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Max and Min ValuesLocal maximum and minimum valuesThe number fc is a■ local maximum value of f on D if fc ≥ fx when x is near c.■ local minimum value of f on D if fc ≤ fx when x is near
Trang 14 APPLICATIONS OF DIFFERENTIATION
4.1 Maximum and Minimum Values
Applied Project: The Calculus of Rainbows
4.2 The Mean Value Theorem
4.3 How Derivatives Affect the Shape of a Graph4.4 Indeterminate Forms and l’Hospital’s Rule
Writing Project: The Origins of l’Hospital’s Rule
Trang 24.9 Computer Algebra (Maple\Matlab)
Trang 34.1 Max and Min Values
Absolute Maximum and Minimum Values
Let c be a number in the domain D of a function f Then f(c) is the
• absolute (global) maximum value of f on D if f(c) ≥ f(x) for every
Trang 44.1 Max and Min Values
Local maximum and minimum values
The number f(c) is a
■ local maximum value of f on D if f(c) ≥ f(x) when x is near c.■ local minimum value of f on D if f(c) ≤ f(x) when x is near c.
Trang 54.1 Max and Min Values
Example Determine
whether the given function has extreme values at
indicated points
Trang 64.1 Max and Min ValuesExistence of extreme values on closed interval
If f is a continuous function on closed interval [a, b] or on a union of finitely many closed interval, then f must have an absolute maximum
value and an absolute minimum value on this interval.
Trang 74.1 Max and Min Values
Fermat’s Theorem
If f has a local maximum or minimum at c, and if f ’(c) exists, then
f ’(c) = 0.
Trang 84.1 Max and Min Values
Trang 94.1 Max and Min ValuesNecessary conditions of extreme values
If f has extreme value at c, then c must be one of the following
1 Critical points of f (f ’(c) = 0 or f ’(c) does not exist),
2 Endpoints of the domain of f
Trang 104.1 Max and Min ValuesEx1. Find the absolute maximum and minimum values of the function
f(x) = x3 – 3x2 + 1 on [– ½, 4].
Trang 114.1 Max and Min ValuesEx2. The Hubble Space Telescope was deployed on April 24, 1990,by the space shuttle Discovery A model for the velocity of the shuttle
during this mission, from liftoff at t = 0 until the solid rocket boosterswere jettisoned at t = 126 s, is given by
v(t) = 0.001302t3 – 0.09029t2 + 23.61t – 3.083
(in feet per second) Using this model, estimate the absolute maximumand minimum values of the acceleration of the shuttle between liftoffand the jettisoning of the boosters.
Trang 124.1 Max and Min ValuesFunctions not defined on closed, finite intervals
If f is continuous on I = (a, b) (or (– ∞, ∞)), and if
(1) If there exists u in I | f(u) > max (L, R) => f has a max value on I.(2) If there exists v in I | f(v) < min (L, R) => f has a min value on I.
Lx
Trang 134.1 Max and Min Values
Example 1 Show that f(x) = x + 4/x has an abs.min value on (0, ∞),
and find that min value.
Example 2 Let Find and classify the critical points of f.f(x)xex2
Trang 144.2 The Mean Value Theorem
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 c in (a, b) such that f’ (c) = 0.
Trang 154.2 The Mean Value Theorem
Trang 164.2 The Mean Value TheoremPhysical meaning: If the moving particle is in the same place at
two different instants t = a and t = b, then there is some instant t = cbetween a and b at which the velocity of the object is zero (In
particular, you can see that this is true when a ball is thrown directlyupward).
Trang 174.2 The Mean Value TheoremEx1. Prove that the equation x3 + x – 1 = 0 has exactly one real root.
Step 1 Prove that a root exists by using the Intermediate ValueTheorem.
Step 2 Argue by contradiction that f(x) = x3 + x – 1 = 0 has more than
one real solution, then using the Rolle’s theorem to give thecontradiction.
Trang 184.2 The Mean Value Theorem
The Mean Value Theorem
Let f be a function that satisfies the following two 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 c in (a, b) such that
' or f(b)f(a)f'(c)(ba)
Trang 194.2 The Mean Value Theorem
There is at least one point on the graph where the tangent line is parallel to the secant line AB.
Trang 204.2 The Mean Value TheoremPhysical meaning: The Mean-value theorem tell us that, at some
time t = c in (a, b), the instantaneous velocity is equal to the average
velocity over this time period.
