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1. What is the condition number of evaluation of the function

f(x) = exp(cos(x))

at the point x?

2. Suppose that f and g are continuously differentiable functions R → R. Let κf (x) denote the

condition number of evaluation of the function f at x. Find an expression for the condition number

of evaluation of the function h(x) = f(g(x)) at x in terms of κf (g(x)) and g

0

(x).

3. Suppose that f and g are continuously differentiable functions R → R. Let κf (x) denote the

condition number of evaluation of the function f at x, and let Let κg(x) denote the condition

number of evaluation of the function g at x. Find an expression for the condition number of

evaluation of the function h(x) = f(x) · g(x) at x in terms of κf (x) and κg(x).

4. Suppose that f and g are continuously differentiable functions R → R. Let κf (x) denote the

condition number of evaluation of the function f at x, and let Let κg(x) denote the condition

number of evaluation of the function g at x. Find an expression for the condition number of

evaluation of the function h(x) = f(x)/g(x) at x in terms of κf (x) and κg(x).

5. What is the Fourier series of the function f(x) = x?

6. What is the Fourier series of the function f(x) = |x|?

(Hint: you can easily find an antiderivative of x exp(inx) using integration by parts).

1

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