Essential Mathematical Skills and Techniques -Problem Classes 1, Study Guides, Projects, Research of Mathematics

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3/14/25, 9:03 PM MAS182 - Problem Classes 1 CURVE SKETCHING Welcome. For all except MASI52, this the first class for this module, so welcome the students, introduce yourself and remind them to attend the Weck I lecture to Team about how the course runs (MASI40 WI DIA LTI, MASIS1 WI DIA LT6, MASI56 Th 5 SG Church). Feel free to tell them the rough format of the course if comfortable, but don’t get drawn into questions about particulars, as they will get these in the lecture (nb extra information provided separately for MAS156). Sminutereview. Using the graph of y= ' —-(or any other suitably easy curve) as an example, remind students briefly + what a graph. is; + the difference between plotting (by calculating) and sketching (by reasoning); + the importance of labelling axes, crossing points and the curve itself; + how to spot asymptotes, and how to reason about what happens near them (including at infinity). Class warm-up. Asking for input at all suitable places, work through sketching the graph of y= xcos x on the board. Start by sketching the envelope y= 2x, before discussing how the graph fluctuates within that envelope. Problems. Choose from the below. 1. General sketching. Sketch @) y=" x? @) y= 2 has © ya est @ y=s0e x= Sb; ©) y=cot x ,,'). 2. Envelopes. Sketch functions of the form y= f(x) cos(x) where @ fo) = kh ) f(x) 5 (© f(x) = cos@2x). Sketchingaquotient, Sketch the function y=“ — (Hints: it may help to start with y=-+'= on the same axes; also, if you find the behaviour at confusing, it may help to inspect values of x very close to zero). Linesand oops. i) (@) Sketch y= mxte, where (@) m>0 and c>0; (b) m>0 and c<0; (©) m<0 and ¢<0; (4) m= 0 and ©>0, () Sketch x? +y?= 1. (Hint: think Pythagoras.) (© Sketch x+y! = 1. (d) What happens to x7"+y?"= 1 as n gets larger? about:blank 1/186