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Assignment on Differential Calculus

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Added on  2021-02-21

Assignment on Differential Calculus

   Added on 2021-02-21

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Differential Calculus
Assignment on Differential Calculus_1
TABLE OF CONTENTTASK 1 ...........................................................................................................................................1A. Differentiate the algebraic function .......................................................................................1B. Differentiating using quotient rule .........................................................................................1C. Differentiating logarithmic function ......................................................................................2D. Differentiating using product rule ..........................................................................................2TASK 2 ...........................................................................................................................................3B. Area enclosed by curves .........................................................................................................3TASK 3............................................................................................................................................4A. Calculating Ɵ1 .......................................................................................................................4B.) Calculating temperature of cooling object ............................................................................5TASK 4............................................................................................................................................6A Least area of lidless box ..........................................................................................................6B Height and radius of cylinder with least surface area .............................................................7
Assignment on Differential Calculus_2
TASK 1 A. Differentiate the algebraic function Equation linking distance and time is given by: x = 3t + 0.5 at² a = acceleration t = time v = velocity a = 1.6 m/s² t = 5 seconds Solution Velocity (v) = Rate of change of distance = dx/dt v = d (3t + 0.5 at² ) / dt v = 3 + 0.5a *2t(a is constant = 1.6, as given) v = 3 + 1.6t at t = 5 seconds velocity = 3 +(1.6*5) = 11 m/second B. Differentiating using quotient rule y= [e^(2t) ] / sin 5t Solution As per quotient rule: If u and v are function of t then du / dv = (du *v ) - (u*dv) / v² where du = du/dt and dv = dv/dtIn the given situation u = e^(2t)v = sin 5t du / dt = 2e^(2t) dv / dt = 5cos 5t On substituting these values we get dy = [2e^(2t) * sin 5t ] - [e^(2t) * 5cos 5t ] / (sin 5t)² dy = e^(2t) [2sin 5t - 5cos 5t] / (sin 5t)² 1
Assignment on Differential Calculus_3
C. Differentiating logarithmic function y = ln [(x-2) (x+1) / (x-1) (x+3) ]Solution Using the formula: d (ln x) / dx = 1/xUsing the property of logarithmic functions we have: ln (a/b) ln a – ln b ln (ab) = ln a * ln b Similarly for function y: ln (x-2) (x+1) / (x-1) (x+3) = ln [(x-2) (x+1) ] - ln [(x-1) (x+3)] It can be further simplified as = ln (x-2) + ln (x+1) - [ln (x-1) + ln (x+3)] y = ln (x-2) + ln (x+1) - ln (x-1) - ln (x+3) Using differentiation property we have: dy /dx = [1 / (x-2)] + [1 / (x+1)] - [1 / (x-1)] - [1 / (x+3) ] D. Differentiating using product rule v = 4t sin 2t when t = 0.5 seconds Solution According to product rule if a and b are two functions of x then d (ab) /dx = a'b + ab' where a' = da /dx and b = db /dx v = 4t sin 2t In the given function of voltage a = t and b = sin 2t On applying differentiation property of products we can write: dv / dt = 4 * [ t * d (sin 2t ) + d (t) * sin 2t ] dt dv / dt = 4 * [ t * 2 * cos 2t ] + [sin 2t ] dv / dt = 4 * [2t cos 2t + sin 2t ]dv / dt = 4 * [2t cos 2t + sin 2t ]At t = 0.5 seconds Rate of voltage change across capacitor = 4 [sin 2(0.5) + 2(0.5) cost 2(0.5)] 2
Assignment on Differential Calculus_4

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