University:
Carleton College
Course:
MATH 111 | lntroduction to Calculus
Academic year:

2024
Views:

308

Pages:

2
Author:

Alecai8y

Question 1 Write each expression without using absolute value symbols. |0| Answer Amount in whole: if x is a real number, then |x| = x when x ≥ 0 |x| = - x when x < 0 According to this definition, x is its own absolute value when it is positive or 0. If x is, however, negative, then its absolute value is -x. As a result, |x| is also non-negative. |x| ≥ 0 for all real numbers x Given: |0| We shall use the definition of absolute value in this situation. |0 | = 0 Final Verdict: Therefore, |0| = 0 Question 2 Given the piecewise function below, select all of the statements that are true. f(x) = {-x + 1, x < 0} {-2, x = 0} {x² - 1, x > 0} A. f(4) = 7 B. f(-1) = 2 C. f(1) = 0 D. f(-2) = 0 Answer Result: B. f(-1) = -(-1) + 1 = 2 C. f(1) = 1² - 1 = 0 f(4) = 4² - 1 = 15 f(-2) = -(-2) + 1 = 3 Question 3 Write a compare piecewise function and step function Answer Calculation: On interval, both piecewise and step functions are defined. However, they express intervals differently. The distinction between them is that the piecewise function's expression on intervals can be any function. When step functions are used, the interval functions are only constant functions. Thus, the step functions are a particular case of a piecewise function. Question 4 Define convex piecewise-linear functions Answer Step 1 Piecewise-linear functions are functions for which there are defined intervals over which the function is linear. ... Note that |x| = max{-x, x} this is therefore a convex piecewise linear functions Step 2 A piecewise linear function is a function composed of some number of linear segments defined over an equal number of intervals, usually of equal size. For example, consider the function over the interval. Question 5 Write a definition of Piecewise-Defined functions Answer Piecewise function is a function built from pieces of different functions over different intervals. For example, we can make a piecewise function f(x) where f(x) = -2 when 90-4 when 1