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

2024
Views:

158

Pages:

3
Author:

Alecai8y

Question 1 Solve absolute value inequality. -3|x+7| ≥ -27 Answer 1 Step 1 An absolute value inequality is an equation of the form | A | < B, or | A | ≥ B, where an expression A (and possibly but not usually B) depends on a variable x. Solving the inequality means finding the set of all x that satisfy the inequality. Usually this set will be an interval or the union of two intervals. Step 2 We have, -3|x+7| ≥ -27 On using the property of inequality -ax ≥ -b ⇒ x ≤ (-b)/(-a), we get the result as -3|x+7| ≥ -27 |x+7| ≤ -27/(-3) |x + 7| ≤ 9...(1) Now, we know that for an inequality |x| ≤ a -a ≤ x ≤ a Using the above concept in equation (1), we get the result as |x+7| ≤ 9 -9 ≤ x+7 ≤ 9 -9-7 ≤ x ≤ 9-7 -16 ≤ x ≤ 2 Hence, values of x lies between [-16,2]. Answer 2 Step 1 To Determine: Solve absolute value inequality. Given: we have -3|x+7| ≥ -27 Explanation: we have -3|x+7| ≥ -27 multiply both the sides with negative sign then the sign of inequality get reverse 3|x+7| ≤ 27 now divide the inequality by 3 in both sides then we have 3|x+7|/3 ≤ 27/3 |x+7| ≤ 9 Step 2 now we can write the above inequality as x + 7 ≤ 9 x≤2 x + 7 ≥ -9 x ≥ -16 so we have Answer: x ≤ 2 and x ≥ -16 Question 2 Why is 1 not considered a prime number? Or, why is the definition of prime numbers given for integers greater than 1? Answer 1 One of the whole "points" of defining primes is to be able to uniquely and finitely prime factorize every natural number. If 1 was prime, then this would be more or less impossible. Answer 2 The main point of talking about prime numbers is Euclid's theorem that every positive integer can be written uniquely as a product of primes. As Justin remarks, this would break horribly if 1 were considered prime, for example we could factor 2 as 2 x 1 x 1 x 1 x 1 x 1. Instead we say that 1 is not a prime, but it is the product of zero primes (see Why is 2⁰ = 1 except when x=0? to understand why any prime multiplied by itself 0 times is 1) so Euclid's theorem works out nicely! Question 3 Need to calculate: The value of F(10) for the function F(x) = { 2x if x < 3 , -5x if x ≥ 3} Answer Formula used: Some functions are defined by different equations for various parts for their domains. Those functions are called piecewise-defined functions. To solve the piecewise-defined function for an input, we will determine that which part of the domain it belongs to and use the appropriate formula for that part of the domain. Calculation: Consider the provided function, F(x) = {2x if x < 3 , -5x if x ≥ 3} The function provided is defined in multiple equations. Therefore, to determine F(10) identify in which equation 10 will lie. 10 lies in the interval x ≥ 3. Therefore, to determine the value of F (10), F(x) = -5x will be used. Substitute x = 10 into the equation F(x) = -5x. F(x) = -5x F(10) = -5 * (10) = -50 Hence, the value of F (10) is -50. Question 4 Define the Piecewise-Defined Functions with an example. Answer Step 1 Definition of Piecewise-Defined functions Step 2 A 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 -9