√(2x+1). medium 4 marks Derivative definition for radical function
• 32 Let\ f(x)=x^2-4x-2, x≠2 3, x=2 Which statement about the continuity of f at x=2 is correct? easy MCQ 5 marks Continuity of piecewise function
• 33 The curve defined implicitly by x^3+x y^3=12 passes through the point (2,1).\n(i) Find dy/dx at (2,1) using implicit differentiation.\n(ii) State whether the tangent at this point is horizontal, vertical, or neither. hard 4 marks Implicit differentiation and slope of tangent
• 34 Let g(x)=x^3-3x^2+2x defined on the interval [0,2]. Verify the hypotheses of Rolle’s theorem and find all c∈(0,2) such that g'(c)=0. medium 3 marks Rolle’s theorem