 By R. M. Johnson

ISBN-10: 1898563063

ISBN-13: 9781898563068

This lucid and balanced creation for first 12 months engineers and utilized mathematicians conveys the transparent realizing of the basics and functions of calculus, as a prelude to learning extra complex services. brief and primary diagnostic routines on the finish of every bankruptcy attempt comprehension earlier than relocating to new material.

• Provides a transparent realizing of the basics and purposes of calculus, as a prelude to learning extra complex functions
• Includes brief, helpful diagnostic workouts on the finish of every chapter

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Those notes shape the contents of a Nachdiplomvorlesung given on the Forschungs institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. ok. Chandrasekharan and Prof. Jurgen Moser have inspired me to put in writing them up for inclusion within the sequence, released via Birkhiiuser, of notes of those classes on the ETH.

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Extra info for Calculus. Introductory Theory and Applications in Physical and Life Science

Sample text

Sec. 8 Differentiate with respect to x the function f(x) = (x2 + l) 3 by the following methods. (i) Expanding the binomial. (ii) Using the function of a function rule. (i) Let y = (x2 + l) 3 . Expanding, we obtain y = x6 + 3xA + 3x2 + 1. dv — = 6 x 5 + 1 2 x 3 + 6x àx = 6x(x2 + I)2. y = u 3 , where u =x2 + 1. ay àx ày du au àx = (3« 2 )(2x) = 3(x2+ l) 2 (2x) = 6x(x2+ l) 2 . 9 Find/'(x) for the following functions. (i) /(x) = ( x 2 + l ) 1 0 . (ii) /(x) = ( * 3 - 3 x ) 4 / Γ (iii) /(*) = J x + - .

1. Sec. 1 d dv du — (uv) = u — + V-. 1a) v(duldx) — u(dv/dx) 3 dx\v/ v · ( 2 - 2a > Proof Let Ax be a small change in x resulting in changes Au in u, Av in v and Ay in y where y = uv. Therefore, y + Ay = (M + ΔΜ) (V + Av) = uv + u Av + v Au + Au Av. Thus, Ay = u Av + v Au + Au Av and Ay Av — = u l· Δχ Δχ v Au Δχ 1- ΔΜ Av — . Δχ Now let Δχ -»■ 0 (also Δ« -* 0, since u is differentiable), and by the definition of a derivative we obtain dy dv du — =u \- v— . 1a) to the product u =yv to give du dv dx dx dy 1 I du u dv dx v \dx v dx — =y l· dy v— .

A body is thrown vertically upwards with a velocity of 20 m/s. If the distance s m above the ground after r s is given by s = 20t — 5t2, find the following. (i) The velocity after t s. (ii) The highest point to which it will rise. (iii) When it will strike the ground. The profit iP made by a factory when it produces an x kg batch of a certain commodity is given by P=l5x2-60 x3 3 . Find the rate at which the profit changes with respect to the number of kilograms produced for the following x values.