Trang 214.2 The Mean Value TheoremEx1. Suppose that f(0) = –3 and f ’(x) ≤ 5 for all values of x How largecan f(2) possibly be?
1 If f ’(x) = 0 for all x in (a, b), then f (x) = C for all x in (a, b);
2 If f ’(x) = g’(x) for all x in (a, b), then f(x) – g(x) = C for all x in (a, b).
Trang 224.2 The Mean Value TheoremEx1. Proof that
Solution Let
Show that F ’(x) = 0 Then F(x) = F(1) =/2.
tan1 1
F()tan1 cot1
Trang 234.3 How Derivatives Affect the Shape of a Graph
Increasing/Decreasing Test
(a) If f ’(x) > 0 on an interval, then
f is increasing on that interval.
Trang 244.3 How Derivatives Affect the Shape of a GraphThe first derivative tests
Trang 254.3 How Derivatives Affect the Shape of a Graph
Trang 264.3 How Derivatives Affect the Shape of a Graph
Ex2. Find the local maximum and minimum values of the function
g(x) = x + 2sinx, 0 ≤ x ≤ 2.
Trang 274.3 How Derivatives Affect the Shape of a Graph
If the graph of f lies above all of its tangents on an interval I,
then it is called concave upward on I.
If the graph of f lies below all of its tangents on I, it is called
Trang 284.3 How Derivatives Affect the Shape of a GraphConcavity 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.
f is concave up on I if it is differentiable there and the
derivative f ’ is an increasing function on I.
f is concave down on I if it is differentiable there and the
derivative f ’ is an decreasing function on I.
Trang 294.3 How Derivatives Affect the Shape of a Graph
TEST FOR CONCAVITY
f ’’(x)f ’’(x)
Miss SmileMr Frown
Trang 304.3 How Derivatives Affect the Shape of a Graph
Ex3. Figure shows a population graph for Cyprian honeybees raisedin an apiary How does the rate of population increase change overtime?
P (thousands)
3691215 18 tin weeks
When is this ratehighest? Over whatintervals is P concaveupward or concavedownward?
Trang 314.3 How Derivatives Affect the Shape of a Graph
Definition A point P on a curve y = f(x) is called an inflection point
if is continuous there and the curve changes its concavity at P.
In view of the Concavity Test, there is a point of inflection at anypoint where the second derivative changes sign.
Trang 324.3 How Derivatives Affect the Shape of a Graph
Ex4. Sketch a possible graph of a function that satisfies the followingthree conditions:
(1) f ’(x) > 0 on (– ∞, 1), f ’(x) < 0 on (1, ∞),
(2) f ’’(x) > 0 on (– ∞, – 2) and (2, ∞), f ’’(x) < 0 on (– 2, 2) and(3) f (x) ‒> – 2 when x ‒> – ∞, f (x) ‒> 0 when x ‒> ∞.
Trang 334.3 How Derivatives Affect the Shape of a GraphThe Second Derivative Test
Suppose f ’’ is continuous near c.
(a) If f ’(c) = 0 and f ’’(c) > 0, then has a local minimum at c.(b) If f ’(c) = 0 and f ’’(c) < 0, then has a local maximum at c.
f (c) f (x) P
cf (c)
f (x) xP
Trang 344.3 How Derivatives Affect the Shape of a Graph
Ex5. Discuss the curve y = x4 – 4x3 with respect to concavity, points ofinflection, and local maxima and minima Use this information to sketchthe curve.
Solution f(x) = x4 – 4x3, f ’(x) = 4x3 – 12x2, f ”(x) = 12x2 – 24x.
f ’(x) = 0 4x2 (x – 3) = 0 => x = 0, x = 3,
f ”(x) = 0 12x (x – 2) = 0 => x = 0, x = 2
Trang 354.3 How Derivatives Affect the Shape of a Graph
Trang 3604 6
4.3 How Derivatives Affect the Shape of a Graph
Ex6 Sketch the graph of the
function y = x2/3(6 – x)1/3.
min
Trang 374.3 How Derivatives Affect the Shape of a Graph
Trang 384.3 How Derivatives Affect the Shape of a Graph
Trang 394.4.Indeterminate Forms and l’Hospital’s Rule
Def 1.
Consider a limit of the form
If both f(x) -> 0 and g(x) -> 0 as x -> a (a - finite or infinite), then
this type of limit is called an indeterminate form of type 0/0.
