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### 1314L14

Course: MATH 1313, Fall 2010
School: U. Houston
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Word Count: 648

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14 Lesson Optimization Now youll work some problems where the objective is to optimize a function. That means you want to make it as large as possible or as small as possible depending on the problem. In many problems, youll state the domain before you work the problem. Once you have the function and its domain, youll find the critical points and see if the critical point(s) fall within the domain of the function....

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14 Lesson Optimization Now youll work some problems where the objective is to optimize a function. That means you want to make it as large as possible or as small as possible depending on the problem. In many problems, youll state the domain before you work the problem. Once you have the function and its domain, youll find the critical points and see if the critical point(s) fall within the domain of the function. You can also use the second derivative test to verify that you have an absolute max or an absolute min in many problems. Example 1: A company that produces digital cameras wants to minimize its production costs. They estimate that their total monthly cost for producing the camera is given by C( x ) = 0.0025x 2 + 80x + 10000 . Find the average cost function. Find the level of production that results in the smallest average production cost. Use the second derivative test to verify that you have found a minimum cost. Lesson 14 - Optimization 1 If the function is not given, the first task is to write a function that describes the situation in the problem. Here are some suggestions to help make this easier: 1. Read the problem carefully to determine what function you are trying to find. 2. If possible, draw a picture of the situation. Choose variables for the values discussed and put them on your picture. 3. Determine if there are any formulas you need to use, such as area or volume formulas. If you have a right triangle in your picture, decide if the Pythagorean Theorem will help. Example 2: A homeowner wants to fence in a rectangular vegetable garden using the back of her garage (which measures 20 feet across) as part of one side of the garden. She has 110 feet of fencing material and wants to use that to build the fence. What should be the dimensions of the garden if she fences in the maximum possible area? Lesson - 14 Optimization 2 Example 3: Suppose you wish to fence in a pasture that lies along the straight edge of a river. You will divide the pasture into two parts by means of a fence that runs perpendicular to the river and parallel two of the sides of the pasture. You have 1500 meters of fencing to use, and you wish to fence in the maximum possible area. Determine the dimensions of the pasture that will provide the maximum area. What is that area? Lesson 14 - Optimization 3 Example 4: If you cut away equal squares from all four corners of a piece of cardboard and fold up the sides, you will make a box with no top. Suppose you start with a piece of cardboard the measures 3 feet by 8 feet. Find the dimensions of the box that will give a maximum volume. Lesson 14 - Optimization 4 Example 5: Postal regulations state that the girth plus length of a package must be no more than 104 inches if it is to be mailed through the US Postal Service. You are assigned to design a package with a square base that will contain the maximum volume that can be shipped under these requirements. What should be the dimensions of the package? (Note: girth of a package is the perimeter of its base.) Lesson 14 - Optimization 5 Example 6: You are assigned to design some shipping materials at minimum cost. The package will be a closed rectangular box with a square base, and must have a volume of 50 cubic inches. The material used for the top costs 35 cents per square inch, the material used for the bottom of the box costs 45 cents per square inch, and the material used for the sides costs 20 cents per square inch. Find the surface are of the box and then find the dimensions of the box that will minimize the cost. From this section, you should be able to Write a function from a verbal description Optimize a function Lesson 14 - Optimization 6
