1. Advanced Mathematics D

2. Chapter Four The Derivatives in Graphing and Application

3. Increase & Decrease Definition Let f be defined on an interval, and let x 1 and x 2 denote points in the interval.  f is increase on the interval if f ( x 1 )< f ( x 2 ) whenever x 1 < x 2  f is decrease on the interval if f ( x 1 )> f ( x 2 ) whenever x 1 < x 2  f is constant on the interval if f ( x 1 )= f ( x 2 ) for all points x 1, x 2

4. Increase & Decrease - Theorem Let f be a function that is continuous on a closed interval [ a,b ] and differentiable on the open interval ( a,b )  If f ’( x )>0, for all x in ( a,b ) => f is increase on [ a,b ]  If f ’( x ) f is decrease on [ a,b ]  If f ’( x )=0, for all x in ( a,b ) => f is constant on [ a,b ]

5. Concavity Definition  If f is differentiable on an open interval I, then f is said to be concave up on I if f ’ is increasing on I  f is said to be concave down on I if f ’ is decreasing on I

6. Concavity - Theorem  If f ’’( x )>0 for all value of x in I, then f is concave up on I  If f ’’( x )<0 for all value of x in I, then f is concave down on I

7. Inflection Points Definition If f is continuous on an open interval containing a value x 0 and if f change the direction of concavity at the point ( x 0, f ( x 0 ) ), then we say that f has an inflection point at x 0 and we call the point ( x 0, f ( x 0 ) ) on the graph of f an inflection point of f

8. Relative Extrema Definition  A function f is said to have a relative maximum at x 0 if there is an open interval containing x 0 on which f ( x 0 ) is the largest value, i.e. f ( x 0 )≥ f(x) for all x in the interval  A function f is said to have a relative minimum at x 0 if there is an open interval containing x 0 on which f( x 0 ) is the smallest value, i.e. f ( x 0 )≤ f ( x ) for all x in the interval

9. Relative Extrema - Theorem Suppose that f is a function defined on an open interval containing the point x 0. If f has a relative extreme at x = x 0, then x = x 0 is a critical point of f ; that is, either f ’( x 0 )=0 or f is not differentiable at x 0

10. First Derivative Test Theorem Suppose that f is continuous at a critical point x 0  If f ’( x ) >0 on an open interval extending left from x 0 and f ’( x )<0 on an open interval extending right from x 0, then f has a relative maximum at x 0  If f ’( x ) 0 on an open interval extending right from x 0, then f has a relative minimum at x 0  If f ’( x ) has the same sign on an open interval extending left from x 0 as it does on an open interval extending right from x 0, then f does not have a relative extreme at x 0

11. Second Derivative Test Theorem Suppose that f is twice differentiable at the point x 0  If f ’( x 0 )=0 and f ’’( x 0 )>0, then f has a relative minimum at x 0  If f ’( x 0 )=0 and f ’’( x 0 )<0, then f has a relative maximum at x 0  If f ’( x 0 )=0 and f ’’( x 0 )=0, then the test is inconclusive

12. Geometric Implications of Multiplicity Suppose that p ( x ) is a polynomial with a root of multiplicity m at x=r  If m is even, then the graph of y=p(x) is tangent to the x -axis at x=r, and no cross to x -axis no inflection point there  If m>1 is odd, then the graph of y=p(x) is tangent to the x -axis at x=r, and cross to x -axis inflection point there  If m =1, then the graph is not tangent to x -axis, cross to x -axis May or may not inflection point

13. About Polynomials Domain: ( -∞,+∞ ) Continuous everywhere Differentiable everywhere – no corners, no vertical tangent line Eventually goes to ∞ without bounds, the sign is determined by the highest term Has at most n x -intercepts, at most n- 1 relative extrema, at most n- 2 inflection points

14. Absolute Extrema Definition  Let I be an interval in the domain of a function f We say that f has an absolute maximum at a point x 0, in I if f ( x )≤ f ( x 0 ) for all x in I We say that f has an absolute minimum at a point x 0, in I if f ( x )≥ f ( x 0 ) for all x in I We say that f has an absolute extreme at a point x 0, in I if it has either an absolute maximum or an absolute minimum at that point

15. Absolute Extrema - Theorem If a function is continuous on a finite closed interval [ a,b ] then f has both an absolute maximum and an absolute minimum on [ a,b ]

16. Absolute Extrema -- Theorem Suppose that f is continuous and has exactly one relative extremum on an interval I say at x 0  If f has a relative minimum at x 0, then f ( x 0 ) is the absolute minimum of f on I  If f has a relative maximum at x 0, then f ( x 0 ) is the absolute maximum of f on I

17. Steps to Graph a Polynomial 1. find all intersection points to x -axis 2. find the intersection point to y -axis 3. find all relative extreme points 4. find increasing and decreasing intervals 5. determine infinite behaviors 6. find all inflection points 7. find convex up and down intervals 8. connect the points

18. Rolle’s Theorem Let f be continuous on the closed interval [ a,b ] and differentiable on the open interval ( a,b ), If f ( a )= f ( b )=0, then there is at least one point c in the interval ( a,b ) such that f ’( c ) = 0.

19. Mean-Value Theorem Let f be continuous on the closed interval [ a,b ] and differentiable on the open interval ( a,b ), then there is at least one point c in the interval ( a,b ) such that f ’( c ) = ( f ( b )- f ( a ))/( b-a )

20. Revisited Theorem Let f be continuous on the closed interval [ a,b ] and differentiable on the open interval ( a,b ),  If f ’( x ) >0 for all x in ( a,b ), then f is increasing on [ a,b ];  If f ’( x ) <0 for all x in ( a,b ), then f is decreasing on [ a,b ];  If f ’( x ) =0 for all x in ( a,b ), then f is constant on [ a,b ].

21. Constant Difference Theorem If f and g are differentiable on an interval I, and if f ’( x ) = g ’( x ) for all x in I, then f-g is constant on I ; that is, there is a constant k such that f ( x ) – g ( x ) = k, or f ( x ) = g ( x ) + k For all x in I.