Unit 5: Analytical Applications of Differentiation

Question 1

Mean Value Theorem

Answer 1

If the function f(x) is:
- continuous on the closed interval [a,b] and
- differentiable on the open interval (a,b),
then, f’(c) = (f(b)-f(a))/(b-a) for some c in the open interval (a,b)

Question 2

Mean Value Theorem condition

Answer 2

f(x) is continuous on [a,b] and differentiable on (a,b)

Question 3

Define The function f(x) is differentiable

Answer 3

f(x) is differentiable. So f’(x) exists for all x in the domain. You can use MVT on f(x) to draw conclusions about f’(x)

Question 4

Define The function f(x) is twice differentiable

Answer 4

f(x) and f’(x) are both differentiable. You can use MVT on f(x) to draw conclusions about f’(x) and use MVT on f’(x) to draw conclusions about f’‘(x)

Question 5

Define The function f(x) is continuously differentiable

Answer 5

Means that the function and all of its derivatives are differentiable. This is the best. You can use MVT on f(n) (x) to draw conclusions about f(n+1) (x)

Question 6

Define Rolle’s Theorem

Answer 6

If the function f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then f’(c) = 0 for some c where c E (a,b)

not really important

Question 7

When to use MVT

Answer 7

• apply the MVT
• prove that f’(x) equals some particular value somewhere on some interval
• show that the function must have a tangent line with some particular slope
• how many times must f’(x) equal some particular value on some interval?