If both f(x) -> ∞ (– ∞ ) and g(x) -> ∞ (– ∞ ) as x -> a (a - finite or
infinite) then this type of limit is called an indeterminate form oftype ∞/∞.
xfax
Trang 404.4.Indeterminate Forms and l’Hospital’s Rule
L’Hospital’s Rule
Suppose f and g are differentiable and g ’(x) ≠ 0 on an open interval Ithat contains a (except possibly at a) Suppose that
is an indeterminate form of type 0/0 or ∞/∞ Then
if the limit of the right side exists (or is ∞ or – ∞).
axa
Trang 414.4.Indeterminate Forms and l’Hospital’s Rule
f and g approach 0 as x -> a If
we were to zoom in toward the
point (a, 0), the graphs would
start to look almost linear.
L’Hospital’s Rule might be true visually
f and g are linear
axa
Trang 424.4.Indeterminate Forms and l’Hospital’s Rule
Examples. Calculate the following limits
1
xx
1 sin
Trang 434.4.Indeterminate Forms and l’Hospital’s Rule
vào Ví dụ 2
Trang 444.4.Indeterminate Forms and l’Hospital’s Rule
INDETERMINATE PRODUCTS
The limit of the form
where f(x) -> 0 and g(x) -> ∞ (or – ∞) as x -> a (a - finite or infinite)
is called an indeterminate form of type 0 ∞.
Solution: Write the product as a quotient or
Use the L’Hospital’s rule.
Trang 454.4.Indeterminate Forms and l’Hospital’s Rule
Ex6. Evaluate
Trang 464.4.Indeterminate Forms and l’Hospital’s Rule
lim lnlim1
x
Trang 474.4.Indeterminate Forms and l’Hospital’s Rule
INDETERMINATE DIFFERENCES
The limit of the form
where f(x) -> ∞ and g(x) -> ∞ as x -> a (a - finite or infinite) is called
an indeterminate form of type ∞ – ∞.
Solution: Convert the difference into a quotient (using commondenominator, rationalization, factoring out a common factor, etc.) thenuse the L’Hospital’s rule.
Trang 484.4.Indeterminate Forms and l’Hospital’s Rule
Ex7. Evaluate
(
Trang 494.4.Indeterminate Forms and l’Hospital’s Rule
Trang 504.4.Indeterminate Forms and l’Hospital’s Rule
Examples. Evaluate the following limits
Trang 514.4.Indeterminate Forms and l’Hospital’s Rule
Solution
Trang 524.4.Indeterminate Forms and l’Hospital’s Rule
Trang 534.5.Summary of Curve Sketching
Some particular aspects of curve sketching:(1) domain, range, and symmetry in Chapter 1;(2) limits, continuity, and asymptotes in Chapter 2;(3) derivatives and tangents in Chapters 2 and 3;
(4) extreme values, intervals of increase and decrease, concavity,points of inflection, and l’Hospital’s Rule in this chapter.
We put all above information together to sketch graphs that reveal the important features of functions.
Trang 544.5.Summary of Curve Sketching
Guidelines for Sketching a Curve y = f(x) by Hands
(1) Domain
(2) Intercepts: the y-intercepts = f (0), the x-intercepts: 0 = f(x).
(3) Symmetry: odd and even functions, periodic function.(4) Asymptotes: horizontal (x -> ±∞),
vertical (y -> ±∞ when x -> a),
slant (f(x) – (ax + b) -> 0 when x -> ±∞)
Trang 554.5.Summary of Curve Sketching
(5) Intervals of Increase and Decrease: f’(x)
(6) Local maximum and minimum: first derivative test(7) Concavity and points of inflection: concavity test(8) Sketch the curve
Trang 564.5.Summary of Curve Sketching
Example. Sketch the graph of
(1) Domain: (–∞, ∞).
(2) The x- and y-intercepts: 0, 0.
(3) Symmetry: odd function, graph is symmetric about the origin.
(4) Asymptotes: oblique asymptote: y = x.
xf
Trang 57inf
Trang 584.6 Graphing with Caculus and Caculators
See from page 318 to page 325
Trang 594.7 Optimization ProblemsProcedure for Solving Extreme-Value Problems
S1. Find out what is given and what must be found.
S2. Make a diagram if appropriate.
S3. Use symbols to denote variables.
