Continuity and Differentiability

Continuity and Differentiability extends the calculus you began in Class 11 and forms the base for applications of derivatives and integrals. In this chapter you will learn that a function is continuous at a point when its limit there exists and equals its value, so its graph can be drawn without lifting the pencil. You will test continuity of polynomial, rational, trigonometric, modulus and greatest integer functions and piecewise functions, use the algebra of continuous functions, and see that every differentiable function is continuous though the converse is false, as the modulus function at x = 0 shows. You will then build a complete toolkit of differentiation: the chain rule for composite functions, derivatives of inverse trigonometric functions, implicit differentiation, derivatives of exponential and logarithmic functions, logarithmic differentiation for functions such as x^x, and differentiation of functions in parametric form. The chapter ends with second order derivatives and proofs of relations such as x^2 y'' + x y' + y = 0, a common board exam question.

Detailed Worksheet: Continuity and Differentiability Section A - Definitions (10 marks) 1. Define continuity of a function f at a point x = k of its domain. (2 marks) 2. Show that the modulus function f(x) = abs(x) is continuous at x = 0. (2 marks) 3. State the chain rule for differentiating a composite function. Use it to differentiate sin(x^2 + 5). (2 marks) 4. Write the derivatives of sin^-1 x and tan^-1 x, stating the domains on which they hold. (2 marks) 5. Every differentiable function is continuous. Is the converse true? Justify with an example. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the value of k so that the function f(x) = kx + 1 for x <= 5 and f(x) = 3x - 5 for x > 5 is continuous at x = 5. (3 marks) 7. Differentiate with respect to x: (i) (3x^2 - 7x + 3)^(5/2) and (ii) cos(sqrt(x)). (3 marks) 8. Find dy/dx if x^2 + xy + y^2 = 100. (3 marks) 9. Differentiate (i) y = x^x and (ii) y = x^(sin x), x > 0, using logarithmic differentiation. (3 marks) 10. Find dy/dx if x = a(theta + sin theta) and y = a(1 - cos theta). Express the answer in terms of theta/2. (3 marks) Section C - Diagrams (10 marks) 11. Draw the graph of f(x) = abs(x). Use the graph to explain why f is continuous but not differentiable at x = 0, by finding the left hand and right hand derivatives. (4 marks) 12. Draw the graph of the greatest integer function f(x) = [x] for -2 <= x < 3. Mark the points of discontinuity and explain why it is discontinuous there. (3 marks) 13. Draw the graph of f(x) = x + 1 for x <= 1 and f(x) = 3 - x for x > 1. Determine from the graph whether f is continuous at x = 1. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. If y = 3 cos(log x) + 4 sin(log x), show that x^2 y'' + x y' + y = 0. (5 marks) 15. Find the values of the constants p and q such that the function f(x) = 5 for x <= 2, f(x) = px + q for 2 < x < 10 and f(x) = 21 for x >= 10 is continuous. (5 marks) 16. Using a suitable substitution, differentiate (i) tan^-1(2x/(1 - x^2)) for -1 < x < 1 and (ii) sin^-1(2x sqrt(1 - x^2)) for -1/sqrt(2) < x < 1/sqrt(2). Explain why the substitution simplifies the work. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show every step, state the rule used, and simplify the final answer.