Mathematics 242 – Principles of Analysis Supplementary Problems – Exam 3 December 1, 2004

I. A) Give the formal ϵ-δ definition of the statement lim_{x→c} f(x) = L.

B) Use the definition to show lim_{x→2} x^{2} − 5x = −6.

II. State and prove the Mean Value Theorem.

III. Let

f(x) = \{ \begin{matrix}1 & if x < 0 \\ 1 + x + x^{2} & if x ≥ 0 \end{matrix}

A) Using the definition of f′(0), determine whether f(x) is differentiable at x = 0.

B) Show using the definition that f(x) is integrable on [−1, 1].

C) Without using the Fundamental Theorem of Calculus, find the value of ∫_{−1}^{1} f(x) dx. Potentially useful information: ∑_{i=1}^{n} i = \frac{n(n+1)}{2}, and ∑_{i=1}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}.

IV. True-False. For any two of the following, determine whether the statement is true or false. If it is true, give a short proof. If it is false give a reason or a counterexample.

A) The function

f(x) = \{ \begin{matrix}\frac{sin(x)}{x} & if x ≠ 0 \\ 0 & if x = 0 \end{matrix}

is continuous at x = 0.

B) If f(x) is a bounded function on [a, b] with the intermediate value property ^{1}, then f(x) is continuous on [a, b].

C) If f(x) is a function whose second derivative f′′(x) exists for all x ∈ R and there are three distinct points x_{1}, x_{2}, x_{3} such that f(x_{1}) = f(x_{2}) = f(x_{3}), then there is some point c where f′′(c) = 0.

1 The “intermediate value property” is the statement that if m = inf{f(x) : x ∈ [a, b]}, M = sup{f(x) : x ∈ [a, b]}, then for all y with m ≤ y ≤ M, there exists x ∈ [a, b] such that f(x) = y.