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Assignment 3 Due : 4pm Friday 21 September MAS182 Applied Mathematics Semester 2, 2012 Mathematics and Statistics Please show all your working. Where possible give the exact answer. 1. Find the...

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Assignment 3 Due : 4pm Friday 21 September MAS182 Applied Mathematics Semester 2, 2012 Mathematics and Statistics Please show all your working. Where possible give the exact answer. 1. Find the equation for the tangent at x = 1 to the curve given by the solutions of (1 - x)y 3 - y x 3 = 5 - 4xy. 2. A grain silo has sprung a leak and the leaking grain is forming a conical pile below the silo. The grain is leaking at a rate of 0.6m3 /hour. If the base width of the pile is always 1.5 times the height of the pile, how fast is the pile increasing in height when it is 2m high? 3. Let f(x) = 5 + (2x + 3) ln(v x + x - 1) (i) Find the derivative of f. (ii) Using the derivative, estimate f(1.1) (Hint: Linear Approximation). 4. Find the following integrals. Make sure you use calculus and show all your working. (i) ? 3x(x XXXXXXXXXX/3 dx (ii) ? x 2 - 2 x 3 - 6x + 1 dx (iii) ? 4 1 e -t/4 + t -2 dt (iv) ? 4 0 v x v x v x + 7 dx. 5. Using calculus and showing all your working find the area bounded by the curves f(x) = x 3 - x 2 + x and g(x) = 2x 2 - x. Notes: • 10% of the marks for this assignment are reserved for presentation. • There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline. 1

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Assignment 3 Due : 4pm Friday 21 September MAS182 Applied Mathematics Semester 2, 2012 Mathematics and Statistics Please show all your working. Where possible give the exact answer. 1. Find the equation for the tangent at x = 1 to the curve given by the solutions of y 3 (1x)y = 54xy: 3 x 2. A grain silo has sprung a leak and the leaking grain is forming a conical pile below 3 the silo. The grain is leaking at a rate of 0.6m /hour. If the base width of the pile is always 1.5 times the height of the pile, how fast is the pile increasing in height when it is 2m high? p 3. Let f(x) = 5+(2x+3)ln( x+x1) (i) Find the derivative of f. (ii) Using the derivative, estimate f(1:1) (Hint: Linear Approximation). 4. Find the following integrals. Make sure you use calculus and show all your working. ? 2 2=3 (i) 3x(x 4) dx ? 2 x 2 (ii) dx 3 x 6x+1 ? 4 t=4 2 (iii) e +t dt 1 ? v 4 p p (iv) x x x+7dx: 0 5. Using calculus and showing all your working ?nd the area bounded by the curves 3 2 2 f(x) = x x +x and g(x) = 2x x. Notes:  10% of the marks for this assignment are reserved for presentation.  There are penalties for late assignments. You must contact your tutor before the due date if you have di?culties making the deadline. 1

Answered Same Day Dec 29, 2021

Solution

David answered on Dec 29 2021
117 Votes
3
1. (1-x)y3 – y/x3 = 5 – 4xy
Substituting x = 1 in the above equation we get,
0 – y = 5 – 4y
y = 5/3
So the tangent passes through the point (1,5/3) on the curve.
Next we should find the slope of the tangent
Differentiating both sides w.r.t x
-y3 + 3y2dy/dx.(1-x) – ((-3/x2)y + 1/x3.(dy/dx)) = -4y -4xdy/dx
-y3 + 3y2(1-x).dy/dx +(3/x2)y -1/x3.(dy/dx)) = -4y -4xdy/dx
dy/dx = (y3-4y-3y/x2)/( 3y2(1-x) -1/x3+4x)
Substituting x= 1 and y = 5/3
dy/dx = (125/27 – 20/3 – 5)/3 = -190/81
Slope of the tangent is m = -190/81
Equation of the tangent is y = mx+c
5/3 = -190/81 + c
C = 190/81+5/3 = 325/81
So the equation of the tangent is y = (-190/81)x + 325/81
190x+ 81y – 325 = 0
2.dV/dt = 0.6 m3/h
V = 1/3 Πr2h
d = 3h/2
= 3h/4
V = 1/3Π(3h/4)2.h = 3Πh3/16
Differentiating...
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