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1314Lesson5
Course: M 1314, Fall 2009School: U. Houston
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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 < -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) > 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
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U. Houston - M - 1300
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U. Houston - M - 1300
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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
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - M - 1300
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U. Houston - MATH - 3303
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U. Houston - MATH - 3303
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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
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U. Houston - MATH - 3303
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U. Houston - MATH - 3304
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U. Houston - MATH - 1314
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U. Houston - MATH - 1314
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U. Houston - MATH - 1314
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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
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