Theorem 1 (Mean Value Theorem). Suppose ? ⊂ R is open and ? : ? →...

Theorem 1 (Mean Value Theorem). Suppose ? ⊂ R is open and ? : ? → R. If ? is differentiable on the open interval (?, ?) ∈ ? then there exists ? ∈ (?, ?) such that ?(?) − ?(?) ? − ? = ? ′ (?).

4. An alternative interpretation of the Mean Value Theorem is that if ? is differentiable on (?, ? + ℎ), then there exists ? ≤ ? ≤ ? + ℎ such that ?(? + ℎ) = ?(?) + ??(?)(ℎ) (1)

a. The Mean Value Theorem fails if ? : R^? → R^? even with ? = 1 and ? = 2. Let ? : [0, 1] → R^2 be given by ?(?) = (? − ?^2 , ? − ?^3 ). Show that Equation (1) fails for ? when ? = 0 and ℎ = 1.

b. On the other hand, let ?(?, ?, ?) = ?? + ?^2 , a = (0, 0, 0) and b = (2, 1, 2). Let ? : [0, 1] → R^3 be the parameterization of the line segment going from a to b. Find (? ∘ ?)(?) and (? ∘ ?) ′ (?).

c. Find a value ?0 ∈ (0, 1) such that ?(b) − ?(a) = (? ∘ ?) ′ (?0).

Expert Solution

Colby Messinger answered 15 hours ago

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