Continuity and Differentiability quiz

The derivative of sin 2x with respect to x is:

The derivative of e^(3x) is:

The function f(x) = abs(x) at x = 0 is:

The derivative of log(sin x) is:

The derivative of tan^-1 x is:

The greatest integer function f(x) = [x] is discontinuous at:

If x = t^2 and y = t^3, then dy/dx is:

The derivative of cos^-1 x is:

The value of k for which f(x) = kx^2 for x <= 2 and f(x) = 3 for x > 2 is continuous at x = 2 is:

If y = x^3, then the second derivative of y with respect to x is:

Examine the continuity of f(x) = 2x + 3 at x = 1. (2 marks)

Differentiate sin(cos(x^2)) with respect to x. (2 marks)

Find dy/dx if 2x + 3y = sin y. (2 marks)

Differentiate e^(sin^-1 x) with respect to x. (2 marks)

Find dy/dx if x = 2at^2 and y = at^4. (2 marks)

Differentiate log(x + sqrt(x^2 + 1)) with respect to x. (2 marks)

Find the second derivative of y = e^x sin x. (2 marks)

Is the function f(x) = 1/x continuous? Explain with reference to its domain. (2 marks)

Differentiate (sin x)^x with respect to x. (2 marks)

Find the points of discontinuity of f(x) = (x^2 - 9)/(x - 3), if any, and explain. (2 marks)

If y = sin^-1 x, show that (1 - x^2) y'' - x y' = 0. (3 marks)

If x = at^2 and y = 2at, find dy/dx and the second derivative of y with respect to x. (3 marks)

Find k if the function f(x) = (sin 2x)/x for x not equal to 0, and f(0) = k, is continuous at x = 0. (3 marks)

Differentiate y = log(log x), x > 1, and find the value of dy/dx at x = e. (3 marks)

Differentiate y = 5^x + x^5 and evaluate dy/dx at x = 1 (leave log 5 as it is). (3 marks)

Read the passage and answer the questions. A taxi company charges Rs 50 for the first 2 km and Rs 15 per km after that. The fare for x km is f(x) = 50 for 0 < x <= 2 and f(x) = 50 + 15(x - 2) for x > 2. (i) Is f continuous at x = 2? (ii) Find the left and right hand derivatives at x = 2. (iii) Is f differentiable at x = 2? (5 marks)

Read the passage and answer the questions. The position of a particle moving along a curve is given by x = cos t + t sin t and y = sin t - t cos t, where t is time. (i) Find dx/dt and dy/dt. (ii) Find dy/dx. (iii) Find dy/dx at t = pi/4. (5 marks)

Read the passage and answer the questions. A population model gives P(t) = 1000 e^(0.05t), where t is in years. (i) Find dP/dt. (ii) Find the second derivative of P. (iii) Show that dP/dt is proportional to P. (5 marks)

Read the passage and answer the questions. A student needs to differentiate y = (x + 1)(x + 2)(x + 3)/(x + 4) for x > -1. The teacher suggests taking logarithms first. (i) Write log y as a sum and difference of logarithms. (ii) Find dy/dx. (iii) Why is this method easier here? (5 marks)

Read the passage and answer the questions. The curve of a bridge arch satisfies x^2 + y^2 = 25 (in metres) for y >= 0. (i) Find dy/dx by implicit differentiation. (ii) Find the slope at the point (3, 4). (iii) Find the second derivative of y at the point (3, 4). (5 marks)

Discuss the continuity of f(x) = x + 2 for x <= 1, f(x) = x - 2 for x > 1 at all points. Draw its graph. (6 marks)

Differentiate y = (cos x)^x + (sin x)^(1/x) with respect to x. (6 marks)

If y = (tan^-1 x)^2, show that (x^2 + 1)^2 y'' + 2x(x^2 + 1) y' = 2. (6 marks)

If x sqrt(1 + y) + y sqrt(1 + x) = 0, for -1 < x < 1 and x not equal to y, prove that dy/dx = -1/(1 + x)^2. (6 marks)

Prove that the greatest integer function f(x) = [x] is not differentiable at x = 1 and x = 2, and draw its graph from -1 to 3. (6 marks)