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AP Calc ch 3
Extrema and Linear Approximation
Terms in this set (28)
occurs at "hills" on the graph. There is an open interval on which it is the maximum of the function.
occurs at "valleys" on the graph. There is an open interval on which it is the minimum of the function
a point where relative extrema can occur where the derivative is 0 or undefined
critical number/critical value
Let f be defined at c. If f'(c) = 0, or if f is not differentiable at c, then c is a ________ __________ of f. c must be in the domain of f, but domain of f'. Relative extrema can only occur at critical points.
Absolute/global extrema informal definition
Informally, THE highest or lowest value a function achieves on its entire domain (or on an interval). Sometimes referred to as the max or the min. Can occur at the endpoints of the interval, or on the interior of the interval (at relative/local extrema).
Absolute/global extrema formal definition
Let f be a function that is defined on an interval that contains c. f(c) is the minimum of f on the interval iff f(c) = - for all x in the interval. f(c) is the maximum of f on the interval iff f(c) = + for all x in the interval.
Intermediate Value Theorem (IVT)
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
The Extreme Value Theorem (EVT)
If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval.
Mean Value Theorem
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a) (c is an element of (a, b)).
First verify and state that the conditions have been met before invoking this theorem.
A function is strictly __________ on an interval when it is either increasing on the entire interval, or decreasing on the entire interval.
The First Derivative Test
Used to classify critical points as relative maxima, relative minima, or neither.
A critical point of f represents a local maximum when...
f changes from increasing to decreasing or f' changes from positive to negative.
A critical point represents a local minimum when...
f changes from decreasing to increasing or f' changes from negative to positive
A critical point of f does not represent an extremum when...
f is increasing on both sides of c or decreasing on both sides of c, or f' is positive on both sides of c or negative on both sides of c
To use the First Derivative Test
Find all critical values and discontinuities. Set up a "sign chart" and fill in information about each interval. Use your sign chart to answer the question. Write a concise statement interpreting your sign chart.
T/F There is a relative maximum or minimum at each critical value.
T/F The relative maxima of the function f are f(1) = 4 and f(3)=10. Therefore f has at least 1 minimum for some value in the interval (1,3).
when f' is increasing, f'' is +
when f' is decreasing, f" is -
when f'' = 0, then f is linear on that interval.
If concavity changes at a point on the graph, where a tangent line exists, then the point is called an _______. Concavity may also change at discontinuities or piecewise breaks (not _________).
A method where the tangent line is very close to the curve near the point of tangency and can therefore be used to approximate function values in that area.
We call the equation of the tangent line the ________ of the function.
when f is concave upward...
The graph of f lies above its tangent lines. The tangent line is an under approximation of the function value.
when f is concave downward...
The graph of f lies below its tangent lines. The tangent line over approximates the function value.
How to linear approximation
First find the equation of the tangent line. Then use the linearization to approximate (plug value into equation). Find the concavity of the equation or graph (if it is concave up it gives an under aprox. if it is convave down it gives and over aprox.).
How to optimization word problems
List out any relevant information and draw a diagram if applicable. Determine your primary equation, the equation for the quantity you want to optimize. If the primary equation has more than 2 variables, find a relevant secondary equation so you can make some substitutions. Use calculus to find the max or min (first derivative test)- remember to consider what domain is reasonable in context. Answer the question including specific language and units.
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