Higher Engineering Mathematics
202 GRAPHS Figure 19.30 Figure 19.31 It is noted from the latter example, that not all func- tions have an inverse. An inverse, ...
FUNCTIONS AND THEIR CURVES 203 C (c) arccos ( − √ 3 2 ) ≡cos−^1 ( − √ 3 2 ) = 150 ◦ = 5 π 6 rador2.6180 rad (d) arccosec( √ 2)=a ...
204 GRAPHS A y 5 4 3 2 1 (^01234) x D x + 2 y = x + 1 x + 2 y = x + 1 − 4 − 3 − 2 − 1 C − 2 − 3 − 4 − 5 B − 1 Figure 19.32 The c ...
FUNCTIONS AND THEIR CURVES 205 C x− 3 2 x +^1 y = x = − 12 y = 12 x− 3 2 x +^1 y = −^8 −^6 −^4 −^2 −10 1 2 468 x y 6 4 2 −^4 −^6 ...
206 GRAPHS Problem 9. Determine the asymptotes par- allel to the x- andy-axes for the function x^2 y^2 =9(x^2 +y^2 ). Asymptotes ...
FUNCTIONS AND THEIR CURVES 207 C y 6 4 2 − 6 − 4 − 2 0 2 4 6 − 2 − 4 − 6 − 8 − 10 x y^ = x− 2 y(x + 1) = (x − 3)(x + 2) y(x + 1) ...
208 GRAPHS Hence y=mx+c=± 1 x+0, i.e. y=x and y=−xare asymptotes. To determine any asymptotes parallel to the x- andy-axes for t ...
FUNCTIONS AND THEIR CURVES 209 C 4 6 y 2 − 4 − 2 2 4 x y = x x^2 + 1 y = x x^2 + 1 y = x − 4 − 6 − 2 0 Figure 19.35 19.8 Brief g ...
210 GRAPHS 19.9 Worked problems on curve sketching Problem 13. Sketch the graphs of (a)y= 2 x^2 + 12 x+ 20 (b) y=− 3 x^2 + 12 x− ...
FUNCTIONS AND THEIR CURVES 211 C Figure 19.37 Problem 15. Sketch the curves depicting the following equations: (a) 4x^2 = 36 − 9 ...
212 GRAPHS (a) Squaring the equation givesx^2 = 4 [ 1 − (y 2 ) 2 ] and transposing givesx^2 = 4 −y^2 , i.e. x^2 +y^2 = 4 .Compar ...
FUNCTIONS AND THEIR CURVES 213 C 7.x^2 y^2 = 9 [ rectangular hyperbola, lying in first and third quadrants only ] 8.x=^13 √ (36− ...
214 GRAPHS Figure 19.39 (Continued) ...
FUNCTIONS AND THEIR CURVES 215 C Graphical solutions to Problems 6 and 7, Exercise 88, page 208 x = − 1 y^ = x −^2 6 (^6) x y 4 ...
Graphs 20 Irregular areas, volumes and mean values of waveforms 20.1 Areas of irregular figures Areas of irregular plane surface ...
IRREGULAR AREAS, VOLUMES AND MEAN VALUES OF WAVEFORMS 217 C (d) Simpson’s rule To determine the areaPQRSof Fig. 20.1: (i) Divide ...
218 GRAPHS From para. (d) above, Area=^13 (3)[(0+0)+4(2. 2 + 4. 5 + 2 .4) +2(3. 3 + 4 .2)] =(1)[0+ 36. 4 +15]=51.4 m^2 Now try t ...
IRREGULAR AREAS, VOLUMES AND MEAN VALUES OF WAVEFORMS 219 C Problem 4. The areas of seven horizontal cross-sections of a water r ...
220 GRAPHS For asine wave, the mean or average value: (i) over one complete cycle is zero (see Fig. 20.7(a)), Figure 20.7 (ii) o ...
IRREGULAR AREAS, VOLUMES AND MEAN VALUES OF WAVEFORMS 221 C Problem 6. Determine the mean value of cur- rent over one complete c ...
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