By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

This quantity includes the extended models of the lectures given by means of the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers amassed listed below are vast surveys of the present study within the mathematics of elliptic curves, and in addition include a number of new effects which can't be chanced on somewhere else within the literature. due to readability and magnificence of exposition, and to the heritage fabric explicitly integrated within the textual content or quoted within the references, the quantity is definitely fitted to examine scholars in addition to to senior mathematicians.

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Example 7 Using the function from the previous example, evaluate f(a) This means that the input value for t is some unknown quantity a. As before, we evaluate by replacing the input variable t with the input quantity, in this case a. 4 Composition of Functions 53 The same idea can then be applied to expressions more complicated than a single letter. Example 8 Using the same f(t) function from above, evaluate f ( x 2) Everywhere in the formula for f where there was a t, we would replace it with the input ( x 2) .

To do this, we will extend our idea of function evaluation. Recall that when we evaluate a function like f (t ) t 2 t , we put whatever value is inside the parentheses after the function name into the formula wherever we see the input variable. Example 6 Given f (t ) t 2 t , evaluate f (3) and f (2) f (3) 3 2 3 f (2) (2) 2 (2) We could simplify the results above if we wanted to f (3) 32 3 9 3 6 f (2) (2)2 (2) 4 2 6 We are not limited, however, to putting a numerical value as the input to the function.

64. 14. The input (years) has changed by 2. 50. 25 dollars per year 2 years Try it Now 1. Using the same cost of gas function, find the average rate of change between 2003 and 2008 Notice that in the last example the change of output was negative since the output value of the function had decreased. Correspondingly, the average rate of change is negative. Example 2 Given the function g(t) shown here, find the average rate of change on the interval [0, 3]. At t = 0, the graph shows g (0) 1 At t = 3, the graph shows g (3) 4 The output has changed by 3 while the input has changed by 3, giving an average rate of change of: 4 1 3 1 30 3 Example 3 On a road trip, after picking up your friend who lives 10 miles away, you decide to record your distance from home over time.