old_final_questions

old_final_questions - The following questions appeared on...

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The following questions appeared on some prior Fnal exams. These are only questions for (optional) extra practice. Disclaimer, these problems do not necessarily constitute a complete or comprehensive list of topics for material covered in this course. 1. Let c ∈ R and deFne A = {x ∈ I | x < c}. Completely prove that sup(A) = c. 2. Suppose c > 1 is a constant and |f (x)| ≤ |x| c for all x ∈ R. Use the δ − deFnition of the derivative to completely prove that f is di±erentiable at 0. 3. Assume functions f , g : R → R are di±erentiable at a point a. Give an accurate δ − proof of the sum rule for derivatives: (f + g) (a) = f (a) + g (a). 4. Suppose that a function f only takes on integer values and lim x→∞ f (x) exists. Prove that there exists a K > 0 so that f (x) is constant on the interval (K, ∞). 5. Suppose that f is di±erentiable at a. Give a complete proof of the fact that f is continous at a. 6. Suppose the function f is deFned for all x ∈ [a, b] and that f is nonde- creasing on [a, b]. (a) Let P = {t 0 , t 1 , . . . , t n } be a partition of [a, b]. Write a formula for L(f, P ) and U (f, P ) in as simple terms as possible. (b) Suppose that Q is a partition of [a, b] into n equal subintervals, i.e.
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old_final_questions - The following questions appeared on...

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