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U. Houston - MATH - 1313
Lesson 15 Exponential Functions as Mathematical Models In this lesson, we will look at a few applications involving exponential functions. Well first consider some word problems having to do with money. Next, well consider exponential growth and decay pro
U. Houston - MATH - 1313
Lesson 16 Antiderivatives So far in this course, we have been interested in finding derivatives and in the applications of derivatives. In this chapter, we will look at the reverse process. Here we will be given the answer and well have to find the proble
U. Houston - MATH - 1313
Lesson 17 Integration by Substitution Sometimes the rules from the last lesson arent enough. In this lesson, you will learn to integrate using substitution. This is related to the chain rule that you used in finding derivatives. Using Substitution to Inte
U. Houston - MATH - 1313
Lesson 18 Area and the Definite Integral We are now ready to tackle the second basic question of calculus the area question. We can easily compute the area under the graph of a function so long as the shape of the region conforms to something for which we
U. Houston - MATH - 1313
Lesson 19 The Fundamental Theorem of Calculus In the last lesson, we approximated the area under a curve by drawing rectangles, computing the area of each rectangle and then adding up their areas. We saw that the actual area was found as we let the number
U. Houston - MATH - 1313
Lesson 20 Evaluating Definite Integrals We will sometimes need these properties when computing definite integrals. Properties of Definite Integrals Suppose f and g are integrable functions. Then: 1. aaf ( x)dx = 0 f ( x)dx = f ( x)dxb a2. 3.ba cf
U. Houston - MATH - 1313
Lesson 21 Area Between Two Curves Two advertising agencies are competing for a major client. The rate of change of the clients revenues using Agency As ad campaign is approximated by f(x) below. The rate of change of the clients revenues using Agency Bs a
U. Houston - MATH - 1313
Lesson 22 Functions of Several Variables So far, we have looked at functions of a single variable. In this section, we will consider functions of more than one variable. You are already familiar with some examples of these.P ( x, y ) = 2 x + 2 y A( P, i,
U. Houston - MATH - 1313
Lesson 23 Partial Derivatives When we are asked to find the derivative of a function of a single variable, f ( x), we know exactly what to do. However, when we have a function of two variables, there is some ambiguity. With a function of two variables, we
U. Houston - MATH - 1313
Lesson 24 Maxima and Minima of Functions of Several VariablesWe learned to find the maxima and minima of a function of a single variable earlier in the course. Although we did not use it much, we had a second derivative test to determine whether a critic
U. Houston - MATH - 1313
1. Given the following graph of a function, f(x).y6 5 4 3 2 1x9 8 7 6 5 4 3 2 1 1 2 3 4 1 2 3 4 5 6 7 8 9 10Find: a. lim f ( x)x 2b.lim f ( x)x 1xc.xlim f ( x)5d.limf ( x) 2+e.lim f ( x)x 1f. f(-2) h. f(2)g. f(1)2. Given the follow
U. Houston - MATH - 1313
Math 1314 Test 1 Review Review Properties of Exponents 1. a 0 = 12. a n =1 n1 an3. a = n am4. a n = n a m 5. a m a n = a m + n am 6. n = a m n , a 0 a 7 . ( a m ) n = a mn 8. (ab) n = a n b nan a 9. = n , b 0 b b 10. For b 1 , b x = b y if and only
U. Houston - MATH - 1313
Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve at a given point on the curve?y 1
U. Houston - MATH - 1313
Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point.We need a way to find the
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x ) f ' ( x) dx Example 1: Use the produ
U. Houston - MATH - 1313
Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = (3 x 2 5 x + 6 ) into functions f ( x) and g ( x ) such that h( x) = ( f o g )( x).4Rule 10: The Cha
U. Houston - MATH - 1313
Lesson 7 Higher Order Derivatives Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is denoted f ' ' (
U. Houston - MATH - 1313
Math 1314 Test 2 ReviewIn numbers 1 7, find the limit. 1.xlim x3 + 2x 2 1 2 x + 32. limxx3 + x 1 x 13. limxx3 x 1 x 14. Given the following graph of a function f, find lim f ( x) if it exists.x 4Math 1314 Test 2 Review1Limits at Infinity: I
Missouri (Mizzou) - IST - 5
Case 2 a. Most should be close-ended, but should allow customers to provide additional comments. Questions: Visit Frequency Food Quality The order taking process Delivery Speed Potential New Services Overall satisfactionb. Asking open-ended question in a
CUNY Queens - ECON - 247
Bus 247 Homework Set 1 Summer 2009 Queens College Professor Bradbury This assignment is due at the beginning of class on Monday June 15. The assignment must be typed and stapled in order to receive credit. (though graphical solutions may be handwritten in
CUNY Queens - ECON - 247
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
CUNY Queens - ECON - 247
Home Work II Exam II Preparation Assignment This assignment is due in class Tuesday April 29. The assignment will not be accepted if it is turned in late. The assignment will not be accepted unless it is typed. 1) Consider the following Game: Compete Comp
CUNY Queens - ECON - 247
HW III Bus 247 Spring 2008 Professor Bradbury Please answer each of the following questions. This assignment must be turned in typed together with your final paper by May 20th. The assignment will be graded for accuracy.1) When firms are able to achieve
CUNY Queens - ECON - 247
Homework III Bus 247/ Fall 2008 Prof. Bradbury This assignment is due at the beginning of class, 12/02/2008. Assignments must be typed and stabled in order to receive credit. 1) The following reaction functions describe the profit maximization problem for
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
CUNY Queens - ECON - 247
S teven Kordvani Business Economics (BUS 247) HW # 1-Fall 2008 Prof. Bradbury 1. The follow equation represents the Total Cost Function for a representative F i rm: TC= C (q) = 480 + 4q^2 a. Fixed Cost = 480 Variable Cost=4q^2b. Expression for Average To
CUNY Queens - ECON - 247
Bus 247 Business Economics Homework I Fall 2008 Prof. Bradbury To the best of your abilities, please answer each question in its entirety. This assignment will be due at the beginning of class on Thursday October, 2. All assignments must be typed and stab
CUNY Queens - ECON - 247
S teven Kordvani Business Economics (BUS 247) HW # 2-Fall 2008 Prof. Bradbury 1. The behavior of average cost which we associate with natural monopoly is falling cost 2. I t makes more sense for one fi rm to satisfy all market demand rather than m any fi
CUNY Queens - ECON - 247
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
CUNY Queens - ECON - 247
1.Profit q2=320 Maximization Reaction Function, 2 firms competing by quantity:q1=R1 q2= 480- q22| q2=R2 q1= 480- q12A. WAY # 1: B. WAY # 2 :q2R1 q2=R2 q1 480- q22 = 480- q12 2q2 32)q2=48023 =12q2=960-480 480 (480 - q22 ) 0 = 480 240-q24 240 = q2 24
CUNY Queens - ECON - 247
Homework III Bus 247/ Fall 2008 Prof. Bradbury This assignment is due at the beginning of class, 12/02/2008. Assignments must be typed and stabled in order to receive credit. 1) The following reaction functions describe the profit maximization problem for
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
