Calculus Week 1-2 PDF

Calculus week 1-2 Preliminaries and functions Sander Dommers Department of Mathematics and Computer Science Contact information dr.ir. Sander Dommers (he/him) Department of Mathematics and Computer Science MetaForu...

Calculus week 1-2 Preliminaries and functions Sander Dommers Department of Mathematics and Computer Science Contact information dr.ir. Sander Dommers (he/him) Department of Mathematics and Computer Science MetaForum 4.071a email: [email protected] If you have any organizational questions, please mail [email protected] 2 Calculus week 1-2 0.1 Functions Definition (Function) x y Let A and B be sets f A function f is a rule A that assigns to each B x ∈ A exactly one y ∈ B Notation: y = f (x) 3 Calculus week 1-2 0.1 Functions Definition (Function) x y Let A and B be sets f A function f is a rule A that assigns to each B x ∈ A exactly one y ∈ B Notation: y = f (x) The set A is called the domain of f The set {f (x) | x ∈ A} ⊆ B is called the range of f 3 Calculus week 1-2 0.1 Functions If the domain of a function is not explicitly specified, we choose the domain ‘as large as possible’ Example √ Consider the function f (x) = x − 1. What are the domain and range of f ? 4 Calculus week 1-2 0.1 Functions If the domain of a function is not explicitly specified, we choose the domain ‘as large as possible’ Example √ Consider the function f (x) = x − 1. What are the domain and range of f ? We need x − 1 ≥ 0 so the domain is [1, ∞) The possible values are all y ≥ 0, so the range is [0, ∞) 4 Calculus week 1-2 0.1 Polynomials Example Examples of polynomials f (x) = 2 g(x) = 7x + 3 h(x) = x2 − 4x t(x) = −6x9 + 2x3 − 4x + 1 5 Calculus week 1-2 0.1 Rational functions Example Examples of rational functions 1 f (x) = , for x 6= 0 x 2x + 1 g(x) = , for x2 − 4x + 3 6= 0 x2 − 4x + 3 x5 + x3 h(x) = 2 , for x2 + 5x + 7 6= 0 x + 5x + 7 6 Calculus week 1-2 0.6 Transformations of functions Definition (Add, subtract, divide, multiply functions) Given a function f with domain Df and a function g with domain Dg Then we define, for x ∈ Df ∩ Dg (f + g)(x) := f (x) + g(x) (f − g)(x) := f (x) − g(x) (f · g)(x) := f (x) · g(x)   f f (x) (x) := provided g(x) 6= 0 g g(x) 7 Calculus week 1-2 0.6 Composition of functions x g (x) Dg g 8 Calculus week 1-2 0.6 Composition of functions f x g (x) f (g (x)) Dg g 8 Calculus week 1-2 0.6 Composition of functions f x g (x) f (g (x)) Dg g f ◦g 8 Calculus week 1-2 0.6 Composition of functions f x g (x) f (g (x)) Dg g f ◦g Definition (Function composition) (f ◦ g)(x) := f (g(x)) Read the symbol ◦ as “after” 8 Calculus week 1-2 0.6 Composition of functions – Domain Domain of (f ◦ g)(x) := f (g(x)): First: x → g(x) =⇒ x should be in the domain of g Second: g(x) → f (g(x)) =⇒ g(x) should be in the domain of f 9 Calculus week 1-2 0.6 Composition of functions – Domain Domain of (f ◦ g)(x) := f (g(x)): First: x → g(x) =⇒ x should be in the domain of g Second: g(x) → f (g(x)) =⇒ g(x) should be in the domain of f Conclusion Df ◦g = {x ∈ Dg | g(x) ∈ Df } 9 Calculus week 1-2 0.6 Composition of functions – Domain Example √ Let f (x) = x2 + 1 and g(x) = x − 2. Find the compositions f ◦ g and g ◦ f and find their domains 10 Calculus week 1-2 0.6 Vertical translation For a constant c, the graph of y = f (x) + c is the same as the graph of y = f (x) shifted c units up y = f (x) + 3 y y = f (x) 6 y = f (x) − 3 4 2 −2 2 4 6 8 10 12 14 x −2 11 Calculus week 1-2 0.6 Horizontal translation For a constant c, the graph of y = f (x − c) is the same as the graph of y = f (x) shifted c units to the right Note the minus sign! y = f (x + 5) y y = f (x) y = f (x − 5) 12 10 8 6 4 2 −8 −6 −4 0 −2 −2 2 4 6 8 x 12 Calculus week 1-2 0.6 Vertical scale transformation For a constant c > 0, the graph of y = c · f (x) is the same as the graph of y = f (x) where the vertical scale is multiplied by c 13 Calculus week 1-2 0.6 Horizontal scale transformation For a constant c > 0, the graph of y = f (c · x) is the same as the graph of y = f (x) where the horizontal scale is divided by c 14 Calculus week 1-2 0.1 Distance between points Formula for the distance between points Let P = (x1 , y1 ) and Q = (x2 , y2 ). Then the distance between P and Q is q d(P, Q) = (x1 − x2 )2 + (y1 − y2 )2 15 Calculus week 1-2 0.1 Distance between points Formula for the distance between points Let P = (x1 , y1 ) and Q = (x2 , y2 ). Then the distance between P and Q is q y d(P, Q) = (x1 − x2 )2 + (y1 − y2 )2 Q y2 d Pythagoras: y1 P (x2 − x1 )2 + (y2 − y1 )2 = d2 x1 x2 x 15 Calculus week 1-2 0.1 Straight lines If you take two points, there is one straight line through these points: y ` y2 Q y1 P x1 x2 x 16 Calculus week 1-2 0.1 Straight lines If you take two points, there is one straight line through these points: y ` y2 We call Q ∆y y − y1 ∆y m := = 2 ∆x x2 − x1 y1 P the slope of the line ` x1 x2 x ∆x 16 Calculus week 1-2 0.1 Straight lines If you take two points, there is one straight line through these points: y ` We call y2 ∆y y − y1 = 2 Q m := ∆y ∆x x2 − x1 y1 the slope of the line ` P x1 x2 x Note that if x1 = x2 the line is vertical and the slope is undefined ∆x 16 Calculus week 1-2 0.1 Equations of lines General form of a line with slope m y = mx + b 17 Calculus week 1-2 0.1 Equations of lines General form of a line with slope m y = mx + b Line through P = (x0 , y0 ) with slope m y = m(x − x0 ) + y0 This is called the point-slope form of a line 17 Calculus week 1-2 0.1 Equations of lines Example Find an equation of the line passing through the points (3, 1) and (4, −2) 18 Calculus week 1-2 0.1 Parallel and perpendicular Theorem Two (nonvertical) lines are parallel if they have the same slope Further, any two vertical lines are parallel. 19 Calculus week 1-2 0.1 Parallel and perpendicular Theorem Two (nonvertical) lines are parallel if they have the same slope Further, any two vertical lines are parallel. Two (nonvertical) lines of slope m1 and m2 are perpendicular or orthogonal (Dutch: loodrecht) whenever the product of their slopes is −1, i.e. m1 · m2 = −1 Also, any vertical line and any horizontal line are perpendicular. 19 Calculus week 1-2 0.1 Equations of lines Example Find the equations of the line passing through the point (−1, 3) and parallel and perpendicular to y = 3x − 2 20 Calculus week 1-2