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### math120chapter3section1

Course: MATH 120, Fall 2008
School: CofC
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Word Count: 1244

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3 Differentiation Chapter Rules In this chapter we learn the main calculational tools that allow us to nd derivatives faster and more easily than by working directly with limits. These will be used in the applications seen for the rest of the semester, starting in this chapter in Sections 3, 10 and 11. By the end of this chapter, you should be able to answer all the Concept Check Questions except 2(k-t), and all...

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3 Differentiation Chapter Rules In this chapter we learn the main calculational tools that allow us to nd derivatives faster and more easily than by working directly with limits. These will be used in the applications seen for the rest of the semester, starting in this chapter in Sections 3, 10 and 11. By the end of this chapter, you should be able to answer all the Concept Check Questions except 2(k-t), and all of the True-False Quiz Questions at the end of the chapter on pages 261. Note that we do not cover Section 8. 3.1 Derivatives of Polynomial and Exponential Functions Our collection of efciently methods for computing derivatives starts with polynomials and exponential functions. Much as with limits, we do this by rst dealing with a few simple functions, and then using rules for handling constant multiples, sums and differences. However, derivatives of products and quotients do not follow quite the same pattern as with limits, so we leave them until the next section. Since the slope of a straight line y = mx + c is the constant m, it is easy to check that the derivative of f (x) = mx + c is m, for any constants m and c. It is often convenient to use notation directly with formulas, without naming the functions, so to illustrate several notations: Theorem (Derivatives of Linear Functions). The derivative of the linear function f (x) = mx + c is f (x) = (mx + c) = d (mx + c) = m. dx The two most basic special cases are when the function is a constant c or just x: (c) = d (c) = 0, dx (x) = d (x) = 1. dx Theorem (The Power Rule). For any positive integer n, (xn ) = dn (x ) = n xn1 dx 3.1.1 CHAPTER 3. DIFFERENTIATION RULES 3.1.2 This includes f (x) = x1 = x and f (x) = x0 = 1, cases seen above. We have also almost seen this for f (x) = x2 in examples above with quadratics. Rather than do that example, let us look at n = 3, which hints at how to do this calculation for any n. Example (A). Calculate the derivative of f (x) = x3 . It is convenient in this case to use the rst formula for the derivative f (a): x3 a3 f (a) = lim . xa x a The cubic in the numerator vanishes for x = a, so it has a factor x a, and in fact the factorization is x3 a3 = (x a)(x2 + x a + a2 ). [Check by expanding!] This gives f (a) = lim (x a)(x2 + x a + a2 ) = lim (x2 + x a + a2 ) = 3a2 . xa xa xa That is, (x3 ) = 3x2 , in agreement with the Power Rule above for n = 3. Proof of the Power Rule (method 1) The key step is the factorization xn an = (x a)(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) This can be checked by expanding the right hand side, distributing the left hand factor: = x(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) = xn + xn1 a + xn2 a2 + + x2 an2 + xan1 = xn an (x a)(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) a(xn1 + xn2 a + xn3 a2 + + xan2 + an1 ) xn1 a xn2 a2 xn3 a3 xan1 an because all the terms in between pair off and cancel out. Much as with x3 , the denition of the derivative gives the derivative of f (x) = xn at x = a as f (a) = xn an xa x a (x a)(xn1 + xn2 a + xn3 a2 + + an1 ) = lim xa xa n1 n2 n3 = lim (x +x a+x a2 + + an1 ) lim =a =a xa n1 n1 + an2 a + an3 a2 + + an1 = nan1 , + an1 + an1 + + an1 (n copies) so f (x) = (xn ) = nxn1 , as claimed. Constant Multiples, Sums and Differences As with limits, we can build up polynomials from these power functions using constant multiples, and sums differences. And the derivatives of these three basic combinations are as simple as with