Exam 3

Question 1

Extreme Value Theorem

Answer 1

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some number c,d in [a,b]

Question 2

Rolle’s Theorem

Answer 2

Let f be a function satisfying that f is continuous on [a,b] , f is differentiable (a,b), and f(a) = f(b)
Then, there is a number c in (a,b) such that f’(c) = 0

Question 3

Mean Value Theorem

Answer 3

Let f be a function that is continuous on [a,b] and differentiable on (a,b)
Then, there is a number c in (a,b) such that
f’(c) = (f(b)-f(a)) / b-a

Question 4

Riemann Sum Definition

Answer 4

For a function f defined on the interval [a,b] the definite integral of f from a to b is the number

integral from a to b f(x) = limit as max Δxi approaches 0 sum of n (top) i=1 f(xi)Δ xi

provided this limit exists. If it does, we say f is integratabtle on [a,b]

Question 5

What is absolute/global maximum

Answer 5

An Absolute Maximum is a value f(c)≥ f(x) for all “x” in the domain

Question 6

What is relative or local maximum

Answer 6

A Local Maximum is a value f(c)≥ f(x) near “c” in the domain

Question 7

What is absolute/global minimum

Answer 7

An Absolute Minimum is a value f(c)≤f(x) for all “x” in the domain

Question 8

What is relative or local minimum

Answer 8

A Local Maximum is a value f(c)≤f(x) near “c” in the domain

Question 9

what is a critical number

Answer 9

a critical number of a function f is a number c in the domain such that either f’(c) = 0 or f’(c) = DNE

Question 10

what does it mean when c is a critical number of f

Answer 10

there is a potential minimum or maximum at c

Question 11

Intermediate Value Theorem

Answer 11

Let a function, f(x) that is continuous on a closed interval [a, b], and L is any value between f(a) and f(b).
Then, there exists a value c within the interval such that f(c) = L.