Thomas' Calculus: Early Transcendentals PDF

8/24/2024 Thomas’ Calculus: Early Transcendentals Fifteenth Edition Chapter 1 Functions Slide - 1  Section 1.1 Functions and Their Graphs Examples...

8/24/2024 Thomas’ Calculus: Early Transcendentals Fifteenth Edition Chapter 1 Functions Slide - 1  Section 1.1 Functions and Their Graphs Examples Exercises 2, 4, 7, 8 3,5,6(only domains), 15, 18,55 Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 2  Functions; Domain Functions; and Domain Range and Range Definition 𝑫 𝒀 A function 𝑓 from a set 𝐷 to a set 𝑌 is a 1 a rule that assigns a unique value 𝑓(𝑥) in 𝑌 2 b to each 𝑥 in 𝐷. 3 c 4 Domain is the value of 𝑥. is the set of all possible values of 𝑓 𝑥 Range as 𝑥 varies throughout the domain. Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 3 Functions; Domain and Range A diagram showing a function as a kind of machine. Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 4 Functions; Domain and Range A function from a set 𝐷 to a set 𝑌 assigns a unique element of 𝑌 to each element in 𝐷. Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 5  Graphs of Functions Example 2 Graph the function 𝑦 = 𝑥 2 over the interval [−2 , 2]. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6  The Vertical Line Test for a Function A curve in the 𝑥𝑦 plane is the graph of a function of 𝑥 if no vertical line intersects the curve more than once. Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 7 Piecewise-Defined Functions Sometimes a function is described in pieces by using different formulas on different parts of its domain. One example is the absolute value function 𝑥 𝑖𝑓 𝑥 ≥ 0 𝑓(𝑥) = 𝑥 = ቐ −𝑥 𝑖𝑓 𝑥 < 0 Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 8 Properties of Absolute Values Suppose 𝑎 > 0 is any real number.Then 1 𝑥 = −𝑥 2 𝑥 ≤ 𝑎 ⟺ −𝑎 ≤ 𝑥 ≤ 𝑎. { 𝑥 ∈ −𝑎, 𝑎 } 3 𝑥 ≥ 𝑎 ⟺ 𝑥 ≥ 𝑎 𝑜𝑟 𝑥 ≤ −𝑎. { 𝑥 ∈ −∞, −𝑎 ∪ 𝑎, ∞ } 4 𝑥 = 𝑎 ⟺ 𝑥 = ±𝑎 𝑎2 = 𝑎 if 𝑎 ≥ 0. 𝑎2 = −𝑎 if 𝑎 < 0.. 𝑎2 = 𝑎 Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9 Example 4  − x, x  0  2 Sketch the function f ( x) =  x , 0  x  1 1, x  1  Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10 Increasing and Decreasing Functions 1 A function 𝑓 is called increasing on an interval 𝐼, if 𝑓(𝑥1 ) < 𝑓(𝑥2 ) whenever 𝑥1 < 𝑥2 in 𝐼. 2 A function 𝑓 is called decreasing on an interval 𝐼, 𝑖𝑓 𝑓(𝑥1 ) > 𝑓(𝑥2 ) , 𝑥1 < 𝑥2 in 𝐼. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11 Example 7 Determine whether the following graph is increasing, decreasing or neither. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12  24 August 2024 13 Example The function whose graph is given is (a) Increasing on −∞, 0. (b) Increasing on 0, ∞. (c) decreasing on −∞, 0. (d) decreasing on ℝ. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13  Even Functions and Odd Functions: Symmetry Even Function Odd Function If 𝑓 −𝑥 = 𝑓(𝑥) ∀ 𝑥 ∈ 𝐷, then If 𝑓 −𝑥 = −𝑓 𝑥 ∀ 𝑥 ∈ 𝐷, then 𝑓 is 𝑓 is called an even function. called an odd function. The graph is symmetric with The graph is symmetric with respect respect to the _ _ _ _ _ _. to the _ _ _ _ _ _. 