Q1 If a function is derivable at a point, is it necessary that it must be continuous at that point?

Solution

Q2 If a function is continuous at a point, is it necessary that it must be derivable at that point?

Solution

No,it is not necessary that the function is derivable at that point where it is continous

Q3 Is the function f(x) = |x| derivable at x = 0?

Solution

Q4 Is the function f(x) = cot x derivable at x = 0?

Solution

Q5 Is the function f(x) = sec x derivable at x = π/2

Solution

Q6 When a function is called derivable?

Solution

A function is derivable at a point where there is a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.

Q7 Which of the following functions are derivable?

1. sin x

Solution

Q8 Examine the function
f(x)=

1+x ,if x≤2

5-x ,if x>2

for differentiability at x=2

Solution

Q9 Examine the following function for continuity at x = 1 and differentiability at x = 2.
f(x)=

5x-4 ,0< x< 1

4x2-3x ,1< x< 2

3x+4 ,x≥2

Solution

Q10 Show that the function f(x) = 2x — |x| is continuous at x = 0 but not differentiable at x=0.

Solution

Q11 Show that f(x) = |x -5| is continuous but not differentiable at x = 5.

Solution

Q12 If the function f(x) = |x-3| + |x-4|, then show that f is not differentiable at x = 3 and x = 4.

Solution

Q13 Show that the function f is continuous at x = 1 for all values of a where
f(x)=

ax2+1,x≥1

x+a ,x< 1

Find its right and left hand derivatives at x = 1. Hence, find the condition for the existence of the derivative at x = 1.

Solution

Q14 Prove that the function f(x)=

x / |x| ,x≠0

1, x=0

,is not differentiable at x=0

Solution