# Hard problem

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• Dec 10th 2007, 12:54 PM
key
Hard problem
1.
a) Let f(x) = 2x Find the first differences between the values of f for x= 0,1,2,3,4, and 5. Write a formula that gives the difference between f(x+1) and f(x) in terms of x.
b) Repeat part (a) but use function g(x)=3x
c) Repeat part (a) but use the function h(x) = 5x
d) Make a conjecture for a general formula that gives the difference between f(x+1) and f(x) for the function k(x) = ax , for a > 1.
2. The average growth rate of exponential function f(x) on an interval a≤x≤b is f(b) – f(a)
b-a
The instantaneous growth rate at a is the number that this fraction approaches (if any) as b → a. Let f(x)=3x, and let b=a+h. Write the instantaneous growth rate at a as the product of 3a and a function of h (that does not involve a). Use a calculator to estimate the number this second factor approaches as h → 0.

3. Let f(x) =bx . For a fixed positive integer n, let g(x)= [f(x)]n , and let h(x) = f(f(f(…(x)…)))

a) Express g(x) as f(?)
b) Express h(x) as f(?)

Everytime it says 3x or something x or a or n it supposed to be that number to the x or the a or the n