Applied Calculus Math 215 - University of Hawaii

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18 CHAPTER 1. SOME BACKGROUND MATERIAL Let us do a higher degree example: Example 1.11. Let p(x) =x 4 −2x 3 +5x 2 −x+3 and x0 =2. Finda line l(x) and a constant A, such that |p(x) − l(x)| ≤A(x−x0) 2 for all x in the interval I =(1,3). (Note that the open interval I contains the point x0 =2.) Expanding p(x) inpowersof(x−2) we find p(x) =(x−2) 4 +6(x−2) 3 + 17(x − 2) 2 + 27(x − 2) + 21. Set l(x) = 27(x − 2) + 21. Then |p(x) − l(x)| = 4 3 2 (x − 2) +6(x−2) + 17(x − 2) = 2 (x − 2) +6(x−2) + 17 2 (x − 2) ≤ (1 + 6 + 17)(x − 2) 2 ≤ 24(x − 2) 2 . In the calculation we used the triangle inequality ((5.9) in Section 5.2 to get the first inequality. If x ∈ (1, 3), then |x − 2| < 1and|x−2| k
1.2. PARABOLAS AND HIGHER DEGREE POLYNOMIALS 19 Then |p(x) − l(x)| = Cn(x − x0) n−2 + ···+C3(x−x0)+C2 (x−x0) 2 ≤ |Cn(x−x0) n−2 |+···+|C3(x−x0)|+|C2| (x−x0) 2 ≤ (|Cn|+|Cn−1|+···+|C2|)(x−x0) 2 for all x ∈ I =(x0−1,x0+ 1). The details of the calculation are as follows. To get the equation, we took |p(x) − l(x)| and factored out (x − x0) 2 . To get the first inequality we repeatedly used the triangle inequality, see (5.9) in Section 5.2. The last inequality follows as (x − x0) k < 1ifk≥1. In summary, for l(x) =C1(x−x0)+C0 and A =(|Cn|+···+|C2|)we have seen that |p(x) − l(x)| ≤A(x−x0) 2 for all x ∈ (x0 − 1,x0 + 1). In the sense of our preview, and the upcoming discussion about derivatives, this means • Therateofchangeofp(x)atthepoint(x0,p(x0)) is C1, theslopeof the line l(x). Exercise 17. For each of the following polynomials p(x) andpointsx0, find the rate of change of p(x) whenx=x0. 1. p(x) =x 2 −7x+2andx0 =4. 2. p(x) =2x 3 +3andx0 =1. 3. p(x) =x 4 −x 3 +3x 2 −8x+4andx0 =−1. Remark 3. You may have noticed, that we began to omit labels on the axes of graphs. One reason for this is, that we displayed more than one function in one graph, and that means that there is no natural name for the variable associated to the vertical axis. Our general rule is, that we use the horizontal axis for the independent variable and the vertical one for the dependent one 1 . This is the rule which almost any mathematical text abides by. In some sciences this rule is reversed. 1 If you like to review the terms independent and dependent variable, then we suggest that you read Section 5.3 on page 268.
Page 1 and 2: Applied Calculus Math 215 Karl Hein
Page 3 and 4: Contents Preface iii 0 A Preview 1
Page 5 and 6: Preface These notes are written for
Page 7 and 8: need the powerful machine developed
Page 9 and 10: Chapter 0 A Preview In this introdu
Page 11 and 12: scrutiny, you detect that the graph
Page 13 and 14: 0.8 0.6 0.4 0.2 -0.2 R(t) 1 2 3 4 t
Page 15 and 16: Chapter 1 Some Background Material
Page 17 and 18: 1.1. LINES 9 In application, we are
Page 19 and 20: 1.1. LINES 11 Exercise 3. Suppose a
Page 21 and 22: 1.2. PARABOLAS AND HIGHER DEGREE PO
Page 23 and 24: 1.2. PARABOLAS AND HIGHER DEGREE PO
Page 25: 1.2. PARABOLAS AND HIGHER DEGREE PO
Page 29 and 30: 1.3. THE EXPONENTIAL AND LOGARITHM
Page 39 and 40: 1.4. USE OF GRAPHING UTILITIES 31 E
Page 41 and 42: 1.4. USE OF GRAPHING UTILITIES 33 a
Page 43 and 44: Chapter 2 The Derivative The deriva
Page 45 and 46: straight line. There is still a dif
Page 47 and 48: 6 4 2 -2 -1 1 2 -2 -4 Figure 2.3: E
Page 49 and 50: Figure 2.4: The radial line is perp
Page 51 and 52: 2.1. DEFINITION OF THE DERIVATIVE 4
Page 53 and 54: 2.1. DEFINITION OF THE DERIVATIVE 4
Page 55 and 56: 2.2. DIFFERENTIABILITY AS A LOCAL P
Page 57 and 58: 2.3. DERIVATIVES OF SOME BASIC FUNC
Page 65 and 66: 2.4. SLOPES OF SECANT LINES AND RAT
Page 67 and 68: 2.4. SLOPES OF SECANT LINES AND RAT
Page 69 and 70: 2.5. UPPER AND LOWER PARABOLAS 61 o
Page 71 and 72: 2.5. UPPER AND LOWER PARABOLAS 63 6
Page 73 and 74: 2.5. UPPER AND LOWER PARABOLAS 65 2
Page 75 and 76: 2.5. UPPER AND LOWER PARABOLAS 67 I
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2.6. OTHER NOTATIONS FOR THE DERIVA
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2.7. EXPONENTIAL GROWTH AND DECAY 7
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2.8. MORE EXPONENTIAL GROWTH AND DE
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2.8. MORE EXPONENTIAL GROWTH AND DE
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2.9. DIFFERENTIABILITY IMPLIES CONT
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2.10. BEING CLOSE VERSUS LOOKING LI
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2.11. RULES OF DIFFERENTIATION 85 t
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2.11. RULES OF DIFFERENTIATION 87 2
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2.11. RULES OF DIFFERENTIATION 89 E
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2.11. RULES OF DIFFERENTIATION 91 2
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2.11. RULES OF DIFFERENTIATION 93 f
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2.11. RULES OF DIFFERENTIATION 95 I
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2.11. RULES OF DIFFERENTIATION 97 T
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2.11. RULES OF DIFFERENTIATION 99 E
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2.11. RULES OF DIFFERENTIATION 101
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2.12. IMPLICIT DIFFERENTIATION 111
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2.12. IMPLICIT DIFFERENTIATION 113
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2.13. RELATED RATES 115 Exercise 80
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2.13. RELATED RATES 117 Exercise 86
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2.14. NUMERICAL METHODS 119 Your ca
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2.14. NUMERICAL METHODS 121 There i
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2.14. NUMERICAL METHODS 123 n xn f(
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2.14. NUMERICAL METHODS 125 Practic
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2.14. NUMERICAL METHODS 127 Solutio
