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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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**Sample text**

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 )).