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1 Functions What is calculus? Functions. Equations of functions. General notation for functions. Notation for increases in functions. Graphs of functions. Using calculators or computers for plotting functions. Inverse functions. Implicit functions. Functions of more than one variable. |
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1 | (10) |
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2 Variations in functions, limits Variations in functions. Limits. Limit of a function of the form XXX. A trigonometric limit, (lim)(sinXXX). A geometric illustration of a limit. Theorems on limits. |
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11 | (12) |
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3 Gradient Gradient of the line joining two points. Equation of a straight line. Approximating to gradients of curves. Towards a definition of gradient. Definition of the gradient of a curve. Negative gradient. |
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23 | (10) |
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4 Rate of change The average change of a function over an interval. The average rate of change of a non-linear function. Motion of a body with non-constant velocity. Graphical interpretation. A definition of rate of change. |
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33 | (11) |
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5 Differentiation Algebraic approach to the rate of change of a function. The derived function. Notation for the derivative. Differentials. Sign of the derivative. Some examples of differentiation. |
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44 | (14) |
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6 Some rules for differentiation Differentiating a sum. Differentiating a product. Differentiating a quotient. Function of a function. Differentiating implicit functions. Successive differentiation. Alternative notation for derivatives. Graphs of derivatives. |
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58 | (23) |
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7 Maxima, minima and points of inflexion Sign of the derivative. Stationary values. Turning points. Maximum and minimum values. Which are maxima and which are minima? A graphical illustration. Some worked examples. Points of inflexion. |
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81 | (22) |
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8 Differentiating the trigonometric functions Using radians. Differentiating sin x. Differentiating cos x. Differentiating tan x. Differentiating sec x, cosec x, cot x. Summary of results. Differentiating trigonometric functions. Successive derivatives. Graphs of the trigonometric functions. Inverse trigonometric functions. Differentiating sin(-1) x and cos(-1) x. Differentiating tan(-1) x and cot(-1) x. Differentiating sec (-1) x and cosec(-1) x. Summary of results. |
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103 | (22) |
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9 Exponential and logarithmic functions Compound interest law of growth. The value of lim (1+1/n)(n). The compound interest law. Differentiating e(x). The exponential curve. Natural logarithms. Differentiating In x. Differentiating general exponential functions. Summary of formulae. Worked examples |
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125 | (14) |
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10 Hyperbolic functions Definitions of hyperbolic functions. Formulae connected with hyperbolic functions. Summary. Derivatives of the hyperbolic functions. Graphs of the hyperbolic functions. Differentiating the inverse hyperbolic functions. Logarithm equivalents of the inverse hyperbolic functions. Summary of inverse functions formulae. |
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139 | (15) |
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11 Integration; standard integrals Meaning of integration. The constant of integration. The symbol for integration. Integrating a constant factor. Integrating x(n). Integrating a sum. Integrating 1/x. A useful rule for integration. Integrals of standard forms. Additional standard integrals. |
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154 | (20) |
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12 Methods of integration Introduction. Trigonometric functions. Integration by substitution. Some trigonometric substitutions. The substitution t=tan 1/2(x). Worked examples. Algebraic substitutions. Integration by parts. |
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174 | (20) |
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13 Integration of algebraic fractions Rational fractions. Denominators of the form ax(2)+bx+c. Denominator a perfect square. Denominator a difference of squares. Denominator a sum of squares. Denominators of a higher degree. Denominators with square roots. |
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194 | (14) |
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14 Area and definite integrals Areas by integration. Definite integrals. Characteristics of a definite integral. Some properties of definite integrals. Infinite limits and infinite integrals. Infinite limits. Functions with infinite values. |
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208 | (19) |
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15 The integral as a sum; areas Approximation to area by division into small elements. The definite integral as the limit of a sum. Examples of areas. Sign of an area. Polar coordinates. Plotting curves from their equations in polar coordinates. Areas in polar coordinates. Mean value. |
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227 | (32) |
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16 Approximate integration The need for approximate integration. The trapezoidal rule. Simpson's rule for area. |
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259 | (7) |
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17 Volumes of revolution Solids of revolution. Volume of a cone. General formula for volumes of solids of revolution. Volume of a sphere. Examples. |
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266 | (8) |
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18 Lengths of curves Lengths of arcs of curves. Length in polar coordinates. |
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274 | (8) |
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19 Taylor's and Maclaurin's series Infinite series. Convergent and divergent series. Taylor's expansion. Maclaurin's series. Expansion by the differentiation and integration of known series. |
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282 | (9) |
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20 Differential equations Introduction and definitions. Type I: One variable absent. Type II: variables separable. Type III: linear equations. Type IV: linear differential equations with constant coefficients. Type V: homogenous equations. |
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291 | (21) |
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21 Applications of differential equations Introduction. Problems involving rates. Problems involving elements. |
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312 | (11) |
| Answers |
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323 | (20) |
| Index |
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343 | |
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