ESSENTIAL CALCULUS CH03 Applications of differentiation

ESSENTIAL CALCULUSESSENTIAL CALCULUS

CH03 Applications of CH03 Applications of differentiationdifferentiation

In this Chapter:In this Chapter:

3.1 Maximum and Minimum Values 3.2 The Mean Value Theorem 3.3 Derivatives and the Shapes of Graphs 3.4 Curve Sketching 3.5 Optimization Problems 3.6 Newton’s Method 3.7 Antiderivatives

Review

Chapter 3, 3.1, P142


1 DEFINITION A function f has an absolute maximum (or global maximum) at c if f(c)≥f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of f on D. Similarly, f has an absolute minimum at c if f(c)≤f(x) for all x in D and the number f(c) is called the minimum value of f on D. The maximum and minimum values of f are called the extreme values of f.


2. DEFINITION A function f has a local maximum (or relative maximum)at c if f(c) ≥f(x) when x is near c. [This means that f(c) ≥f(x) for all x in some open interval containing c.] Similarly, f has a local minimum at c if f(c)≤f(x) when x is near c.





3. 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 minimumvalue f(d) at some numbers and d in [a,b] .







4. FERMAT’S THEOREM If f has a local maximum or minimum at c , and if f’(c)exists, then f’(c)=0.




6. DEFINITION A critical number of a function f is a number c in the domain of f such that either f’(0)=0 or f’(c) does not exist.


7. If f has a local maximum or minimum at c, then c is a critical number of f.


THE CLOSED INTERVAL METHOD To find the absolute maximum and minimum values of a continuous function 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 absol

ute maximum value; the smallest of these values is the absolute minimum value.


5-6 ▓Use the graph to state the absolute and local maximum and minimum values of the function.


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 in (a,b) such that f’(c)=0.







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 in (a,b) such that1

or, equivalently,

2

ab

afbfcf

)()(

)('

))((')()( abcfafbf


5. THEOREM If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b).


7. 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 where c is a constant.


7. Use the graph of f to estimate the values of c that satisfy the conclusion of the Mean Value Theorem for the interval [0,8].



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.


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’ does not change sign at c (that is, f’ is

positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c.









Chapter 3, 3.32, P158

DEFINITION If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If f the graph of lies below all of its tangents on I, it is called concave downward on I.



DEFINITION 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 from concave downward to concave upward at P.


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.


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.


11. In each part state the x-coordinates of the inflection points of f. Give reasons for your answers.(a) The curve is the graph of f.(b) The curve is the graph of f.(c) The curve is the graph of f.


12. The graph of the first derivative f’ of a function f is shown.(a) On what intervals is f increasing? Explain.(b) At what values of x does f have a local maximum or minimum? Explain.(c) On what intervals is f concave upward or concave downward? Explain.(d) What are the x-coordinates of the inflection points of f? Why?



Guidelines for sketching a curve

A.Domain

B.Intercepts

C.Symmetry

D.Asymptotes

E. Interrals of Increase or Decrease

F. Local Maximum and Minimum Values

G.Concavity and Points of Intlection

H.Slcetch the Curve



FIRST DERIVATIVE TEST FOR ABSOLUTE 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 maximum value of f.


APPLICATIONS TO BUSINESS AND ECONOMICS

Cost function c(x): the cost of producing x units of a certain product

Marginal Cost function c’(x): the rate of change of C with respect to x

Price function C Demand function p(x): the price per unit that the company can change if it sells x units.

Revenue function R(x): R(x)=xp(x)

Marginal Revenue function R’(x)=the rate of change of R with respect of x

Profit function P(x): P(x)=R(x)-C(x)

Marginal Profit function P’(x)=the derivative of P(x)




)('

)(1

n

nnn Xf

xfxX



Newton’s method fails and a better initial approximation x1 should be chosen.


4.For each initial approximation, determine graphically what happens if Newton’s method is used for the function whose graph is shown.

(a)x1=0 (b) x1=1 (c) x1=3(d)x1=4 (e)x1=5


DEFINITION A function F is called an antiderivative of f on an interval I if F’(x)=f(x) for all x in I.


1.THEOREM If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is

F(x)+C

where C is an arbitrary constant.



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ESSENTIAL CALCULUS CH03 Applications of differentiation