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### Math103_9_3_Deriv

Course: KFS 103, Fall 2009
School: CSU Northridge
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Word Count: 601

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The 9.3 Derivative The student will learn about: Rate of change Slope of the tangent line The derivative Existence/Nonexistence of the derivative 1 Difference Quotient: Slope The difference quotient that follows gives the average rate of change of the function passing through P and Q: 4 slope 3 2 f (a + h) f (a + h ) ! f (a ) h f (a + h) f (a) -1 1 f (a) O (a) 1 (a + h) 2 h 2 Example 1 The revenue is...

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The 9.3 Derivative The student will learn about: Rate of change Slope of the tangent line The derivative Existence/Nonexistence of the derivative 1 Difference Quotient: Slope The difference quotient that follows gives the average rate of change of the function passing through P and Q: 4 slope 3 2 f (a + h) f (a + h ) ! f (a ) h f (a + h) f (a) -1 1 f (a) O (a) 1 (a + h) 2 h 2 Example 1 The revenue is given by R (x) = x (75 3x) for 0 ! x ! 20. What is the change in revenue if production changes from 9 to 12? What is the average rate of change in revenue if production changes from 9 to 12? 3 The Instantaneous Rate of Change. Consider the function y = f (x) only at the point (a, f (a)) . The limit of the difference quotient that follows gives the instantaneous rate of change of the function passing through (a, f (a)): The instantaneous rate of change = f (a + h) " f (a) lim h!0 h 4 4 Visual Interpretation 3 f (a + h) 2 Tangent Slope of tangent. Instantaneous Rate of Change. 1 f (a) -1 lim f (a + h ) ! f (a ) h"0 h O (a) 1 (a + h) 2 Let h approach 0 5 3 h Definition of the Derivative Given y = f (x), the slope of the graph at the point (a, f (a)) is given by h!0 lim f (a + h) " f (a) h Provided that the limit exist. We define it to be the derivative of f at x, denote it by f (x). We define the tangent line to y=f(x) at the point (a, f (a)) to be the line through this point of slope equal to f (x). f (x) is the instantaneous rate of change at x (e.g. velocity). If f (x) exists for each x in the open interval b), (a, then f is said to be differentiable over (a, b). 6 Four-Step Process To find f (x) we use a four-step process Step 1. Find f (x = h) Step 2. Find f (x + h) f (x) Step 3. Find f ( x + h) ! f ( x) h Step 4. Find lim f ( x + h) ! f ( x) h h"0 7 Example 2 Find the derivative of f (x) = x 2 3x f (x + h) = (x + h) 2 3(x + h) f (x + h) f (x) f ( x + h) ! f ( x) h 8 Example 3 Find the slope of the graph of f (x) = x 2 3x at x = 0, x = 2, and x = 3. From example 2 we found the derivative of this function at x to be f (x) = 2x - 3 9 Example 4 R(x) = 60x .02x2 Find R (x) 10 Nonexistence of the Derivative. The existence of a derivative at x = a depends on the existence of a limit at x = a; that is, on the existence of f (a) = f (a + h ) ! f (a ) h h"0 lim If the limit does not exist at x = a, we say that the function is nondifferentiable at x = a, or f (x) does not exist. f (x) = x |x+h|"|...

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MEAN= -16.88689, 804.1 290.0 755.0 289.4 749.2 287.8 735.2 287.7 725.4 285.5 697.2 282.0 686.0 281.2 660.5 281.1 634.2 281.0 623.2 277.9 620.6 277.6 595.0 277.0 554.5 276.1 544.9 274.6 533.0 273.6 528.0 273.3 524.8 273.2 520.5 272.9 519.0 271.7 510.9
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2.5 2 1.5 1 0.5246810
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path p-constant V -constant T -constant S-constant line free cycleW Q pV nCp T 0 nCV T nRT ln (Vf /Vi ) W nCV T 0 1 (p + pi )V W + E 2 f 0 0 area area Note: nRT Cp = C V + R CV =Eint S nCV T nCp ln (Tf /Ti ) nCV T nCV ln (Tf /Ti ) 0 nR ln (Vf /Vi