Continuity and Differentiability — Practice

Question 1

Check whether f(x)=3x+2 if x<=1, 5x if x>1, is continuous at x=1.

Accepted Answer

LHL = 3(1)+2 = 5. RHL = 5(1) = 5. f(1) = 3(1)+2 = 5 (first branch, since x<=1 includes 1). Answer: All three match, so f is continuous at x=1.

Question 2

Differentiate sin(3x^2) w.r.t. x.

Accepted Answer

By the chain rule: d/dx[sin(3x^2)] = cos(3x^2).6x. Answer: dy/dx = 6x.cos(3x^2).

Question 3

Find dy/dx if x^2+y^2=25.

Accepted Answer

Differentiating both sides: 2x+2y.dy/dx=0. Answer: dy/dx = -x/y.

Question 4

Differentiate y=x^(cos x) w.r.t. x.

Accepted Answer

Taking log: log(y)=cos(x).log(x). Differentiating: (1/y)(dy/dx) = -sin(x).log(x) + cos(x)/x. Answer: dy/dx = x^(cos x).[cos(x)/x - sin(x).log(x)].

Question 5

If x=t^2, y=t^3, find dy/dx.

Accepted Answer

dx/dt=2t, dy/dt=3t^2. Answer: dy/dx = 3t^2/2t = 3t/2.

Question 6

Find the second order derivative of x^4.

Accepted Answer

First derivative: 4x^3. Second derivative: 12x^2. Answer: d^2y/dx^2 = 12x^2.

Question 7

Find k so that f(x)=kx+5 if x<=2, x^2-1 if x>2, is continuous at x=2.

Accepted Answer

LHL/f(2) = 2k+5. RHL = 4-1=3. Setting 2k+5=3 gives 2k=-2. Answer: k=-1.

Question 8

Show that f(x)=|x-2| is not differentiable at x=2.

Accepted Answer

For x>2, f(x)=x-2, so the right-hand derivative is 1. For x<2, f(x)=2-x, so the left-hand derivative is -1. Answer: Since the left-hand derivative (-1) is not equal to the right-hand derivative (1), f is not differentiable at x=2.

Practice — Continuity and Differentiability

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