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### 1314Lesson5

Course: M 1314, Fall 2009
School: U. Houston
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Word Count: 229

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1314 Math 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 Rule d [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x) f ' ( x ) dx Example 1: Use the product rule to find the derivative if f ( x) = (7 x 2 5)( x 3 + 6). Example 2: Find the derivative if f ( x) = xe x . Example 3: Find the derivative if...

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1314 Math 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 Rule d [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x) f ' ( x ) dx Example 1: Use the product rule to find the derivative if f ( x) = (7 x 2 5)( x 3 + 6). Example 2: Find the derivative if f ( x) = xe x . Example 3: Find the derivative if f ( x) = x 2 ln x . Example 4: Find the derivative if f(x) = 3x2ex. Rule 9: The Quotient Rule d f ( x) g ( x) f ' ( x) f ( x) g ' ( x) = , g ( x) 0 dx g ( x) [g ( x)]2 It is easier for some students to remember using this this device: d hi lo de hi hi de lo = dx lo lo lo where de hi refers to the derivative of the numerator and de lo refers to the derivative of the denominator. Example 5: Find the derivative if f ( x) = 5x + 6 . 9x + 6 Example 6: Find the derivative if f ( x) = x . x +1 2 Example 7: Find the derivative if f ( x) = 4 ln x x Example 8: Find the derivat...

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U. Houston - M - 1314
Math 1314 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
U. Houston - M - 1314
Math 1314 Lesson 10 Applications of the First Derivative Determining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f Definition: A function is increasing on an interval (a, b) if, for any two numbers x1 and x2 in (a,
U. Houston - M - 1314
Math 1314 Lesson 13 Absolute Extrema In earlier sections, you learned how to find relative (local) extrema. These points were the high points and low points relative to the other points around them. In this section, you will learn how to find absolut
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 1 1. Find lim f ( x) given the graph of f (x). (10.4.3)x32. Evaluate lim 3 x 2 - 2 (10.4.35)x 3x 2 + 4x + 3 (10.4.57) x 1 x2 -1 2 x - 1, x &lt; -1 4. Evaluate lim f ( x) if f ( x) = 2 (10.4.18) x -1 x + 2, x -1 5. Eval
U. Houston - MATH - 1314
Math 1314 Daily Poppers Lesson 3f ( x + h) - f ( x ) . h 2. True or False: If you need to find an instantaneous rate of change, you will find the derivative.1. True or False: The formula for the average rate of change is3. Find ( x + h ) . (10.6
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 4 1. Find the derivative: f ( x) = 12 . (11.1.2)2. Find the derivative: f ( x) = x 6 . (11.1.4) 3. Find the derivative: f ( x) = 5 x . (11.1.13)4. Find the derivative: f ( x) = x 4 - 7 x 2 + 5. (11.1.17) 5. Find the deriva
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 5 1. Find the derivative: f ( x) = 5 x( x 2 + 4) . (11.2.1) 2. Find the derivative: f ( x) = (6 x - 3)( x + 4) . (11.2.3)3. Find the derivative: f ( x) = xe x (13.3.7) 4. Find the derivative: f ( x) = x ln x . (13.4.15) 5. S
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 6 1. If f ( x) = 3 x + 4 and g ( x) = x 2 , then find g o f .2. Find the derivative: f ( x) = (4 x 5) . (11.3.1)63. Find the derivative: f ( x) = x 2 + 2 . (11.3.12) 4. Find the derivative: f ( x) =( x 5 )26. (11.