𝑓 𝑥 = 𝑥 2 is even 𝑓 𝑥 = 𝑥 3 is odd Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14  Special Properties Note even ± even even odd ± odd odd odd ± even neither even × even even ÷ odd × odd even ÷ odd × even odd ÷ Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15  Determine whether each of the following Example 8 functions is even, odd or neither even nor odd. (a) 𝑓 𝑥 = 𝑥 2 (b) 𝑓 𝑥 = 𝑥 2 + 1 (c) 𝑓 𝑥 = 𝑥 (d) 𝑓 𝑥 = 𝑥 + 1 Exercise 55 Note 1 ℎ 𝑡 = (a) 𝑓 𝑥 = 𝑐 𝑖𝑠 _ _ _ _ _ 𝑡−1 (b) 𝑓 𝑥 = 𝑥 is _ _ _ _ _ (c) 𝑓 𝑥 = 𝑥 𝑛 𝑖𝑠 ቊ −− − 𝑖𝑓 𝑛 is even _ _ _ _ _, 𝑖𝑓 𝑛 is odd Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16  ALGEBRAIC FUNCTIONS Any function constructed from Polynomial Functions polynomials using algebraic Rational Functions operations (such as +, −,×,÷, 𝒏 ) Radical Functions starting with polynomials. Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 17  Common Functions Linear Functions A function of the form 𝒇(𝒙) = 𝒎𝒙 + 𝒃, where 𝑚 and 𝑏 are fixed constant is called a linear function. The function 𝑓(𝑥) = 𝑥 where 𝑚 = 1 and 𝑏 = 0 is called the identity function. When 𝑚 = 0 , then 𝑓(𝑥) = 𝑏 is called the constant function. (a) Lines through the origin with (b) A constant function with slope 𝑚. slope 𝑚 = 0 Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 18  Power Function is a function of the from 𝑓 𝑥 = 𝑥 𝑎 a is a constant we have 3 cases 1 𝑎 = 𝑛 ( +𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒: 𝑥 1 , 𝑥 2 , 𝑥 5 , … ) 𝑫 = ℝ = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑹 = ℝ = (−∞, ∞) 𝑹 = [𝟎, ∞) 𝑹 = (−∞, ∞) 𝑹 = [𝟎, ∞) 𝑹 = (−∞, ∞) 𝒏 𝒊𝒔 𝒐𝒅𝒅 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏 𝒏 𝒊𝒔 𝒐𝒅𝒅 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏 𝒏 𝒊𝒔 𝒐𝒅𝒅 Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 19  1 1 2 𝑎 = −𝑛 (𝑒𝑥𝑎𝑚𝑝𝑙𝑒: , 2 , …. ) 𝑥 𝑥 reciprocal 𝑫 = (−∞, 𝟎) ∪ (𝟎, ∞) 𝑫 = (−∞, 𝟎) ∪ (𝟎, ∞) 𝑹 = (−∞, 𝟎) ∪ (𝟎, ∞) 𝑹 = (𝟎, ∞) Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20  1 1 1 3 𝑎= ( +𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒: 𝑥 , 𝑥 , … ) 2 3 𝑛 Root function 𝑫 = 𝟎, ∞ 𝑫 = ℝ = (−∞, ∞) 𝑫 = 𝟎, ∞ 𝑫 = (−∞, ∞) 𝑹 = 𝟎, ∞ 𝑹 = ℝ = (−∞, ∞) 𝑹 = 𝟎, ∞ 𝑹 = 𝟎, ∞ Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21  24 August 2024 22 Polynomial Function A function 𝑓(𝑥) is called a polynomial if constant f(x) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0 x is a variable Leading coefficient coefficients Examples 𝑛: non-negative integer 2 2 𝑎𝑛 , 𝑎𝑛−1 , … , 𝑎1 , 𝑎0 ∈ ℝ (1) 𝑓 𝑥 = 𝑥 5 + 𝑥 + 1 is a polynomial 3 Domain of f(x) = ℝ = −∞, ∞. deg.( 𝒇 𝒙 ) = Degree of the polynomial = 𝑛. (2) 𝑔 𝑥 = 𝑥 + 2𝑥 −1 + 𝑥 2∕3 is not a polynomial Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 22 1 Constant Function 3 Quadratic Function 𝑓 𝑥 =𝑐 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Deg. 𝑓(𝑥) = Deg. 𝑓(𝑥) = 𝐷𝑓 = 𝑅𝑎𝑛𝑔𝑒 = 𝑎>0 𝑎 0 , 𝑎 ≠ 1. Domain= Domain= Range= Range= 𝟏 𝒙 𝒂>𝟏, 𝒚= 𝟐𝒙 , 𝒚 = 𝒆𝒙 𝟎 < 𝒂 < 𝟏, 𝒚= 𝟐 Increasing or decreasing !! even or odd ? Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 34  ‫العدد‬ Logarithmic Functions 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 The power to which we The base 𝒂 is ‫الأساس‬ raise 𝒂 to get 𝒙 ‫القوى‬ a +ve constant ≠ 𝟏 Note 𝑦 = 𝑙𝑜𝑔𝑒 𝑥 = ln 𝑥 1 𝑙𝑜𝑔 functions are the inverse functions 2 of the exponential functions. Domain= Range= 3 𝑙𝑜𝑔𝑎 1 = ⋯ , 𝑙𝑜𝑔𝑎 𝑎 = ⋯ Increasing or decreasing !! even or odd ? Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 35  HOMEWORK 1,2, 17, 19, 49, 53 Copyright © 2024 Pearson Education, Ltd. All Rights Reserved Slide - 36