old_final_questions

# old_final_questions - The following questions appeared on...

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

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.

Unformatted text preview: The following questions appeared on some prior final 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 define 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- definition of the derivative to completely prove that f is differentiable at 0. 3. Assume functions f , g : R → R are differentiable 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 differentiable at a . Give a complete proof of the fact that f is continous at a . 6. Suppose the function f is defined for all x ∈ [ a,b ] and that f is nonde- creasing on [ a,b ]. (a) Let P = { t ,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.) in as simple terms as possible....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

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