Unit 4 - Applications of Derivatives

extreme value theorem

if a function is continuous from [a,b], then there must be an abs. max and an abs. min

critical points

the points where f’(x) DNE or =0.

these are where max/mins will occur

how to find abs max/min

1. find critical points of f(x)

2. plug crit point x’s into f(x)

3. include the endpoints as x values

4. whichever gives you the smallest y-value is the min, and largest is the max

local extremes vs absolute extremes

locals don’t include domain endpoints

absolutes do

the first derivative test

• how we find where a function is increasing/decreasing

• also how we find local mins/maxes

• not every critical point is a min/max

1. find crit points

2. divide into intervals based on crit points

3. find where f‘ is positive or negative by using test values

4. use first derivative test for local max/mins

concavity

• the function is concave up if it looks like a smiley face, concave down if looks like frowny face

• f(x) is concave up if f’’(x) > 0

• f(x) is concave down if f’’(x) < 0

critical point vs inflection point

• crit points: refers to the point where the function may change its direction and f’ = 0 or undefined

• inflection points: refers to the point wher concavity may change, and f’’ = 0 or undefined

second derivative test

1. if f’(c) = 0 and f’’(c) < 0, then f has a local max at x = c

2. if f(c) = 0 and f’’(c) > 0 then f has a local min at x = c

3. Note: a critical point that is also an inflection point is neither a local max or min

how to do optimization problems

1. draw a diagram

2. determine the quantity that must be min/maxed

3. introduce a “master equation” that includes the special quantity

4. write another equation that involves the variables of the master equation (and likely something given to you by the question)

5. rewrite the second equation to isolate one of the variables, then substitute it into the master equation

6. simplify, then find the abs max/min (derive, and then solve for the variable, then use it to find the other one)

graphs of f’(x)

just remember what it represents. like fully write out “f’(x) = slope of f(x)” and then what a value of 0 would mean.

linear approximation formula

Estimate = f(a) + f‘(a) • (x-a)

(first term of taylor series)

• x = is the value they give you

• a = is a nearby “nice value”

• f = is the function x was plugged into (sin0.1 = sinx, x = 0.1, a = 0)