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 7 1. Find the second derivative: f ( x) = 6 x 2 - 9 x + 3 . (11.5.1) 2. Find the second derivative: f ( x) = 2e x . 1 . (11.5.15) 3. Find the second derivative: f ( x) = ( x - 4 )24. Find the second derivative: f ( x) = ln(
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 8 1. Find the slope of the tangent line to f ( x) = 3 x 2 - 4 when x = 1. (10.6.20) 2. Write an equation of the tangent line to the graph of f ( x) = -2 x 2 + 5 at the point (-2, -3). (10.6.23) 3. Let f ( x) = x 3 - 8 x 2 . F
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 9 Use this information for questions 1 9. Suppose a company's weekly cost incurred in producing x of its products is given by the function C ( x) = .000002 x 3 - .02 x 2 + 150 x + 80,000 and the weekly demand for the product
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 10 1. Suppose f ( x) = x 2 5 x . State the interval(s) where the function is increasing and the interval(s) where it is decreasing. (12.1.13) 2. Suppose f ( x) = x 4 5 x 3 + 6 . State the interval(s) where the function is i
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 11 1. Suppose f ' ' ( x) &gt; 0 for all x in the domain of f. Then which of these statements is true? (12.2.16 21) o f is concave upward for all x in the domain of f o f is concave downward for all x in the domain of f o f has
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 13 1. Suppose f ( x) = x - 2 x . Find any critical points. (12.4.28)2. Suppose f ( x) = x - 2 x . Find the absolute maximum of f on the interval [0, 9]. (12.4.28) 3. Suppose f ( x) = x - 2 x . Find the absolute minimum of f
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 14 1. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 25 inches long and 10 inches wide, (12.5.4) a. writ
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 15 1. Write the general form of a function expressing exponential growth. 2. Write the general form of a function expressing exponential decay. 3. Find the rate of change of the exponential growth function. 4. At the beginnin
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 16 1. Find the antiderivative: (x2- 4 x + 6 dx)(14.1.24)2. Find the antiderivative: e x1 + dx x(14.1.44) x2 - 7x + 4 dx 3. Find the antiderivative: x 4. Find the antiderivative: x dx 3 6 x 2
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 17 1. Rewrite the problem (4 x - 7) dx using a substitution.5(14.2.1)2. Rewrite the problem 5 5 x 2 + 6 dx using substitution. (14.2.7) 3. Find the indefinite integral: 4. Find the indefinite integral:55 x 2 + 6 d
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 18 1. Suppose f ( x) = 2 x 2 + 6 and you are asked to use Riemann sums to approximate the area under the curve on the interval [0, 8]. (14.3.13) a. if you are asked to use four subintervals of equal length, find x . b. if you
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 20 1. Evaluate: x(x1 02+ 6 dx)2(14.5.1)2. Evaluate:31x 3 x 2 2 dx6(14.5.4)3. Evaluate: (5 x 3) dx1 0(14.5.9)4. Evaluate:41e 3 x dx(14.5.16)5. Evaluate:1 0x 2x + 12dx
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 21 1. Write the integral(s) needed to find the area between the x axis and f ( x) = x 3 3 x 2 . (14.6.1)2. Write the integral(s) needed to find the area between the x axis and f ( x) = x 3 x . (14.6.5)3. Write the integ
