old_final_questions

# old_final_questions - The following questions appeared on...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

old_final_questions - The following questions appeared on...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online