By Hugh Neill, Douglas Quadling, Julian Gilbey
Written to check 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 intricate numbers.
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Extra info for Advanced Level Mathematics: Pure Mathematics 2 and 3
Thus, in formal function notation, exp:x Hex. CHAPTER 4: DIFFERENTIATING EXPONENTIALS AND LOGARITHMS 53 Many calculators have a special key, often labelled [ex], for finding values of this function. If you want to know the numerical value of e, you can use this key with an input of 1, so that the output is e 1 = e. 718 281828 .... 1 Find the equations of the tangents to the graph y =ex at the points (a) (0,1), (b) (l,e). (a) Since dy =ex, the gradient at (0,1) is e 0 = 1. The equation of the tangent is dx therefore y- f = l(x - 0), which you can simplify to y = x + 1.
6 Equations and inequalities You know that log 2 2 =1 and log 2 4 = 2 , but how can you find log 2 3? Suppose that log 2 3 =x. Then from the definition, 2x =3. So the problem is to solve an equation where the unknown appears in the index. The trick is to use logarithms and to write the equation as log2' """log3. This is often de<>cribed as 'taking logarithms of both sides of the equation'. You can now use the power rule to write this as x log 2 = log 3 . the [LOG] key on the calculator, this is xx 0301...
Xi- (a) Calculate the y-coordinate of the point where the graph cuts the y-axis. (b) Determine the gradient of the graph where x < -5. (OCR, adapted) 13 A graph has equation y = x +I 2x -11. Express y as a linear function of x (that is, in the form y = mx + c for constants m and c) in each of the following intervals for x . (a) x>i (b) x