# Calculus Made Easy: Being a Very-Simplest Introduction to by Silvanus P. Thompson F.R.S. (auth.)

By Silvanus P. Thompson F.R.S. (auth.)

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Additional info for Calculus Made Easy: Being a Very-Simplest Introduction to those Beautiful Methods of Reckoning which are Generally called by the Terrifying names of the Differential Calculus and the Integral Calculus

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So, therefore, this fluxional notation is less informing than the differential notation, and has in consequence largely dropped out of use. But its simplicity gives it an advantage if only we will agree to use it for those cases exclusively where time is the independent variable. In that dy . du. case y will mean dt and u. w11l mean dt , and x will mean d 2x dt 2 • 51 WHEN TIME VARIES Adopting this fluxional notation we may write the mechani· cal equations considered in the paragraphs above, as follows : X, distance velocity V=X, acceleration a=v=x, force f=mv=mx, work w=xxmx.

Our next step is to find out what effect on the process of differentiating is caused by the presence of constants, that is, of numbers which don't change when x or y changes its value . Added Constants. Let us begin with some simple case of an added constant, thus: Let y=x3 +5. Just as before, let us suppose x to grow to x+dx andy to grow to y+dy. Then: y+dy=(x+dx)3+5 =x3 + 3x2dx + 3x(dx) 2 + (dx) 3 + 5. Neglecting the small quantities of higher orders, this becomes y+dy=x3 +3x2 • dx+5. 22 WHAT TO DO WITH CONSTANTS Subtract the original y =x3 23 + 5, and we have left: dy=3x 2 dx.

Find the rate of vari- ation of the pressure with the temperature at 100° C. Since P=(40+t) 5 . dP 140 I ' dt 5(40+t)4 (140)5 ' 40 CALCULUS MADE EASY so that when t = 100, 5 dP X (I40) 4 5 I dt= (I40) 5 =14o=2s= 0 "036 · Thus, the rate of variation of the pressure is, when t = IOO, 0·036 atmosphere per degree centigrade change of temperature. EXERCISES III (See page 240 for Anawers) (I) Differentiate x2 x3 (a) u=I+x+Ix2+Ix2x3+ •.. · (b) y=ax2+bx+c. (c) y=(x+a) 3 • (d) y=(x+a) 3 • (2) If w =at- ibt2 , find dw dt.