limits. CHAPTER 3. DIFFERENTIATION RULES 3.1.3 Theorem (The Constant Multiple Rule). If a differentiable function f is multiplied by a constant c, its derivative is multiplied by the same constant: d d [cf (x)] = c [f (x)], or (cf ) (x) = cf (x). dx dx Theorem (The Sum Rule). The sum of two differentiable functions f and g is differentiable, with the sums derivative the sum of summands derivatives: d d d [f (x) + g(x)] = f (x) + g(x), or (f + g) (x) = f (x) + g (x). dx dx dx Theorem (The Difference Rule). The difference of two differentiable functions f and g is differentiable, with its derivative the difference of their derivatives: d d d [f (x) g(x)] = f (x) g(x), or (f g) (x) = f (x) g (x). dx dx dx Warning: These are the only three rules that are as simple and guessable as for limits! Example (5, page 177). Compute the derivative of x8 + 12x5 4x2 + 10x3 6x + 5. The same approach works for differentiating any polynomial. By the way, this shows that all polynomials are differentiable. Example (6, page 178). Find the points on the curve y = x4 6x2 + 4 where the tangent is horizontal. Derivatives of Other Power Functions 1 d 1/2 Example 3 in Section 2.8 show that x has derivative . That is, x = (1/2)x1/21 . This dx 2x ts the power rule, but for power 1/2, not a positive integer. In fact, the rule works for all real powers: Theorem (The Power Rule, Generalized Version). For any real number a, da x = axa1 , or (xa ) = axa1 . dx This is most easily shown later when we know how to differentiate exponential functions and compositions of functions. Example (A). Differentiate 3/x, and use this to check the slope computed in Example 2 of Section 2.7. Derivatives of the Natural Exponential Function In Section 1.5 (page 13 of these notes, page 6 of the text) we dened the number e so that the slope of y = ex at point (0, 1) is 1. That is eh 1 lim = 1. h0 h CHAPTER 3. DIFFERENTIATION RULES This choic...

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CofC - MATH - 120
CHAPTER 2. LIMITS AND DERIVATIVES2.7.12.7Derivatives and Rates of ChangeNote: this section revisits ideas seen earlier in the Preview of calculus and Section 2.1, now done more completely using what we know about limits and revisits examples
CofC - MATH - 120
CHAPTER 2. LIMITS AND DERIVATIVES Innite Limits2.4.2We need a slightly different measure for f (x) being close to innity, and what we use is f (x) &gt; M for large M ; likewise having f (x) &lt; M for large negative M measures closenees to . Denition.
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Introductory Calculus (Math 120) Syllabus, Notes and Study GuideSection 90 Monday, Tuesday and Thursday, 4-5:15pm in Maybank 224. These notes are keyed to the textbook Single Variable Calculus: Early Transcendentals (6th ed.) by James Stewart or the
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CHAPTER 2. LIMITS AND DERIVATIVES2.4.12.4The Precise Denition of a LimitxaWe have worked with limits so far using the intuitive idea that lim f (x) = L means As x gets close to a, f (x) gets close to L. First we put this in terms of guarante
CofC - MATH - 120
Chapter 2Limits and DerivativesIn this chapter we meet the two central ideal of calculus, describing rates of change (like instantaneous velocity), and the mathematical tool of limits used to compute such quantities exactly from a collection of ap
CofC - MATH - 120
A Preview of CalculusWe will look at areas, tangents, velocity and a bit about innite sequences of numbers, all of which will be studied in more detail during the semester. The last topic mentioned in this chapter of the text, The Sum of a Series, i
CofC - MATH - 220
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Calculus 2 Quiz 8 Tuesday March 27, 2
CofC - MATH - 220
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Calculus 2 Quiz 3 Tuesday January 30, 2007Give so
CofC - MATH - 220
Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Calculus 2 Quiz 2 Tuesday January 23, 2007Give so
CofC - MATH - 220
MATH 220 TEST 2, TUESDAY MARCH 20, 2007 Write your name on your formula sheet as well as here and hand it in with your work. Read all questions carefully before you start working on any of them. It is usually a good idea to start with the ones tha