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 22 1. Suppose f ( x, y ) = x 2 y - 4 xy + 6 y 2 . Find f (-1, 2) .(16.1.3)2. Suppose f ( x, y ) = 3 x - 4 y x + 5 xy 2 . Find f (4, - 2 ). (16.1.5)3. Suppose f ( x, y, z ) = 2 xy + 3 xz - 4 yz. Find f (0,1,2). 4. Suppose
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 23 1. Find f x : f ( x, y ) = x 2 + 3 xy + 2 y 2 2. Find f y : f ( x, y ) = x 2 + 3 xy + 2 y 2 (16.2.23) (16.2.23) (16.2.4)3. Find the first partial derivatives: f ( x, y ) = 3 - 5 x 2 + 7 y 2 4. Findf : f ( x, y ) = x 2 e
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 24 1. Suppose f ( x, y ) = x 2 - 4 xy + 2 y 2 + 4 x + 8 y - 1 . a. find the first partial derivatives b. find the critical points c. find the second order partial derivatives d. find f xx (a, b) for each critical point e. fin
U. Houston - M - 1300
Math 1300Section 1.2 NotesMake each sentence true by using &lt; or &gt;. 1) -9 _ -8 4) 4 _ 5 2) 0 _-7 5) -5 _11 3) -3 _ -23 6) _ 2/3Adding Integers: o Same signs add and keep the sign o Different signs subtract and take the sign of the number with
U. Houston - M - 1300
Math 1300Section 1.3 NotesGCF (Greatest Common Factor) 1) Write each of the given numbers as a product of prime factors. 2) The GCF of two or more numbers is the product of all prime factors common to every number. Examples: 1. Find the GCF of 24
U. Houston - M - 1300
Math 1300 Rules for Exponents: Multiplying Powers: Section 1.4 Notesa m a n = a m+n am Dividing Powers: = a m-n n a 1 1 Negative Powers: a - m = m and = a n (Note: a 0 ) a a-n Zero Power Rule: a0 = 1 If no power is shown, then the exponent
U. Houston - M - 1300
Math 1300 Rules for Exponents: Multiplying Powers: Section 1.4 Notesa m a n = a m+n am Dividing Powers: = a m-n n a 1 1 Negative Powers: a - m = m and = a n (Note: a 0 ) a a-n Zero Power Rule: a0 = 1 If no power is shown, then the exponent
U. Houston - M - 1300
Math 1300Section 1.5 NotesOrder of Operations: P E M D A SExamples: 1.(2 - 4)3 - (-1 + 6) 2 22.3(-6) + 0.5(-60) (32 - 2) + (-7 - 2)3.-7 + 24.-6 25. 1 - 226. 2 + 3(4 - 5) 2 4 1Math 13001 , 2Section 1.5 NotesE
U. Houston - M - 1300
Math 1300Section 1.6 NotesSolving Linear Equations Steps: 1) Distribute if the equation has parentheses 2) Combine any like terms 3) Isolate the variable by doing addition/subtraction before multiplication/division Examples: 1. x + 2 =82. 4
U. Houston - M - 1300
Math 1300Section 1.7 NotesSolving Linear Inequalities An inequality is similar to an equation except instead of an equal sign &quot;=&quot; you find one of the following signs: &lt;, , &gt;, or . Now &gt; and &lt; are strict inequalities, and and are inequalities th
U. Houston - M - 1300
Math 1300Section 1.8 NotesSolving Absolute Value Equations: To solve and equation involving absolute values, use the following property: If C is positive, then |x| = C is equivalent to x = C. Special cases for |x| = C: Case 1: If C is negative
U. Houston - M - 1300
Math 1300Section 2.1 NotesThe Coordinate Plane:3 2 1 X -3 -2 -1 0 -1 -2 -3 YPlotting: When plotting (a, b) on a graph, go in `alphabetical' order. Go a units in the xdirection (horizontally) first, then in go b units the y-direction (verticall
U. Houston - M - 1300
Math 1300Section 2.1 NotesThe Coordinate Plane:3 2 1 X -3 -2 -1 0 -1 -2 -3 YPlotting: When plotting (a, b) on a graph, go in alphabetical order. Go a units in the xdirection (horizontally) first, then in go b units the y-direction (vertically)
