Functions

Question 1

Derivative of an inverse function
d/dx (f^-1 (x))

Answer 1

1/ f ‘ (f^-1 (x))

Question 2

Rolle’s Theorem:
If f is continuous on [a, b], differentiable on (a, b), and…

Answer 2

f(a) = f(b), then there exists c = (a, b) such that f ‘ (c) = 0

Question 3

Mean Value Theorem for Derivatives:
If f is continuous on [a, b], differentiable on (a, b), then…

Answer 3

There exists a value of c = (a, b) such that f ‘ (c) = (f (b) - f (a))/b - a

Question 4

Extreme Value Theorem:
If f is continuous on a closed interval then…

Answer 4

f must have both an absolute maximum and an absolute minimum on the interval.

Question 5

Intermediate Value Theorem:
If f is continuous on [a, b], then…

Answer 5

f must take on every y-value between f(a) and f(b)

Question 6

If a function is differentiable at a point, then…

Answer 6

It must be continuous at that point (Differentiability implies continuity)

Question 7

Four ways in which a function can fail to be differentiable at a point:

Answer 7

-Discontinuity
-Corner
-Cusp
-Vertical tangent line