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MthT 430 Chapter 10a Projects - Derivatives

In Class November 28, 2007

1.

Let F(x) be a function such that

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domain(F) = R.

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For all x,y, F(x + y) = F(x) ·F(y).

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F(0) ≠ 0.

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F is differentiable at 0 and F^′(0) = π.

Show that, for every a, F is differentiable at a and find a formula for F^′(x). Here formula is an expression in terms of F or a familiar function.

2.

Let G(x) be a function such that

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domain(G) = R⁺ ≡ {x | x > 0}.

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For all x,y > 0, G(x ·y) = G(x) + G(y).

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G(1) = 0.

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G is differentiable at 1 and G^′(1) = 1.

Show that, for every a > 0, G is differentiable at a, and find a formula for G^′(x), x > 0. Here formula is an expression in terms of G or a familiar function.

3.

Let E be a function such that

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E is differentiable for all x,

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E is an even function.

Show that

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E^′ is an odd function,

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E^′(0) = 0.

4.

S and C are functions such that

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For all x, S and C are differentiable,

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S^′ = πC (for all x, S^′(x) = πC(x)), C^′ = − πS.

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S(0) = 0, C(0) = π.

Find a formula for S⁽ⁿ⁾(0), n = 0,1,2, …

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