Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lecture Notes 1: Review of Power, Trigonometric, Exponential, Logarithmic, Ablsolute Value and Piecewise Functions: Basics Instructor: Anatoliy Swishchuk Department of Mathematics & Statistics University of Calgary, Calgary, AB, Canada MATH 265 L01 Winter 2017 Outline of Lecture 1. A Brief History of Calculus 2. What is a Function? 3. Power, Trigonometric, Exponential, Logarithmic Functions 4. Absolute Value and Piecewise Function A Brief History of Calculus Wilhelm Leibniz (July 1, 1646-November 14, 1716) and Isaac Newton (December 25, 1642-March 20, 1726) are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton’s time, the fundamental theorem of calculus was known. A Brief History of Calculus Wilhelm Leibniz Isaac Newton What is a Function? A function y = f (x) is a rule for determining y when we are given a value of x. The number f (x) is the value of f (x) at x and is read "f of x". For example, the rule y = f (x) = 2x + 1 is a function. Any line y = mx + b is called a linear function. Functions can be defined in various ways: by an algebraic formula or several algebraic formulas, by a graph, or by an experimentally determined table of values. Given a value of x, a function must give at most one value of y. Thus, vertical lines are not functions. What is a Function? Domain, Restrictions The interval of x-values at which we are allowed to evaluate the function is called the domain of the function. The range of f is the set of all possible values of f (x) as x varies throughout the domain. √ Example: Domain for f (x) = x is [0, +∞), √ range for f (x) = x is [0, +∞). Restrictions for the Domain: 1) We cannot divide by zero, and 2) We cannot take the square root of a negative number. Review of Power Functions : Definition, Examples A function of the form y = xa, where a is a constant, is called a power function. Several cases: i) a = n, where n is a positive integer: f (x) = xn; thus, for n = 1 - linear function, for n = 2 - parabola; ii) a = 1/n, where n is a positive intege: f (x) = x1/n-root function; iii) a = −1 : f (x) = x−1 = 1/x-reciprocal function (hyperbola). Review of Trigonometric Functions Consider a right angle triangle with Hypotenuse (hyp), Opposite (opp) side, Adjacent (adj) side, and opposite to opp angle θ. Then, there are six basic trigonometric functions: opp sin (abbreviation for Sine): sin θ = hyp adj cos (abbreviation for Cosine): cos θ = hyp sin θ tan (abbreviation for Tangent): tan θ = opp = adj cos θ 1 csc (abbreviation for Cosecant): csc θ = hyp = opp sin θ 1 = sec (abbreviation for Secant): sec θ = hyp adj cos θ adj θ cot (abbreviation for Cotangent): cot θ = opp = cos sin θ Review of Trigonometric Functions (cont’d) Angles can be measured in degrees (o) or in radians (abbreviation is ’rad’). The angle given by a complete revolution contains 360o degrees, which is the same as 2π rad. Therefore, π rad = 180o. From here it follows that 180 o ) ≈ 57.3o 1 rad = ( π For example, π6 = 30o, and 2π = 120o , 3 1o = π rad ≈ 0.017rad 180 π = 90o , and so on. 2 Review of Exponential Functions An exponential function is a function of the form f (x) = ax, where a is a positive constant. There are three kinds of exponential functions depending on whether a > 1, a = 1, or 0 < a < 1. Review of Exponential Functions (cont’d) Main Properties: • only defined for positive a • always positive: ax > 0 for all x • Exponent Rules: x 1) axay = ax+y ; 2) (ax)y = axy = ayx = (ay )x; 3) aay = ax−y ; 4) axbx = (ab)x. • If a > 1, then ax → +∞ as x → +∞, and ax → 0, as x → −∞. Review of Inverse Functions We need an inverse function to define the logarithmic function as inverse to the exponential function. An inverse is a function that serves to ’undo’ another function: if f (x) produces y, then putting y into the inverse of f produces the output x. A function f (x) is called one-to-one if every element of the range corresponds to exactly one element of the domain. The Horizontal Line Test: A function is one-to-one if and only if there is no horizontal line that intersects its graph more than once. (Example: f (x) = x2 is not one-to-one function). Review of Inverse Functions (cont’d) Notation for the inverse function to f (x) : f −1(x). Explanation: if f maps x to y, then f −1(x) maps y to x. Cancellation Formulas for the Inverse Function: f −1(f (x)) = x and f (f −1(x)) = x. Guidelines for Computing Inverse Function: 1. Write down y = f (x) 2. Solve for x in terms of y 3. Switch the x0s and y 0s 4. The result is y = f −1(x) Review of Logarithmic Functions The logarithmic function with base a (denoted as loga x) is defined as an inverse function to the exponential function (we consider only the case a > 1) ax : loga x = f (x) ⇔ af (x) = x. The cancellation formulas for logarithmic function: loga(ax) = x (f or every x in R) and Properties of Logarithmic Function (x, y > 0): •loga(xy) = loga x + loga y; •loga( xy ) = loga x − loga y; •loga(xn) = n loga x. aloga x = x (x > 0) Review of Logarithmic Functions: The Natural Logarithmic Function (cont’d) The logarithmic function with the base e ≈ 2.71828 is called the natural logarithmic function and denoted as ln x = loge x. It is referred to as ’natural log.’ Change of Base Formula: ln x loga x = ln a Absolute Value The absolute value of a number x is written as |x| and represents the distance x from zero. We define it as ( |x| = x, −x, if if x≥0 x < 0. Properties of |x| : 1. |x| ≥ 0; 2. |xy| = |x||y|; 3. |1/x| = 1/|x|, when x 6= 0; 4. √ | − x| = |x|; 5. |x + y| ≤ |x| + |y|- triangle inequality; 6. x2 = |x|. Piecewise Function We call function a piecewise function if it is defined by different formulas in different parts of its domain. Examples. 1. ( f (x) = 1 − x, x2 , if if x≤1 x > 1. 2. Absolute value function |x| is another example of piecewise function. Reference 1) Calculus: Early Transcendental, 2016, An Open Text, by David Guichard: https : //lalg1.lyryx.com/textbooks/CALCU LU S 1/ucalgary/winter2016/math265/Guichard − Calculus − EarlyT rans − U of Calgary − M AT H265 − W 16.pdf 2) Optional Textbook: Essential Calculus, Early Transcendental, 2013, by J. Stewart, 2nd edition, Brooks/Cole