CUNY Queens - ECON - 247
Steven Kordvani Business Economics (BUS 247) HW # 4-Fall 2008 Prof. Bradbury 1. When fi rms are able to achieve perfect price discrimination: P1= 3 5 10 a. The value of consumer surplus is zero because all the consumer surplus is t ransferredto the fi rm
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
*)F* 0 901401*)F 9 *)F 20~1500 *pR3 B B1 *pR3 901402*F@+9014036 H|*)F @| 9 (1)X* Qa B*pR3 (2)X* Qa r p*)F Q 6 X* a 3 *F@+ B(*pR3 1~6 *pR3 B Q )X a5 *F@+901404 * XFF @ F (9 1234 94 321)H * 2088H 9 901405 C * _ ` F* ABCpR B *ABC 9AC B9 (
Fudan University - ECON - 2965
0 90150119 E H*i ! d ! *iEd H 8.99901502a=1+2+.+109 b=12+22+.+1029 c=13+23+.+103 9 d=14+24+.+104 9 1~10 9 * 2 S D a 1~10 9 * T + 9 S=1 2+1 3+1 4+.+8 10+9 109 T=1 2 3 4+1 2 3 5+1 2 3 6+.+6 8 9 10+7 8 9 109 9 a,b,c,d S9 T 949014034 4 1 4 4 9 4 4 2 4
Fudan University - ECON - 2965
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Fudan University - ECON - 2965
E Gv*/Gv*A0 3 p5 2 42 DBC901701* 9 cC `ABCD ` C c =39 =49 =59 ABCD 9P901702a1,a2,.,a2001 9 2001,2002,2003,.4000,4001 = * ^ E (1)(2001 1) (2002 2) (2003 3) . (4001 2001) a a a a Xb (2)(1 1) (2 2) (3 3) . (2001a2001) a a a cb 9017039C + c ` h
Fudan University - ECON - 2965
0 901801 X * (1) u (2) X] aX xx 12 3 + = 0x x a x+a ax xax 901802 9 n+1 X] 9 A,B,C C L 1, 5 , 5 2 , 5 3 , 5u n * * 2L 6 F A 4 n+1 99 901803 L C BH x DH u u a E9 F9 G9 H9 I9 J 9 DFJ * L 16 4 F A bx AB x AD x EF x BD x(1) a = b (2) a = b + 1 (3) a = b
Fudan University - ECON - 2965
*E* 0 901901100 ) q * 7 b @ G E 50 E * c E * q * ) q * ) x a b c 5 c x c a1 2 3 x 345* abc * x901902p cfw_ y x z x w cfw_* dq = 0.123 xyzw = 0.123xyzw23 xyzw23 xyzw 10 p pxq cfw_ dxx9019034k * + 1 y (1) 9 n d 1,5,9, *y ny an x a1 + a2 + a3 +
Fudan University - ECON - 2965
*@* 0 912001 * I Yx ^ p (k 0 2 9 * 6 2 1 * 48 k3 2@9 912002 ZI * 8 k 3 8 q * 8 q p q p a&lt;b * a,k b3 8 p, q 9 * a,k b3 8 , p* 0 0 k a,k b0 9 912003 CK @e &quot; 6 @GM*p@ @je * h |*4 (1) @U * I (2) 4 * * @I U* 83 k* 83 k912004 &quot; @e 1 1 @ @ 1 8 @ @ U I@
Fudan University - ECON - 2965
0 912101 u (1)9 n (2)9 n nff ( n) f * n n * i `- *9E f ( n) n * (N *9E f ( n)f ( 2307) = 2 + 3 + 0 + 7 = 12 ff 9121029 * 1 : * * N (9E 9 912103 *7 9 912104 )91 N 6(917 + B 24 9f 912105 9 * n : * nf180* n : * n 6 j( 8 i*9E )i*9E 9 (1)9 n = 7
Fudan University - ECON - 2965
0 912201 zC &lt; * M&lt; C z 2 2 2 (1) AH + BK +CP = H B 2+ KC 2 +PA2 (2) AH+BK+CP=HB+KC+PA ABC 9 MH,MK,MP9 9 H ,K,P &lt; z9 912202 * . E z = * . 1E 2 2912203 W1 9 W 2 9 W 3z C 9 &lt; W1 9 W 2 9 W 3 9 *z= N @ * A&lt; C z 829 359 21 9 zC &lt; W1 9 W 2 9 W 3 9 A9 B 9 C C
Fudan University - ECON - 2965
0 912301 9 9 ABCD AB + BM AD + DN AM = AN M9 N9 MAN=45099 912302 ABC E9 F 9 P9 / P: B c) =m9 =n9 / F: B = r9 m9 n9 a9 b9 c 9 EB : r 9 (9 = a 9 = b9 =D99 912303 (1)a9 b9 c9 d9 eR9 a+b9 c+d 9 b+c9 d+e 9 c+d9 e+a 9 d+e9 a+b 9 a9 b9 c9 d9 e(2)a9 b9 c9 d9
Fudan University - ECON - 2965
0 912401 (20021 )2 1 20022002 1 * (1 n1 =n (n1) (n2) 2 1 , n )1 912402 ( HH HH H H H * A B @+ F* ACB t 2 @ B A *C C 912403 1 2x2 11[x]+12=0 1 ([x] * x x=3.8[x]=31 x=-0.4[x]=-11 x=7[x]=7)1 912404A BCD= =10 ABC=100 0 1 CDA=130 0 1=9124058 7 6 5
Fudan University - ECON - 2965
E a * p * x 0 912501 h n n 4 * p8 K n n * p8 K1 912502 * 1 a @a E @ 7 E @a 1.a 2. * 3. &gt; * bd * a a E ac * 0h K 1912503 (1) A BCD A BCD &gt; * * x (2) A BCD A BCD &gt; * * x 2 U + ! x( 3 U + ! x(D F Cx 1) K hx 1) K hAE BB912504 *OH 24 * &gt;*4 *1H O (1)
Fudan University - ECON - 2965
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Fudan University - ECON - 2965
0912701 1 1 t1, t2, t3, t4, t2003 1 t1=2 , tn+1 = , n=1,2,3,1 912702 1 1A 1 S1 M1,M2 1 300 1 (1 0 = = e1 H d D e )H D ( d e S H D S R9Ed 0 ) S S 1 912703 a,b He D a,b He D 7H de D 7H d e D M21d M2H eDM1K M1A V(1) (2)10a+b 5a+4ba-2b 4a-b7 79127
Fudan University - ECON - 2965
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