Calculus 1 Notes PDF

# Calculus 1 ## Mohammad Al-Smadi ### College of Arts and Sciences ### Department of Mathematics ## Chapter (0): Revision ### Sets A set is a collection of elements. #### Examples 1. The set of natural numbers: IN = {1, 2, 3, ...} 2. The set of integers: Z = {-2, -1, 0, 1, 2, 3, ...} 3. The set of...

# Calculus 1 ## Mohammad Al-Smadi ### College of Arts and Sciences ### Department of Mathematics ## Chapter (0): Revision ### Sets A set is a collection of elements. #### Examples 1. The set of natural numbers: IN = {1, 2, 3, ...} 2. The set of integers: Z = {-2, -1, 0, 1, 2, 3, ...} 3. The set of rational numbers: Q = { a / b, a, b ∈ Z, b ≠ 0} For example: 2 / 3, -3 / 2, √25 = 5, -1/2, 0, 3 √2 ∉ Q. 4. The set of irrational numbers. I, For example: √2, π, e ≈ 2.7, x = 3.14. 5. The set of real numbers: IR = IN ∪ Z ∪ Q ∪ I Note: IN ⊂ Z ⊂ Q ⊂ IR 6. The set of whole numbers: W = {0, 1, 2, ...} # Intervals * (a, b): a < x < b open * (a, b]: a < x ≤ b half-open * [a, b): a ≤ x < b half-open * [a, b]: a ≤ x ≤ b closed * (-∞, b]: x ≤ b * (-∞, b): x < b * [a, ∞): x ≥ a * (a, ∞): x > a * (-∞, ∞) = IR ### Properties of inequalities Let a, b, c, d ∈ IR * a < b ⇒ a + c < b + c * a < b ⇒ a - c < b - c * a < b ⇒ ac < bc if c > 0 (positive) * a < b ⇒ ac > bc if c < 0 (negative) * a < b ⇒ a / c < b / c if c > 0 * a < b ⇒ a / c > b / c if c < 0 * If a < b ⇒ |a| > |b|, where a & b are both positive or both negative. * If a < b and b < c then a < c. * If a < b and c <d then a + c < b + d ### Examples 1. 7 ≤ 2 - 5x ≤ 9 * Solution: 5 ≤ -5x ≤ 7 -1 ≥ x > -7/5 x ∈ (-7/5, -1] 2. x² - 3x > 10 * Solution: x² - 3x -10 > 0 (x - 5)(x + 2) > 0 (x - 5)(x + 2) = 0 Our zeros are x = 5 and x = -2. (-3 - 5)(-3 + 2) > 0, (0 - 5)(0 + 2) < 0, (6 - 5)(6 + 2) > 0 Solution is x ∈ (-∞, -2) ∪ (5, ∞) = IR - [-2, 5] 3. x² - 9 ≤ 0 * exercise 4. 3 + 7x ≤ 2x - 9 * exercise