U. Houston - M - 1300
Math 1300Section 2.2 NotesThe Distance and Midpoint Formulas: The distance between two points A = (x1 , y1 ) and B = (x 2 , y 2 ) is given by the distance formula d ( A, B) = (x 2 - x1 )2 + ( y 2 - y1 )2 . 1. Find the distance between the two poi
U. Houston - M - 1300
Math 1300Section 2.2 NotesThe Distance and Midpoint Formulas: The distance between two points A = (x1 , y1 ) and B = (x 2 , y 2 ) is given by the distance formula d ( A, B) = (x 2 - x1 )2 + ( y 2 - y1 )2 . 1. Find the distance between the two poi
U. Houston - M - 1300
Math 1300Section 2.3 NotesSlope and Intercepts of a Line: The slope, m, of a line that passes through the points A = (x1 , y1 ) and B = (x 2 , y 2 ) is y 2 - y1 rise = given by the slope formula m = . x 2 - x1 runPositive SlopeNegative Slope
U. Houston - M - 1300
Math 1300Section 2.3 NotesSlope and Intercepts of a Line: The slope, m, of a line that passes through the points A = (x1 , y1 ) and B = (x 2 , y 2 ) is y 2 - y1 rise = given by the slope formula m = . x 2 - x1 runPositive SlopeNegative Slope
U. Houston - M - 1300
Math 1300Section 2.4 NotesEquations of a Line: Forms of Lines: 1. The standard form of a linear equation is given by Ax + By = C where A and B cannot both be equal to zero. 2. The point-slope form of a linear equation is given by y y1 = m ( x x
U. Houston - M - 1300
Math 1300Section 2.5 NotesParallel and Perpendicular Lines: Two lines with slopes m1 and m2 are parallel if and only if m1 = m2.3 2 1 X -3 -2 -1 0 -1 -2 -3 YTwo lines with slopes m1 and m2 are perpendicular if and only if m1m2 = -1.012
U. Houston - M - 1300
Math 1300Section 2.5 NotesParallel and Perpendicular Lines: Two lines with slopes m1 and m2 are parallel if and only if m1 = m2.3 2 1 X -3 -2 -1 0 -1 -2 -3 YTwo lines with slopes m1 and m2 are perpendicular if and only if m1m2 = -1.012
U. Houston - M - 1300
Math 1300Section 2.6 NotesFunctions and Domains: Definition: A relation in mathematics is a set of one or more ordered pairs. It can be described by: 1. A set of ordered pairs: {(-3, -1), (-2, 1), (-1, 1), (1, 3), (3, 1), (3, 2), (0, 3)} 2. Graph
U. Houston - M - 1300
Math 1300Section 2.7 NotesFunctions and Domains: Definition: The graph of a function f is the set of all points (x, y) in the coordinate plane where the x-coordinates are the elements of the domain of f and where the ycoordinates are given by y =
U. Houston - M - 1300
Math 1300Section 2.7 NotesFunctions and Domains: Definition: The graph of a function f is the set of all points (x, y) in the coordinate plane where the x-coordinates are the elements of the domain of f and where the ycoordinates are given by y =
U. Houston - M - 1300
Math 1300Section 3.1 NotesPolynomials A polynomial is an expression made up of terms called monomials. A monomial is an expression made up of one real-number coefficient and (at least) one variable to some whole-number power. Examples are -3; 7x;
U. Houston - M - 1300
Math 1300 Operations with PolynomialsSection 3.2 NotesAddition of Polynomials The sum of two polynomials is found by combining like terms. To add like terms, add the coefficients and do not change the variable and exponents in common. Example: Pe
U. Houston - M - 1300
