MATH1013 Tutorial 1 General Information Teaching Assistant Leung Ho Ming Office: Rm 3489 Tel: 3469 2017 Email: [email protected] Tutorial Homepage All materials used in the tutorials will be found at http://ihome.ust.hk/~malhm Consultation Hours You will find me outside class in math support center or by appointment (preferably through email). Math support center hours: http://www.math.ust.hk/~support/mschours_fall_2014.html Time and Venue T10B Mo 05:00PM - 05:50PM Rm 3006 Lift 3-4 T06C Tu 03:00PM - 03:50PM Rm 3006 Lift 3-4 T10C Tu 05:00PM - 05:50PM Rm 2503, Lift 25-26 T10A Tu 06:00PM - 06:50PM Rm 2406, Lift 17-18 Useful Resources WolframAlpha http://www.wolframalpha.com/ Capable of doing different kinds of calculations, examples can be found at http://www.wolframalpha.com/examples/ Desmos https://www.desmos.com/ Free graphing website. Winplot http://math.exeter.edu/rparris/winplot.html Freeware that works in Windows desktop environment, with more options and ability to plot 3-d graphs. Prepared by Leung Ho Ming Homepage: http://ihome.ust.hk/~malhm 1 Sets and Intervals We give a brief introduction to some of the preliminaries. A set is a collection of objects, usually defined by listing its elements explicitly or by specifying a property. For example, A = {1, 3, 4} and B = {x : x is an even number} are sets. We write a ∈ S if a belongs to the set S. If a is not in S, we write a ∈ / S. For instance, 2 ∈ / A but 2 ∈ B. The union and intersection of two sets are respectively A ∪ B = {x : x ∈ A or x ∈ B} and A ∩ B = {x : x ∈ A and x ∈ B} . Following the preceding example, we have A ∪ B = {1, 3, 0, 2, −2, 4, −4, 6, −6, . . . } and A ∩ B = {4}. Given two numbers a < b, we define the open interval (a, b) and closed interval [a, b] respectively as (a, b) = {x : a < x < b} and [a, b] = {x : a ≤ x ≤ b} . The half-open intervals (a, b] or [a, b) are defined in a similar way. The notation (a, b) may sometimes represent an ordered pair in the plane, but the meaning can be understood by reading the contexts in most cases. We also have (−∞, a) = {x : x < a} and (a, ∞) = {x : x > a} , where (−∞, a] and [a, ∞) are defined similarly. R = (−∞, ∞) = {x : −∞ < x < ∞} is the set of all real numbers. As an example of set operations on intervals, we have [0, 3) ∪ (2, 4) = [0, 4) and [0, 3) ∩ (2, 4) = (2, 3) . 1. Express the solutions of the inequalities using interval notation. (a) 1 < 3x + 4 ≤ 16 (b) (x + 1)(x − 2)(x + 3) ≥ 0 2 Absolute Value The absolute value of a number x is defined as ( x, |x| = −x, x≥0 . x<0 √ √ The function returns the magnitude of a number, which is always non-negative. For example 2 = 2 and |−7| = −(−7) = 7. Some commonly used properties of |x| includes (but not limited to): |xy| = |x| |y| if c > 0 : n |xn | = |x| √ x2 = |x| |x| < c if and only if − c < x < c − |x| ≤ x ≤ |x| for all x ∈ R |x| > c if and only if x < −c or x > c |x| = c if and only if x = ±c |x + y| ≤ |x| + |y| 2. Solve the equation |x + 3| = |2x + 1|. 3. Sketch the graphs of the given equations. (a) |x + y| = 1 (b) |x + y| + |x − y| = 2 3 Radian Measure In Calculus (or almost everywhere unless underwise specified), we will measure angles in radians(rad) instead of degrees(◦ ). The size of an angle θ in radian is measured as follows: Given a circle of radius r, a sector of the circle whose interior angle is θ, the size of θ is the ratio of the length of the arc to that of the radius. For instance, a πr π 90◦ = right-angle is since the arclength of a quarter circle is 2πr · . More generally, the two units are related ◦ 2 360 2 by π rad = 180◦ . ◦ π 180 ≈ 57.3◦ and 1◦ = ≈ 0.0175 rad. We will usually omit rad in writing. The following is For example, 1 rad = π 180 a table of correspondence of some common angles: Degrees 0◦ 30◦ 45◦ 60◦ 90◦ 120◦ 135◦ 150◦ 180◦ 270◦ 360◦ Radians 0 π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6 π 3π 2 2π The sine, cosine and tangent values of common angles are summarized as follows: sin 0 π 6 0 1 2 √ cos 1 3 2 tan 0 1 √ 3 π 4 π 3 √ √ 2 2 √ 2 2 3 2 1 1 2 0 √ 1 π 2 3 undefined It is assumed in most computer programs that angles input are measured in radians. When you use your pocket calculators, make sure you have switched to radian mode before evaluating any trigonometric functions. 4