. Differential and integral calculus, an introductory course for colleges and engineering schools. P X these points the tangent is _L to OX and consequently f(x) is the second figure the derivative is discontinuous at D, for atthat point the tangent springs from the position T+ to the positionT~, and f(x) springs from the value tana to the value (x) is here discontinuous like the functions in Art. 48. Pointssuch as D are termed angular points. 50. How a Function Changes Sign. When a function changessign its graph passes from one side of the x-axis to the other it is geome
. Differential and integral calculus, an introductory course for colleges and engineering schools. P X these points the tangent is _L to OX and consequently f(x) is the second figure the derivative is discontinuous at D, for atthat point the tangent springs from the position T+ to the positionT~, and f(x) springs from the value tana to the value (x) is here discontinuous like the functions in Art. 48. Pointssuch as D are termed angular points. 50. How a Function Changes Sign. When a function changessign its graph passes from one side of the x-axis to the other it is geometrically evident that a curve can pass from oneside of OX to the other side in one of the following ways only.(See figures on next page.) (a) By crossing OX as in Fig. (a). (6) By a leap or spring as in the case of the curve of example 2,Art. 48. Fig. (b). (c) By passing through infinity as in Fig. (c). This is thecase of the tangent curve y = tan x within any interval that con- 7T 7T St 3x , tarns a;=^or — «> or — > or ——> etc., etc. (d) A curve may, as in Fig. (d), lie on b
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