# Calculus 1A, College of the Desert, Chapter 4.txt

 .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } Critcal 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. Closed interval method To find the absolute maximum and minimum values of a continuous function F on a closed interval [a,b]:1) Find the values of F at the critical numbers of F in (a,b).2) Find the values of F at the endpoints of the interval.3) 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. Roll's Therom Let F be a function that satisfies the following three hypotheses:1) F is contiuous 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 Guidlines for sketching a curve (a-h) A. DomainB. InterceptsC. Symmetry (even/odd)D. AsymptotesE. Increasing/DecreasingF. Local min and max's valuesG. Concavity & points of inflectionH. Sketch Mean Value Therom Let F be a function that satisfies the following hypotheses:1) F is continuouse 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)OR EQUIVALENTLY,F(b) - F(a) = F'(c)(b-a) Theorem 5 If F'(x) = 0 for all x in an interval (a,b), then F is consistant on (a,b). Corollary If F'(x) = g'(x) for all x in an interval (a,b), then F - g is constant on (a,b); that is, F(x) = g(x) + c is a constant. 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. First derivitive test Suppose that "c" is a critical number of a continuous function F.A) If F' changes from positive to negitive at c, then F has a local maximum at c.B) If F' changes from negitive to positive at c, then F has a local minimum at c.C) If F' does not change sign at c (for example, if F' is positive on both sides of c or negative on both sides), then F has no local maximum or minimum at c. Concavity 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. 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. Inflection point A point P on a curve y = F(x) is called an INFLECTION POINT if F is continuous there and the curve changes from concave upward to concave downward or vis versa at P. Secound Derivitive 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. L'Hospital's Rule Suppose F and g are differentaible and g'(x) =/ 0 on an open interval I that contains a (except possibly at a). Suppose thatlim F(x) = 0. And lim g(x) = 0x-->a. x-->aOR THATlim F(x) = +- inf. x-->a. ANDLim g(x) =+-infx-->a(In other words, we have an indeterminate form of type 0/0 or inf/inf) Then Lim (F(x)/g(x))=Lim(F'(x)/g'(x))x->a. x->a if the limit on the right side exists (or is inf or - inf). 1st Derivitive Test for Absalute Extreme Values Suppose that c is a critical number of a continuous function F defined on an interval.A) If F'(x) > 0 for all x < c and F'(x) < 0 for all x > c, then F(c) is the absolute maximum value of F.B) If F'(x) < 0 for all x < c and F'(x) > 0 for all x > c, then F(c) is the absolute minimum value of F. .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } AuthorMattyj1388 ID33032 Card SetCalculus 1A, College of the Desert, Chapter 4.txt Descriptiongeneral info Updated2011-06-23T22:24:27Z Show Answers