Advanced Level Mathematics: Pure Mathematics 2 and 3 by Hugh Neill, Douglas Quadling, Julian Gilbey PDF

By Hugh Neill, Douglas Quadling, Julian Gilbey

ISBN-10: 0521530121

ISBN-13: 9780521530125

ISBN-10: 0521696356

ISBN-13: 9780521696357

Written to compare the contents of the Cambridge syllabus. natural arithmetic 2 corresponds to devices P2 and P3. It covers algebra, logarithmic and exponential services, trigonometry, differentiation, integration, numerical resolution of equations, vectors, differential equations and complicated numbers.

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X 1 - c. Therefore loge b- logb Historically logarithms were important because for many years,_ before calculators and computers were available, they provided the most useful form of calculating aid. With a table of logarithms students would, for example, find the cube root of 100 by looking up the value of log 100 and dividing it by 3. By the nth root rule, this gave log~, and the cube root could then be obtained from a table of the inverse function. 666 666 6 ... 641. 588 83 .... But of course you don't need to do this, since your calculator has a special key for working out roots directly.

3x-1 3 Note that this integral is only valid if x > ~,since ln(3x -1) only exists if 3x -1 > 0. >7 f". J~~,c~}~ยท, ~ ~ . ,i," ~~~"' ,\ >' 1 Carry out the following indefinite integrations, and state the values of x for which your answer is valid. x (d) (f) J 4 dx 1+2x (g) J-4-dx -1-2x (h) J --dx 2x-1 1 2 Calculate the area under the graph of y = - from x (a) x=3tox=6, (b) x=4tox=8, (c) x=~tox=l, 1 J~ -dx 4x+3 (d) x =a to x = 2a, a>O. 4 59 CHAPTER 4: DIFFERENTIATING EXPONENTIALS AND LOGARITHMS 3 Calculate the areas under the following graphs.

6931.. The method of finding the derivative for any other exponential function bx is exactly the same. The only difference is that the numerical value of the constant f'{O) is different for different values of the base b . For the general exponential function f{x) =bx, where b > 0 and b "# 1, the derived function is f'{x) =constant x bx, where the value of the constant, which depends on the base b, is equal to f'(O). 1 Show that, for any exponential function, the graph of y =bx bends upwards. 2 2 ; = (f'{0)) bx.

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Advanced Level Mathematics: Pure Mathematics 2 and 3 by Hugh Neill, Douglas Quadling, Julian Gilbey

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