# Practice Problems

Site: | Saylor Academy |

Course: | MA005: Calculus I |

Book: | Practice Problems |

Printed by: | Guest user |

Date: | Tuesday, July 23, 2024, 5:27 AM |

## Description

Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

## Problems

In problems 1–3, state each answer in the form "If is within ________ units of..."

1. . What values of guarantee that is (a) within 1 unit of 7?

3. What values of guarantee that is within unit of ?

5. For problems 1-3, list the slope of each function and the (as a function of ). For these linear functions , how is related to the slope?

7. You have been asked to cut three boards (exactly the same length after the cut) and place them end to end. If the combined length must be within 0.06 inches of 30 inches, then each board must be within how many inches of 10?

9. . What values of guarantee that is within unit of 8? within units?

11. . What values of guarantee that is within 1 unit of 2? Within units?

13. You have been asked to cut four pieces of wire (exactly the same length after the cut) and form them into a square.

If the area of the square must be within 0.06 inches of 25 inches, then each piece of wire must be within how many inches of 5?

In problems and the function and value for are given graphically. Find a length for that satisfies the definition of limit for the given function and value of .

In problems 19–21, use the definition to prove that the given limit does not exist.

(Find value for for which there is no that satisfies the definition).

## Answers

1. (a) If is within unit of 3.

3. (a) If is within unit of 2.

5. Problem 1: slope Problem 3: slope .

General pattern: slope for linear functions

7. Each board must be within inches of 10 inches in length.

13. Each piece of wire must be within inches of 5 inches.

15. & 17. See Figures

There is no value of that is both larger than and smaller than so the limit does not exist.

21. Take (or smaller) and suppose is within 1 of .

There is no value of that is both larger than and smaller than so the limit does not exist.

23. This proof is very similar to the proof of the second theorem on page 9.

Assume that and . Then, given any , we know and that there are deltas for and and , so that

if , then ("if is within of a, then is within of ", and

if , then ("if is within of , then is within of M").