Mastering Functions: Maximize Graphing Efficiency

1. Calculus 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 x1 and x2 denote points in the interval. • f is increase on the interval if f (x1)< f (x2) whenever x1 < x2 • f is decrease on the interval if f (x1)> f (x2) whenever x1 < x2 • f is constant on the interval if f (x1)= f (x2) for all points x1 , x2

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)<0, for all x in (a,b) => 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 x0 and if f change the direction of concavity at the point (x0, f (x0) ), then we say that f has an inflection point at x0 and we call the point (x0, f (x0) ) 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 x0 if there is an open interval containing x0 on which f(x0) is the largest value, i.e. f(x0)≥f(x) for all x in the interval • A function f is said to have a relative minimum at x0 if there is an open interval containing x0 on which f(x0) is the smallest value, i.e. f(x0)≤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 x0. If f has a relative extreme at x= x0, then x= x0 is a critical point of f ; that is , either f ’(x0)=0 or f is not differentiable at x0

10. First Derivative Test • Theorem Suppose that f is continuous at a critical point x0 • If f ’(x) >0 on an open interval extending left from x0 and f ’(x)<0 on an open interval extending right from x0, then f has a relative maximum at x0 • If f ’(x) <0 on an open interval extending left from x0 and f ’(x)>0 on an open interval extending right from x0, then f has a relative minimum at x0 • If f ’(x) has the same sign on an open interval extending left from x0 as it does on an open interval extending right from x0, then f does not have a relative extreme at x0

11. Second Derivative Test • Theorem Suppose that f is twice differentiable at the point x0 • If f ’(x0)=0 and f ’’(x0)>0, then f has a relative minimum at x0 • If f ’(x0)=0 and f ’’(x0)<0, then f has a relative maximum at x0 • If f ’(x0)=0 and f ’’(x0)=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 nx-intercepts, at most n-1 relative extrema, at most n-2 inflection points

14. Absolute Extrema • Definition • LetI be an interval in the domain of a function f • We say that f has an absolute maximum at a point x0, in I if f (x)≤f (x0) for all x in I • We say that f has an absolute minimum at a point x0, in I if f (x)≥f (x0) for all x in I • We say that f has an absolute extreme at a point x0, 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 x0 • If f has a relative minimum at x0, then f (x0) is the absolute minimum of f on I • If f has a relative maximum at x0, then f (x0) 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.