Review

1
Review

• 4.5 Applied Optimization Problems
• Example
• Guidelines to Solve Optimization Problems

2
Guidelines to Solve Optimization Problems

• Read the problem carefully. Identify what is
  given and what is to be optimized.
• Draw a picture if possible.
• Introduce variables. Use the given information to
  relate variables
• Write an equation for the quantity that is to be
  optimized.
• Try to express the quantity as a function of a
  single variable.
• Find the optimal value by the calculus technique.

3
Example

• (5/page 285) You are planning to make an open
  rectangular box from an 8 in.-by-15 in. piece of
  cardboard by cutting congruent squares from the
  corners and folding up the sides. What are
  dimensions of the box of the largest volume you
  can make this way, and what is its volume?
• Solution xside of square cut from the corners.

4

• vvolume
• We want to maximize v.
• v(15-2x)(8-2x)x
• (120-46x4x2)x
• 120x-46x24x3
• v120-92x12x2
• 120-92x12x20
• 4(30-23x3x2)0
• 30-23x3x20
• (3x-5)(x-6)0

5

• x6, x5/3
• v(6)(15-12)(8-12)6-72
• V(5/3)(15-10/3)(8-10/3)(5/3)(35/3)(14/3)(5/3)
• 2450/27
• So v is maximized at x5/3. The maximal volume is
• 2450/27. The dimensions of the box are
• 35/3, 14/3, 5/3.

6
4.8 Antiderivatives

• Finding Antiderivatives
• Definition
• A function F is an antiderivative of f if
  F(x)f(x).
• The process of recovering a function F(x) from
  its derivative f(x) is called antidifferentiation.
  
• We use capital letters such as F to represent an
  antiderivative of f, G to represent an
  antiderivative of g, and so forth.
• 2) Examples

7
Examples

• a) f(x)3x2, F(x) x3 is an antiderivative of
  f(x).
• F(x) x3 1,
• F(x) x3 2.
• In fact, F(x) x3 C, where C is a constant, is
  an antiderivative of f(x).

8
Examples

• b). f(x)x4,
• F(x) x4 is an antiderivative of f(x).
• Similarly, the general solution is
• F(x) x4C, where C is any constant.

9
Examples

• c). Find an antiderivative F of f(x)3x2 and
  F(0)5.
• Solution F(x) x3 C
• Since F(0)5, then
• 03 C5 (Substituting x0 )
• C5
• So F(x) x3 5.

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Antiderivative Formulas

• 2. Antiderivative Formulas
• f(x)xn,
• F(x) xn1C
• f(x)sin(kx)
• F(x)- cos(kx)C
• f(x)cos(kx)
• F(x) sin(kx)C
• f(x)sec2x
• F(x)tanxC

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Antiderivative Formulas

• 5) f(x)csc2x
• F(x)-cot xC
• f(x)sec x tan x
• F(x)sec x C
• f(x)csc x cotx
• F(x) -csc x C

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More Examples

• Find the general antiderivatives of each of the
  following functions.
• f(x)x6
• F(x) x7C
• f(x)
• x-2
• F(x) x-21C
• -x-1C
• C

13
Examples

• c) f(x)cos 2x
• F(x) sin 2x C
• f(x)sin (x/2)
• F(x) -2cos (x/2)C

14
Antidifferentiation Rules

• 3. Antiderivative Linearity Rules
• function general antiderivative
• 1) kf(x), kF(x)C
• 2) f(x)g(x) F(x)G(x)C
• 3) f(x)-g(x) F(x)-G(x)C

15
Examples

• Find the general antiderivative of
• f(x)3x4-6x
• Solution F(x)3(1/5) x5-6(1/2)x2C
• 3/5 x5-3x2C

16
Initial Value Problems and Differential Equations

• 4. Differential Equatons
• 1) Definiton
• An equation involving a derivative is called a
  differential equation.
• 2) Examples
• is a differential equation.

17
Initial Value Problems and Differential Equations

• 3) Solution
• Find y whose derivative is 3x2.
• yx3C
• 4) Initial value problem
• y(2) 7
• 23C7, C-1
• So yx3-1 is the solution.

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Example

• (68/page 316)
• , y(0)-1.
• Solution y10x-(1/2)x2C
• 10(0)-(1/2) 02C-1
• C-1
• So y10x-(1/2)x2-1 is the solution.

19
Indefinite Integrals

• 5. Indefinite Integrals
• 1) Defintion
• The collection of all antiderivatives of f is
  called the indefinite integral of f, and is
  denoted by
• The function f is called the integrand of the
  integral.
• It is a special symbol to represent the general
  antiderivative.

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Examples

• 2) Examples
• a)
• b)
• c)

21
Practice

• 2, 4, 6, 20, 70 on page 314-316.

22
Homework

• 4.8