CofC - MATH - 221
SECTION 14.6. DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR3414.6Directional Derivatives and the Gradient VectorHomework Exercises 7-13, 14*, 19-26, 28, 30*, 31, 32, 50-56. It is natural to ask about the rate of change of a function f (x, y
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SECTION 16.6. PARAMETRIC SURFACES AND THEIR AREAS7616.6Parametric Surfaces and Their AreasHomework Exercises 1-4, 11, 12*, 13-16, 29, 30*, 31-41 Just as not all plane curves can be conveniently described as the graph of a function y = f (x) a
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Course Objectives and OrganizationCourse Objectives and Expected Student OutcomesThe main objective of this course is to combine ideas of calculus and geometry to deal with functions whose values are a point in the plane or space (a vector), and fu
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SECTION 15.7. TRIPLE INTEGRALS5115.7Triple IntegralsHomework Exercises 3-9, 10*, 11-19, 29, 30*, 35, 36. Note, in 30, you may omit the two orders where dx comes rst. The ideas used to dene double integrals and then evaluate then in terms of i
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Chapter 13Vector FunctionsSince a position in space can be described by a vector r = x, y, z and position can be a function of time, it is natural to consider vector functions like r(t) = f (t), g(t), h(t) = f (t)i + g(t)j + h(t)k, functions whose
CofC - MATH - 221
CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE312.2VectorsThe earliest meaning of a vector relates to movement from one place to another (like a mosquito as a vector for malaria), and this leads to the geometrical idea of a vector describing t
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SECTION 16.5. CURL AND DIVERGENCE7316.5Curl and DivergenceHomework Exercises 1-7, 8*, 12*, 13,14,15*,16-20 The curl of a vector eld on R3 is an important quantity in the description of uid ow and electromagnetic elds, and is also related to w
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SECTION 16.3. THE FUNDAMENTAL THEOREM FOR LINE INTEGRALS6716.3 The Fundamental Theorem for Line IntegralsHomework Exercises 1, 3, 4*, 5-8*, 9-12*, 13-20*, 23 For a function of a single variable f (x), the Fundamental Theorem of Calculus says tha
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CHAPTER 13. VECTOR FUNCTIONS1613.2Derivatives and Integrals of Vector FunctionsDerivatives We can build derivatives of vector functions from derivative of components, but the denition can also be done from rst principles, with difference quot
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SECTION 15.4. DOUBLE INTEGRALS IN POLAR COORDINATES4815.4Double Integrals in Polar CoordinatesHomework Review polar coordinates in Section 10.4, and do Section 15.4. Exercises 7, 8*, 9-11,12*, 13-16, 29-32. Perhaps the single most common shap
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Chapter 14Partial DerivativesMany physical quantities depend on several other quantities like temperature depending on position specied by latitude, longitude altitude and time. A quantity can often be expressed in terms of a formula involving sev
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CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE612.3The Dot Producta b = a1 , a2 , a3 b1 , b2 , b3 = a1 b1 + a2 b2 + a3 b3 . (1)There are two useful notions of a product of two vectors: the rst is the dot productThis is also known as the s
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MATH 545 COMPUTATIONAL PROJECT 1 EXAMPLEBRENTON LEMESURIER, MONDAY MARCH 12This is an example of what Project 1 nal report could look like: this document plus the les newton.m, newtonrun.m, newton.txt, quasinewton.m, quasinewtonrun.m, quasinewton.
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MATH 545 COMPUTATIONAL PROJECT 2: INTEGRATION, ODES, AND RICHARDSON EXTRAPOLATIONDUE FRIDAY MARCH 23, WITH POSSIBILITY OF REVISIONS BRENTON LEMESURIER, MARCH 12, 20071. The form of project reports and submissions The goal with this and all project