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2.14. NUMERICAL METHODS 129 t y(t)
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2.14. NUMERICAL METHODS 131 stabili
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2.14. NUMERICAL METHODS 133 peratur
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2.15. SUMMARY 135 • Quotient rule
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Chapter 3 Applications of the Deriv
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3.2. CAUCHY’S MEAN VALUE THEOREM
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3.3. THE FIRST DERIVATIVE AND MONOT
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3.4. THE SECOND AND HIGHER DERIVATI
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3.5. THE SECOND DERIVATIVE AND CONC
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3.6. LOCAL EXTREMA AND INFLECTION P
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3.7. THE FIRST DERIVATIVE TEST 167
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3.7. THE FIRST DERIVATIVE TEST 169
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3.8. THE SECOND DERIVATIVE TEST 171
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3.9. EXTREMA OF FUNCTIONS 173 and a
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3.9. EXTREMA OF FUNCTIONS 175 200 1
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3.9. EXTREMA OF FUNCTIONS 177 P (25
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3.9. EXTREMA OF FUNCTIONS 179 We le
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3.9. EXTREMA OF FUNCTIONS 181 We de
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3.10. DETECTION OF INFLECTION POINT
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3.10. DETECTION OF INFLECTION POINT
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3.11. OPTIMIZATION PROBLEMS 187 3 2
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3.11. OPTIMIZATION PROBLEMS 189 sha
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3.11. OPTIMIZATION PROBLEMS 191 is
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3.11. OPTIMIZATION PROBLEMS 193 (
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3.11. OPTIMIZATION PROBLEMS 195 or
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3.11. OPTIMIZATION PROBLEMS 197 Exe
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3.12. SKETCHING GRAPHS 199 is conca
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3.12. SKETCHING GRAPHS 201 The valu
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Chapter 4 Integration We will intro
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0.6 0.5 0.4 0.3 0.2 0.1 1 2 3 4 5 6
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4.1. UPPER AND LOWER SUMS 207 So, i
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4.1. UPPER AND LOWER SUMS 209 1.25
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4.1. UPPER AND LOWER SUMS 211 4 3 2
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4.2. INTEGRABILITY AND AREAS 213 4.
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4.2. INTEGRABILITY AND AREAS 215 Pr
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4.2. INTEGRABILITY AND AREAS 217 th
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4.3. SOME ELEMENTARY OBSERVATIONS 2
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4.3. SOME ELEMENTARY OBSERVATIONS 2
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4.4. INTEGRABLE FUNCTIONS 223 4.4 I
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4.5. ANTI-DERIVATIVES 225 In this e
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4.6. THE FUNDAMENTAL THEOREM OF CAL
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4.7. SUBSTITUTION 235 We explain th
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4.7. SUBSTITUTION 237 Solution: We
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4.7. SUBSTITUTION 239 The first ide
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4.7. SUBSTITUTION 241 u = π/2. (Fo
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4.8. AREAS BETWEEN GRAPHS 243 x-coo
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4.9. NUMERICAL INTEGRATION 245 In a
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4.9. NUMERICAL INTEGRATION 247 In t
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4.9. NUMERICAL INTEGRATION 249 an a
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4.10. APPLICATIONS OF THE INTEGRAL
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4.10. APPLICATIONS OF THE INTEGRAL
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4.11. THE EXPONENTIAL AND LOGARITHM
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Chapter 5 Prerequisites from Precal
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5.1. THE REAL NUMBERS 265 A typical
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5.2. INEQUALITIES AND ABSOLUTE VALU
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5.3. FUNCTIONS, DEFINITION AND NOTA
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5.4. GRAPHING EQUATIONS 275 Example
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5.5. TRIGONOMETRIC FUNCTIONS 277 1
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5.5. TRIGONOMETRIC FUNCTIONS 279 40
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5.5. TRIGONOMETRIC FUNCTIONS 281 op
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5.5. TRIGONOMETRIC FUNCTIONS 283 an
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5.5. TRIGONOMETRIC FUNCTIONS 285 A
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5.6. INVERSE FUNCTIONS 287 Observe
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5.6. INVERSE FUNCTIONS 289 5 4 3 2
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5.6. INVERSE FUNCTIONS 291 Exercise
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5.7. NEW FUNCTIONS FROM OLD ONES 29
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Applied Calculus Math 215 - University of Hawaii

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