By E. A. Maxwell

This can be the 1st quantity of a chain of 4 volumes overlaying all phases of improvement of the Calculus, from the final yr in school to measure commonplace. The books are written for college students of technology and engineering in addition to for professional mathematicians, and are designed to bridge the distance among the works utilized in colleges and extra complex experiences, with their emphasis on rigour. This quantity is worried with the fundamental rules and purposes of differentiation and integration on the subject of algebraic and trigonometric services, yet except for logarithmic and exponential capabilities. Integration starts off at the 'Riemann imperative' foundation, and the remedy of curves combines accuracy with simplicity, with out shirking the awkward difficulties of signal. every one part has examples; on the finish of every bankruptcy there are difficulties from school-leaving and open scholarship examinations.

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**Example text**

Hence the equation of the tangent is y- 32 = 20(a-4), or 20x-y = 80-32 = 48, and the equation of the normal is or z+202/= 644. EXAMPLES VIII [Compare exx. 3, 4, 6 on p. ] 1. Find the equation of the tangent and of the normal to the curve y = 5x2 at each of the points (3,45), ( - 2,20), (0,0). 2. Find the equation of the tangent and of the normal to the curve y = x* at each of the points (2, 8), (— 1, — 1), (0,0). 3. Find the equation of the tangent and of the normal to the curve y = xz — 8 at the points where it crosses (i) the x-axis, (ii) the i/-axis.

But, by elementary trigonometry, cos x = sin {\IT + #), so that y' = sin (|TT + a;). Thus the effect of differentiating is to add \TT to the independent variable. Proceeding in this way, we have y" = sin(7r + #), 2/'" = sin (§77 +a*), = sinl- 36 EVALUATION OF DIFFERENTIAL COEFFICIENTS In the same way, if y = cos#, yW = cos I - TT + x J. CiilLL ^ d 5 *y 7 O d3 J 7 y t. Dr each of the following functions: o I < aa; aar aar 1. a:5. 2. a;3. 3. X. 4. # s i n # . 5. a: cos x. 2 6. x sin a;. 7. sin 2a;.

First of all, we 'borrow' from Pure Mathematics some theorems about limits which seem obviously true but which are not too easy to prove rigorously. (i) The limit of the sum of two functions individual limits. Thus lim {f(x) + g(x)} = limf(x) is the sum of their + lim g(x). This result can be extended t o a n y number of functions. For example, we can prove t h a t lim andA so 1 — x2 = 2, Lx r hm lim 1 — xz = 3, lx lim 1 — x* = 4; lx ( l - z ) + (l-a;) + (l-:s) — 1— ' = ft9. (ii) The limit of the product of two functions is the product of their individual limits.