U5 Calc

If f(x) is increasing, f’(x) is…

+Positive

If f(x) is decreasing, f’(x) is

-negative

f(x) has a critical point if f’(x) is

0 or undefined

f(x) has a relative max if f’(x) is

0(CP) and changes from pos to neg

f(x) has a relative max if f”(x) is

-negative (as long as f’ is 0)

f(x) has a relative min if f’(x) is

0, changing from -to+

f(x) has a relative min if f”(x) is

+positive when f’ is 0

f(x) has an inflection point if f’ has

a MAX/inc→dec ORRR a MIN/dec→inc

f(x) has an inflection point if f” is

0 and is going from +→- OR -→+

f(x) is concave up if f’ is

increasing

f(x) is concave up if f” is

+positive

f(x) is concave down if f’ is

decreasing

f(x) is concave down if f” is

-negative

extreme value theorem

if f is continuous on an interval, there is a guaranteed to be an absolute maximum and minimum

mean value theorem

if f is continuous and differentiable on an interval [a,b] then there is at least one number x=c in (a,b) where f’( c ) = AROC

IROC=AROC at least once if it’s continuous and differentiable

how to find if a function is inc/dec

find when f’(x)=0, make a sign chart, sample points to find the sign between the zeros, justify (f is inc cuz f’ is + or vice versa)

To identify relative extreme use the first derivative test which is…

find the derivative, find the 0s of the derivative, determine if it’s going + to - or - to +, min if -+ and max if +-

identifying absolute extrema, use candidates test which is…

find all critical points, list all critical points and ENDPOINTS in a table, evaluate the function f(x) at these points and select answer as the xvals with the greatest/lowest yvals

second derivative test is…

if f’(a) = 0 and f”(a) is +, then f(x) has a relative min at x=a

if f’(a) = 0 and f”(a) is -, then f(x) has a relative max at x=a

first derivative HAS to be 0 (not undefined)

what are the steps for optimization

1. write an equation for the quantity you want to maximize or minimize (what is the largest area: A=lw)

2. use constraints to find a relationship between the variables (60 feet of fencing to make 3 sides (one side is done): 2w+l=60)

3. rewrite first ran in terms of ONE variable (60=2w+l → l=60-2w, sooo A=w(60-2w))

4. Find critical points and use candidates test (include endpoints!)

5. identify what the question is asking and give appropriate answer based on question (if it asks for area say: largest area is 450 ft². if it ask for what width gives the biggest area you would say: 15 feet as the width would give the largest area (of 450 ft²)