Chapter 4 Flashcards
What is the definition of a absolute maximum(minimum)?
Let c be a number in the domain D of a function f. Then f(c) is the absolute maximum(minimum) value of f on D if f(c) >/= (=) f(x) for all x in D.
What is the definition of a local maximum(minimum)?
The number f(c) is a local maximum/minimum value of f if f(c) >/= (=) f(x) when x is near c.
What is the Extreme Value Theorem?
If f is continuous on a closed interval (a,b), then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in (a,b)
What is Fermat’s Theorem?
If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c)=0
What is the definition of a critical number?
A critical number of a function f is a number c in the domain of f such that either f’(c)=0 or f’(c) does not exist. If f has a local maximum or minimum at c, then c is a critical number of f
What is the closed interval method for finding the absolute maximum and minimum values of a continuous function f on a closed interval (a,b)?
- Find the values of f at the critical numbers of f in (a,b)
- Find the values of f at the endpoints of the interval.
- The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
What is Rolle’s Theorem?
Let f be a function that satisfies the following three hypotheses:
1.f is continuous on the closed interval (a,b)
2.f is differentiable on the open interval (a,b)
3.f(a) =f(b)
Then there is a number c in (a,b) such that f’(c)=0
What is The Mean Value Theorem?
Let f be a function that satisfies the following hypotheses:
1.f is continuous on the closed interval (a,b)
2.f is differentiable on the open interval (a,b)
Then there is a number c in (a,b) such that f’(c)=(f(b)-f(a))/(b-a)
What is the Increasing/Decreasing Test?
(a) If f’(x)>0 on an interval, then f is increasing on that interval.
(b) If f’(x)<0 on an interval, then f is decreasing on that interval
What is the First Derivative Test?
Suppose that c is a critical number of a continuous function f.
(a) If f’ changes from positive to negative at c, then f has a local maximum at c.
(b) If f’changes from negative to positive at c, then f has a local minimum at c.
(c) If f’ is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c.
What is the definition of concave upward and concave downward?
If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.
What is the Concavity Test?
(a) If f”(x)>0 for all x in I, then the graph of f is concave upward on I.
(b) If f”(x)<0 for all x in I, then the graph of f is concave downward on I.
What is the definition of an inflection point?
A point P on a curve y=f(x) is called an inflection point if f is continous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.
What is the Second Derivative Test?
Suppose f” is continuous near c.
(a) If f’(c)=0 and f”(c)>0, then f has a local minimum at c.
(b) If f’(c)=0 and f”(c)<0, then f has a local maximum at c.
What is L’Hopsitals’s Rule?
Suppose f and g are differentiable and g’(x) /=/ 0 on an open interval I that contains a (except possibly at a). Suppose that
limf(x) as x->a =0, and limg(x) as x->a =0
OR
limf(x) as x->a =+/-infinity, and limg(x) as x->a =+/-infinity
(In other words, we have an indeterminate form of type 0/0 or infinity/infinity)
Then lim f(x)/g(x) as x->a =limf’(x)/g’(x) as x->a
