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ESSENTIAL CALCULUS CH03 Applications of differentiation. 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. - PowerPoint PPT Presentation
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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
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.
Chapter 3, 3.1, P143
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.
Chapter 3, 3.1, P143
Chapter 3, 3.1, P143
Chapter 3, 3.1, P143
Chapter 3, 3.1, P143
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] .
Chapter 3, 3.1, P143
Chapter 3, 3.1, P143
Chapter 3, 3.1, P143
Chapter 3, 3.1, P144
Chapter 3, 3.1, P144
Chapter 3, 3.1, P144
4. FERMAT’S THEOREM If f has a local maximum or minimum at c , and if f’(c)exists, then f’(c)=0.
Chapter 3, 3.1, P145
Chapter 3, 3.1, P145
Chapter 3, 3.1, P146
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.
Chapter 3, 3.1, P146
7. If f has a local maximum or minimum at c, then c is a critical number of f.
Chapter 3, 3.1, P146
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.
Chapter 3, 3.1, P147
5-6 ▓Use the graph to state the absolute and local maximum and minimum values of the function.
Chapter 3, 3.1, P147
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.
Chapter 3, 3.2, P149
Chapter 3, 3.2, P150
Chapter 3, 3.2, P150
Chapter 3, 3.2, P150
Chapter 3, 3.2, P150
Chapter 3, 3.2, P151
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
Chapter 3, 3.2, P153
5. THEOREM If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b).
Chapter 3, 3.2, P154
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.
Chapter 3, 3.2, P154
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].
Chapter 3, 3.3, P156
Chapter 3, 3.3, P156
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.
Chapter 3, 3.3, P157
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, 3.3, P157
Chapter 3, 3.3, P157
Chapter 3, 3.3, P157
Chapter 3, 3.3, P157
Chapter 3, 3.3, P158
Chapter 3, 3.3, P158
Chapter 3, 3.3, P158
Chapter 3, 3.3, P158
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.
Chapter 3, 3.3, P159
Chapter 3, 3.2, P159
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.
Chapter 3, 3.3, P159
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.
Chapter 3, 3.3, P160
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.
Chapter 3, 3.3, P162
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.
Chapter 3, 3.3, P162
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?
Chapter 3, 3.3, P162
Chapter 3, 3.3, P162
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
Chapter 3, 3.3, P163
Chapter 3, 3.5, P172
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.
Chapter 3, 3.5, P175
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)
Chapter 3, 3.6, P180
Chapter 3, 3.6, P180
Chapter 3, 3.6, P180
)('
)(1
n
nnn Xf
xfxX
Chapter 3, 3.6, P181
Chapter 3, 3.6, P180
Newton’s method fails and a better initial approximation x1 should be chosen.
Chapter 3, 3.6, P183
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
Chapter 3, 3.7, P185
DEFINITION A function F is called an antiderivative of f on an interval I if F’(x)=f(x) for all x in I.
Chapter 3, 3.7, P185
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.
Chapter 3, 3.7, P185
Chapter 3, 3.7, P186
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