Derivatives

Question 1

For a function f to be differentiable at x=a, the function has to be..

Answer 1

If f is differentiable at a point x0, then f must also be continuous at x0. In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold:

Question 2

A continuous function need not be differentiable. True or False?

Answer 2

True. A continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

Question 3

Example of of a function which is continous but not differentiable.

Answer 3

An ordinary cusp on the cubic curve (semicubical parabola) x³ – y² = 0, which is equivalent to the multivalued function f(x) = ± x^3/2. This relation is continuous, but is not differentiable at the cusp.

Question 5

Rolle’s theorem

Answer 5

If a real-valued function ƒ is continuous on aclosed interval [a, b], differentiable on the open interval(a, b), and ƒ(a) = ƒ(b), then there exists a c in the open interval (a, b) such that f’(c) = 0.

Indian mathematician Bhāskara II (1114–1185) is credited with knowledge of Rolle’s theorem.[1]

Question 7

Mean Value Theorem

Answer 7

For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c.

Question 8

Proof of Mean Value Theorem

Answer 8

1. Let g(x) = f(x) − rx, where r is a constant.
2. Since f is continuous on [a, b] and differentiable on (a, b), the same is true for g.
3. Choose r so that g satisfies the conditions of Rolle’s theorem.
4. Since g is differentiable and g(a) = g(b), there is some c in (a, b) for which g′(c) = 0, and it follows from the equality g(x) = f(x) − rx that,

Question 9

Which theorem do we use to prove the Mean Value Theorem

Answer 9

Rolle’s theorem

Question 10

If f is differentiable at x = a then

Answer 10

The limit h approaching 0, (f(a+h) - f(a))/h

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