Differential Mathematics PDF

# Differential Mathematics ## Set's * **Natural Number** N = {1,2,3,...} * **Integers Number** Z = {...-3,-2,-1,0,1,2...} * **Rational Number** Q = {a/b : a,b are integers number and b≠0} * **Irrational Number** I: such as √2 and π are numbers which are not rational. * **Real Numbers** R: The set...

# Differential Mathematics ## Set's * **Natural Number** N = {1,2,3,...} * **Integers Number** Z = {...-3,-2,-1,0,1,2...} * **Rational Number** Q = {a/b : a,b are integers number and b≠0} * **Irrational Number** I: such as √2 and π are numbers which are not rational. * **Real Numbers** R: The set of rational and irrational numbers (R = Q U I) * **Complex numbers** C = {x+iy : x,y are real numbers and i = √(-1)} **Clearly, N ⊆ Z ⊆ Q ⊆ R ⊆ C** ## Functions **Definition:** A relation f: X → Y is called a function if and only if for each element x ∈ X, there exists a unique element y ∈ Y such that y = f(x). * If y = f(x), then the set of all possible input (x-values) is called the domain of f and denoted Df or Dom(f). * And the set of outputs (y-values) that result when X varies over the domain is called the range of f and denoted by Rf or Ran(f). **Example:** Find the domain and range of the following function: 1. f(x) = 2 2. f(x) = x²-4 3. f(x) = √(x-2) 4. f(x) = |x| 5. f(x) = (x²-4)/(x+2) **Solution:** 1. f(x) = 2: * Df = R * Rf = R 2. f(x) = x²-4: * Df = R * Let y = x²-4 ⇒ x² = y + 4 * ⇒ y+4 ≥ 0 ⇒ y ≥ -4 * ∴ Rf = [-4, ∞) 3. f(x) = √(x-2): * x-2 ≥ 0 ⇒ x ≥ 2 * Df = (2, ∞) * Rf = [0, ∞) 4. Df = R and Rf = [0, ∞) 5. f(x) = (x²-4)/(x+2): * x+2 = 0 ⇒ x = -2 * Df = R \ {-2} * Since f(x) = (x²-4)/(x+2) = (x-2)(x+2)/(x+2) = x-2 for x ≠ -2 ⇒ f(x) = x-2 * ⇒ y = x-2 ⇒ y = -2-2 = -4 * ∴ Rf = R \ {-4} ## Homework: Find Domain(f): 1. f(x) = ³√(2x-6) 2. f(x) = (2x+1)/((x+1)(x-4)) 3. f(x) = (x²-x-6)/(x²) ## Graph of Function A function f establishes a set of ordered pairs (x, y) of real numbers. The plot of these pairs (x, f(x)) in coordinate system is the graph of f. **Example:** Sketch a graph of function f(x) = x² **Solution:** * Df = R | x | y | (x,y) | |----|-----|--------------------| | -4 | 16 | (-4, 16 ) | | -3 | 9 | (-3, 9 ) | | -2 | 4 | (-2, 4 ) | | -1 | 1 | (-1, 1 ) | | 0 | 0 | (0, 0) | | 1 | 1 | (1, 1) | | 2 | 4 | (2, 4) | | 3 | 9 | (3, 9) | | 4 | 16 | (4, 16) | **Example:** Sketch the function f(x) = x²+1 **Solution:** * Df = (-∞, ∞) | x | y | (x,y) | |----|-----|--------------------| | -2 | 5 | (-2, 5) | | -1 | 2 | (-1, 2) | | 0 | 1 | (0, 1) | | 1 | 2 | (1, 2) | | 2 | 5 | (2, 5) | **Example:** f(x) = √x. Sketch and find Df, Rf. **Solution:** * Df ≥ 0 * Df = [0, ∞), Rf = [0, ∞) | x | y | (x,y) | |----|-----|--------------------| | 0 | 0 | (0, 0) | | 1 | 1 | (1, 1) | | 4 | 2 | (4, 2) | | 9 | 3 | (9, 3) | ## Homework: Sketch and find Df, Rf f(x) = 3x + 1