Math 1300 Section 2.2 Note: For the final result, use a reduced radical form (i.e., write 8 as 2 2 ).Test 2 ReviewUse the Pythagorean Theorem to find the missing side of each of the following right triangles.1. 2. 3.If a = 5, c =7, find b. If
U. Houston - M - 1300
Section 2.2 1. b = 24 = 2 6 2. c = 125 = 5 5 3. a = 57 5 241 1 = 241 5. d = 100 10 17 6. d = 5 5 7. d = 8 641 1 8. d = = 641 100 10 9. d = 109 3 10. ,4 10 1 5 11. , 5 4 1 1 12. , 2 5 4. d =Section 2.3 13. m = undefined15 2 32 15. m
U. Houston - MATH - 3303
Sep 7-3:21 PM1Sep 7-4:05 PM2Sep 7-4:07 PM3Sep 7-4:12 PM4Sep 7-4:14 PM5Sep 7-4:15 PM6Sep 7-4:21 PM7Sep 7-4:24 PM8Sep 7-4:26 PM9Sep 7-4:27 PM10Sep 7-4:30 PM11Sep 7-4:33 PM12Sep 7-4:36 PM13Sep 7-4:44 P
U. Houston - MATH - 3303
Nov 16-3:55 PM1Nov 16-4:07 PM2Nov 16-4:08 PM3Nov 16-4:10 PM4Nov 16-4:14 PM5Nov 16-4:18 PM6Nov 16-4:24 PM7Nov 16-4:27 PM8Nov 16-4:29 PM9Nov 16-4:33 PM10Nov 16-4:35 PM11Nov 16-4:39 PM12Nov 16-4:42 PM13
U. Houston - MATH - 3303
Some good websites for continued fractions.http:/archives.math.utk.edu/articles/atuyl/confrac/intro.html http:/home.att.net/~srschmitt/script_fractions.html http:/perl.plover.com/yak/cftalk/
U. Houston - MATH - 3303
Homework Module 3 3303 Name: email address: phone number:Who helped me:Who I helped:Homework rules: Front side only. Keep the questions and your answers in order. If you send it pdf, send it in a single scanned file. (dog@uh.edu) If you turn
U. Houston - MATH - 3303
Oct 5-3:53 PM1Oct 5-3:58 PM2Oct 5-4:01 PM3Oct 5-4:07 PM4Oct 5-4:10 PM5Oct 5-4:14 PM6Oct 5-4:16 PM7Oct 5-4:18 PM8Oct 5-4:21 PM9Oct 5-4:23 PM10Oct 5-4:26 PM11Oct 5-4:30 PM12Oct 5-4:32 PM13Oct 5-4:39 P
U. Houston - MATH - 3304
Derivatives HomeworkName: MyUH id number:Who helped you?Who did you help?Turn in the homework to 651 PGH; ask the receptionist to date stamp it and put it in my mailbox or Send it to me as a single pdf file by email.This is a 100 point as
U. Houston - MATH - 3304
Limits HomeworkName: PS id:This is a 112 point assignment. It WILL be on the mid-semester. Please turn it in personally in 651 PGH or via email in a single pdf file.1Problem 1 Why is limxa10 points x 2 + 6x + 5 21 ? = 2 x2 - 1 3Include a
U. Houston - MATH - 1314
Math 1314 Homework Assignment 000 Problem 4 Solve for x: ln( x 2 5) = 2
U. Houston - MATH - 1314
Math 1314 Homework Assignment 012 Problem 1 Suppose f ( x) = x 3 3 x 2 9 x + 27 . a. Find all rational zeros of the function and indicate their multiplicities. b. Use the guide to curve sketching to graph the function.
U. Houston - MATH - 1314
Math 1314 Assignment #15 1. At the beginning of a population study, there were 2.3 million people living in a major city. Five years later, the population had grown to 3.1 million people. a. Assuming exponential growth, what is the projected populati
U. Houston - MATH - 1314
Math 1314 Homework Assignment 018 1. Use Riemann sums, 4 subintervals and right endpoints to approximate the area under f ( x) = 3 x 2 + 5 on the interval [0, 8]. 2. Use Riemann sums, 4 subintervals and midpoints to approximate the area under f ( x)
NYU - B - 4000
Behavioral FinanceTerm: Prof.: Office: Office hours: Spring 2004 Jeffrey Wurgler (jwurgler@stern.nyu.edu) Tisch 9-09 (1) Tuesday 9-9:30pm and longer as needed (starting 1/27) (2) Thursday 1-1:30pm and longer as needed (starting 1/22) (3) Email for a
U. Houston - M - 1330
Simple Trigonometric Identities:1.cot = cos csc 2.tan = sin sec 1 sin2 x 3. = cos x cos x2 2 4. cos (tan + 1) = 15. cot x + tan x = sec x csc x