Unit 5: Analytical Applications of Differentiation Flashcards
What makes something an absolute maximum/minimum?
f is a function on interval I. If there is a number u in I for which f(u)>=f(x) for all x in I, then there is an absolute maximum at f(u). For minimum, use <=
Endpoints can satisfy this definition, if they are >= or <= all other values.
What makes something a local extrema?
f is a function on interval I. u and v are numbers in I. If there is an open interval in I containing u so that f(u) >= f(x) for all x in this open interval, then f has a local maximum at u.
Endpoints never satisfy this definition, as there is no open interval that contains them.
Steps to find critical numbers of f.
Find f’. Find where f’ = 0 or where f’ does not exist (not differentiable). These values are the critical numbers.
Extreme value theorem.
If f is continuous on a closed interval [a,b], then f has an absolute maximum and an absolute minimum on [a,b]
If f has a local maximum or a local minimum at c, f’(c) = 0 or f’(c) does not exist.
Mean value theorem.
If f is continuous on [a,b] and differentiable on (a,b), then there is one value c in the open interval (a,b) for which f’(c) = secant line between f(a) and f(b)
How do you find where a function is increasing and decreasing?
If f’(x) > 0 on (a,b), then f is increasing on (a,b)
If f’(x) < 0 on (a,b), then f is decreasing on (a,b)
What is the first derivative test?
Finds the local minimum and maximum values.
AP justification: f has a local maximum/minimum at x=a because f’(x) changes from positive/negative to negative/positive at x=a.
How do you determine concavity?
If f’‘(x) > 0 on open interval (a,b) then f is concave up.
If f’‘(x) < 0 on open interval (a,b) then f is concave down.
Justify using f’(x) unless otherwise specified. Ex: f(x) is concave up on (a,b) because f’(x) is increasing on (a,b).
Definitions of concavity?
Concave up: The graph of f lies above each of its tangent lines throughout (a,b)
Concave down: The graph of f lies below each of its tangent lines throughout (a,b)
What is an inflection point?
If the concavity of f changes at (c,f(x)) then (c,f(c)) is an inflection point of f.
How do you find an inflection point?
Find where f’‘(x)=0 or where f’‘(x) does not exist. Test for concavity on both sides of these numbers. If concavity changes, there is an inflection point, if not, there is no inflection point.
What is the second derivative test?
First, do not use the second derivative test unless specified. You will not be given marks for it otherwise.
Where c is a critical number of f that lies in (a,b): If f’‘(c) > 0, then c is a local maximum value. If f’‘(c) > 0, then f(x) is a local minimum value.
This essentially checks for the concavity of the point at f(c), and uses it to determine whether it is a maximum or minimum.
How do you graph a function using calculus?
Find domain of f and intercepts.
Identify asymptotes and examine end behavior.
Find critical numbers, and their tangent lines.
Find where f is increasing and decreasing.
Find where f has local maximum and minimum values.
Test for concavity, identify inflection points.
Graph using gathered info. Additional points may help (as well as their tangents)
In optimization problems, what variable (that you are optimizing) do you usually use?
M. State what M is equal to, as with all variables.
What is the quadratic formula?
Where y = ax^2+bx+c
(-b +- sqrt(b^2-4ac)) / 2a
