Calculus: Introductory Theory and Applications in Physical by R. M. Johnson

By R. M. Johnson

This lucid and balanced creation for first 12 months engineers and utilized mathematicians conveys the transparent figuring out of the basics and purposes of calculus, as a prelude to learning extra complicated services. brief and primary diagnostic routines at bankruptcy ends attempt comprehension earlier than relocating to new fabric.

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Write down the value of h^o\ h ) when* = 2. 4. Determine the gradient of the curve y = 2x3 — 3x + 1 at the point (1,0). 5. The position s o f a moving particle at time t is given by s = It — St2. Show that the acceleration is constant. Limits and Differentiation 52 [Ch. 1 6. Determine the equation of the tangent to the curve y = 7 — x2 at the point (2, 3). 7. Given that f(x) = (2x2 - 3 ) 4 , calculate/'(-l). 8. Differentiate with respect to t the function /(/) = hyfT-l). 9. Write down the value of < cos (x + h) — cos x \ ; Γ' h ) when* = π/2.

Iv) a = 1. Evaluate the following. (i) / sin 20 liml — θ-ο\ Θ (ii) lim r-«-o\ tan t/ (iii) lim χ-*π/2\(π/2)— x t cos* Sketch the graph of the function y = fix) where Ί+jc, Λ:<0, fix) •2+x, x>0. Evaluate lim {/(*)} and lim {/(*)}. ) 6. The function /(*) is said to be continuous at x = a if lim {/(*)} exists and is equal to /(a). ) (i) Confirm that the function defined in problem 5 is continuous for all x except x = 0. (ii) Sketch roughly the graph of y = \/x2 and explain why it is not continuous when* = 0.

ST s/Q Ί — —j ^ -"fp 1 1 f Ί 2 X Fig. 3. A distance versus time curve. /(Ί+Λ)-/(ίι) where h = t2 ~t\. The instantaneous velocity at the instant r = ft can be obtained approximately by calculating the average velocity over a small interval starting at r = r ( . e. 2) shows that the instantaneous velocity at time :, is/"Cri ) · t n e value of f'(t) when t — t^. e. 3) It should be noted that the acceleration a, of the vehicle is defined as the rate of change in velocity vi/ith respect to time (typical units are metres per second/second (m/s 2 )).

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