Unit 5

Question 1

Explain MVT and how to use it

Answer 1

Has to be:
- f is continuous on the closed interval [a,b]
- f is differentiable on the open interval (a,b)
if those two are known then some value c exists such that f(b) - f(a) / b - a
How to use
1. make sure it is continuous on interval
2. make sure it is differentiable on interval
3. do the formula for the numbers in the interval and plug those numbers into the original equation ad subtract for the numerator
4. subtract denominator from the interval values
5. whatever f’(c) equals make that equal to the derivative of the function
6. that is your x value and then plug that x value into original equation to get y value

Question 2

Extreme Value Theorem and how to use it

Answer 2

if f is continuous over closed interval [a,b] then there WILL be an absolute max and abs min
f(c), the mins y value <= f(x) <= f(d), the maxs y value

Question 3

Rolle’s Theorem and how to use it

Answer 3

has to be:
- f is continuous on the closed interval [a,b]
- f is differentiable on the open interval (a,b)
- if f(a) = f(b), then there is at least one number c in (a,b) that makes the first derivative equal to zero.
to use it:
1. use the two values from the interval and plug them into the equation and they should both equal the same value, the third rule
2. differentiate the function and make it equal to zero
3. solve for x
4. the x-value is the value c that satisfies the first derivative equal to zero, your answer

Question 4

How to find critical points

Answer 4

make the first derivative equal to zero and solve for the variable
- those r the critical values
- if there are two different variable, implicitly differentiate

Question 5

absolute min and max

Answer 5

absolute min is the lowest y value at any critical point
absolute max is the highest y value at any critical point

Question 6

relative min and max

Answer 6

relative min is the y value at the other critical values that are not the absolute min, when the graph has a horizontal tangent line on the lowest point
relative max is the y value at the other critical values that are not the absolute max, also when the graph has a horizontal tangent line but at the highest point

Question 7

how to determine increasing and decreasing of a function

Answer 7

if f’(x) > 0 for all x in (a,b) then f is increasing on [a,b]
if f’(x) < 0 for all x then f is decreasing on [a,b]
on a graph of f’(x):
if finding where f(x) is increasing or decreasing then the interval above the graph is increasing
interval below the graph is decreasing
if f’(x) = 0 for all x in (a,b) then f is constant on [a,b]

Question 8

first derivative test and how to use it