Derivatives of high orders:
-let f(x) have a derivative at every point x in an open interval (a,b).
The derivative f’(x) is a funtion defined on (a,b).
It may happen that f’(x) has a derivative at apoint x in (a,b).
This is the second derivative of f(x), which is denoted by f”(x) or d2y/dx2
-similary the nth order derivative of f(x) is the derivative of the (n-1)th order derivative of f(x)
ffelsőindexn(x)=ffelsőindex(n-1)(x)
dfelsőindexn y/dxfelsőindex n = [dfelsőindex(n-1) y / dxfelsőindex(n-1)]
Increasing & Decreasing Funtions
-If f (x) > 0 on the interval c<x<d, the funtion is increasing
-If f (x)<0 on the interval c<x<d, the funtion is decreasing
-m line graph
+ rising /
- falling \
0 horizontal __
Critical Points:
-a point (a, f(a) ) is a critical point if f (a) =0 or f(a) does not exist (underfined)
Why?
-a Local Minimum is a critical point whose y coordinate is less than any other point in its
neighbourhood.
(a, f(a) ) is a local minimum if there exist an epsilon>0 such that f(x)_> f(a) when lx-al<epsilon
-a Local Maximum is a critical point whose y coordinate is greater than any other point in its
neighbourhood.
(b f(b) )is a local maximum if there exist an epsilion>0 such that f(x)_<f(b) when lx-bl<epsilon
First derivative test:
-suppose y=f(x) is continous everywhere ont he interval c<x<d and that (a, f (a) )is the only critical
point in the interval.
-choose xL and xR caluse of x int he interval, respectively left and right of the critical point. f’a=0
-f’(xL) f’(xR)
(a, f(a) )
+
Local Maximum
+
Local Minimum
Second derivative test:
-second derivative f”(x) determines whether first derivatives f’(x) is increasing or decreasing.
-if f”(x)>0 on an interval (c,d) derivative f’(x) is increasing and the curve of f(x) is convex on interval.
-if f”(x)<0 on an interval (c,d) derivative f’(x) is decreasing and the curve of f(x) is concave on interval.
-If f’(a) equals zero…then:
1. The critical point (a f(a) ) is a local maximim if f”(a)<0
2. The critical point (a,f(a)) is a local minimum if f”(a)>0
3. The test fails when f”(a)=0!!!
Inflection points:
-a point (a f(a) ) is an inflection point if the graph of the function y=f(x) changes from concave up to
concave down, or vice versa at the point
- if (a f(a) ) is an infelction point, then f”(a) =0 and the signs of the second derivates change int he
neighbourhood of the point (a f(a) ).
-Then int he sign of f”(xL) is different from that of f”(xR), (a f(a)) is an inflection point, otherwise it is
not.
Graphing procedure:
-determine the domain of f
-nothe any symmetries of f
f(x)=f(-x) is called even ; f(x) =-f(x) is called odd symmetry.
-locate any points where f is not defined and determine the behaviour of f near these points.
-evaluate the limits of f as x->végtelen and x->-végtelen
-locate the intersection points as y=f(x) =0 and y=f(0)
-locate the local maxima and minima of f and determine the intervals on which f is increasing and
decreasing
-locate the inflection points of f and determine the intervals on which f is convex or concave.
-draw the graph of the f