S4. Express the quantity Q to be minimized (maximized) as a function
of variables
Trang 604.7 Optimization Problems
S5. If Q depends on n variables, where n > 1, find (n – 1) constrains.
S6. Use these constrains to express Q as a function of only one
S7. Find the required extreme value of Q.
Trang 614.7 Optimization ProblemsEx1. A rectangular animal enclosure is to be constructed having oneside along an existing long wall and the other three sides fenced If100m of fence are available, what is the largest possible area for theenclosure?
xA = xy
Trang 624.7 Optimization ProblemsEx2. A cylindrical can is to be made to hold 1 L of oil Find thedimensions that will minimize the cost of the metal to manufacture thecan.
Trang 634.7 Optimization ProblemsEx3. Find the point on the parabola y2 = 2x that is closest to the
point (1, 4).
Trang 644.7 Optimization ProblemsEx4. A lighthouse L is located on a small island 5km north of a pointA on a straight east-west shoreline A cable is to be laid from L to pointB on the shoreline 10km east of A The cable will be laid through thewater in a straight line from L to a point C on the shoreline between Aand B, and from there to B along the shoreline The part of the cablelying in the water costs $5,000/km and the part along the shorelinecosts $3,000/km.
(a) Where should C be chosen to minimize the total cost of the cable?(b) Where should C be chosen if B is only 3km from A?
Trang 654.7 Optimization Problems
5 km
x km(10 – x) km
Trang 664.7 Optimization Problems
Applications to Business and Economics
(1) cost function: C(x) – the cost of producing x units of a certain
(2) marginal cost: C’(x).
(3) demand (price) function: p(x) – the price per unit that the
company can charge if it sells x units.
Trang 674.7 Optimization ProblemsEx5. A store has been selling 2,000 chairs per month at $50 each Amarket survey indicates that, for each $5 rebate offered to buyers, thenumber of chairs sold will increase by 500 a month.
(a) Find the demand function and revenue function.
(b) How much of a rebate should the store offer to maximize itsrevenue?
Trang 684.7 Optimization ProblemsSolution.
(a) Suppose x is the number of chairs sold per month.Increase in number of chairs: x – 2000
Trang 694.8 Newton’s Method
PROBLEMS
Trang 704.8 Newton’s Method
We use Newton’s method for finding root of the equation f(x) = 0.
f(x1)
Trang 71
Trang 724.8 Newton’s Method
(4) Similar formulas produce x4 from x3, then x5 from x4, and so on.
The formula producing xnfrom xn – 1 is
If the numbers xnbecome closer and closer to r as n becomes large,then we say that the sequence converges to r and write
lim
Trang 734.8 Newton’s Method
1. When using Newton’s method to solve an equation of the form g(x)= h(x), we must rewrite it in the form f(x) = g(x) – h(x) = 0, and applythe Newton’s method to f.
Trang 744.8 Newton’s MethodEx1. Given equation: x3 – 2x – 5 = 0 Initial guess x1= 2.
(a) Find the third approximation x3 to the real root of this equation.(b) Find the real root correct to four decimal places
• f(x) = x3 – 2x – 5, f’ (x) = 3x2 – 2, using the first derivative test, it iseasy to see that the equation has only one real solution.
• As f(2) f(3) < 0 then this solution lies somewhere between 2 and 3.
Hence our initial guess is seemed to be appropriate.
Trang 764.8 Newton’s MethodEx2. Use Newton’s method to find (2)1/6 correct to eight decimalplaces.
Trang 774.9 AntiderivatiesDef 1. An anti-derivative of a function f on an interval I is anotherfunction F satisfying
Trang 784.9 Antiderivaties
FEW SIMPLE INDEFINITE INTEGRALS
Trang 794.9 Antiderivaties
FEW SIMPLE INDEFINITE INTEGRALS
Trang 804.9 Antiderivaties
Differential Equations and Initial-Value Problems
Def 3.A differential equation is an equation involving derivatives
of an unknown function.
Ex1 a) y’ + 2y = 1,b) xy” – sin(x) = x
c) y’’’ – 3xy” + 2y = 0…
Trang 814.9 Antiderivaties
Solution of a DE: Any function whose derivatives satisfy the DE. Order of DE: The order of the highest-order derivative in DE. General solution: A function that contains all solutions of DE
Aparticular solution: One of the possible solutions of DE.
An initial-value prob: DE + sufficiently enough prescribed values.
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