Stochastic Modeling

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By David Machin (auth.)

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3 involving higher powers of x, for example y + ... 6) n where n is taken in this context to take any positive integer value. 6 is then known as a polynomial of degree n. The terms a 0 , a 1 , ... ,an are the coefficients of the polynomial and need to be specified if the graph is to be drawn. If n = 1, 2, 3, 4, then we have respectively linear, quadratic, cubic and quartic functions of x. 1 Show that the minimum value of the function y occurs when x = 3. Now y x 2 - 6x + 9 (x - 3) 2 hence if x = 3, y = 0.

3465. 0l6); (e) log 10 (5 ). 4096) 114 , (2~)-l/ 2 using logarithms. 3; (c) (d) 2. 718. (5) Find loge2 and loge3 from tables and hence find loge6 and log (6 1 / 2 ) without further use of tables. e (6) Sketch the curve v = ex for values of x between -2 and +2 (that is -2 < x like? ~ 2). What does the curve of y (7) Sketch the curve y = = l/2eX look logex for 0 < x < 10. What can you say about the slope of any tangent to this curve? (8) A mother cell produces 2 daughter cells and the mother cell remains.

Thus (n + 1)! = n! , to obtain (n + 1)! we multiply n! by (n + 1). Alternatively we can express this equation as n! = (n + 1) ! (n + 1) so that if we define O! 1 then this is consistent, since 41 O! (0 + 1)! (0 + 1) 1! 1. 2 SERIES AND THE ~ NOTATION Suppose we now wish to add the first five positive integers together. Then T5 = 1 + 2 + 3 + 4 + 5 = 15 It is clear that the total or the sum of the series depends on the number of terms in the series and this is why we have given T the suffix 5. 2 is known as a finite series because the series terminates after n terms.

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