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4 Pages

### wk10calcsoln

Course: MAT 1102, Fall 2009
School: Allan Hancock College
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Word Count: 445

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S1 MAT1102 2007 Calculus Tutorial &amp; Computer Lab Week 10 Solutions This week we wrap up optimisation from Study Book Section 4.6 and then go on to Study Book Section 4.9 on antiderivatives. First Hour - The Tutorial: 1. Use the second derivative test to find the coordinates of the extrema of y = f (x) = x3 - 12x + 1. Use second derivative test. y = 3x2 - 12 = 0 = 0 when x = 2. y = 6x and so y &gt; 0...

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S1 MAT1102 2007 Calculus Tutorial & Computer Lab Week 10 Solutions This week we wrap up optimisation from Study Book Section 4.6 and then go on to Study Book Section 4.9 on antiderivatives. First Hour - The Tutorial: 1. Use the second derivative test to find the coordinates of the extrema of y = f (x) = x3 - 12x + 1. Use second derivative test. y = 3x2 - 12 = 0 = 0 when x = 2. y = 6x and so y > 0 when x = 2 i.e. (2, -15) is a local minimum. y < 0 when x = -2 i.e. (-2, 17) is a local maximum. 2. The bending moment M of a beam, supported at one end, at a distance x from the support is given by: 1 1 M = wLx - wx2 2 2 where L is the length of the beam, and w is the uniform load per unit length. If L = 5 metres and w = 40 kg/m, Find the point on the beam where the moment is greatest. 1 2 Algebra & Calculus I Semester 1, 2007 M = 100x - 20x2 . Use the second derivative test. M = 100 - 40x M = -40 < 0 local maximum M = 0 gives x = 100/40 = 5/2 = 2.5. 3. A box with a square base and open top is to be made with a volume of 32000 cm3 . Find the dimensions of the box which minimise the amount of material to be used. Let the dimensions be x, x, y, hence x2 y = 32000 and so y = 32000/x2 . Minimise surface area S. S = x2 + 4xy = x2 + 4x 32000/x2 x2 = + 128000/x. Differentiate w.r.t. x S = 2x - 128000/x2 . S = 0 when 2x = 128000/x2 , x3 = 64000, x = 40. Second derivative test: S = 2 + 256000/x3 > 0, hence a minimum of S at x = 40. Hence the dimensions of the box are x = 40, and y = 32000/x2 = 32000/1600 = 20. Calculus Tutorial Week 10 3 4. List the antiderivatives of each of the standard functions: Function xn , n = -1 1/x ex cos x sin x sec2 x sec x tan x 1 1-x2 1 1+x2 Antiderivative xn+1 +C n+1 ln |x| + C ex + C sin x + C - cos x + C tan x + C sec x + C arcsin x + C arctan x + C 5. Find the antiderivatives of each of the following functions: (a) t2 + t t3 3 + t2 2 +C (b) t2 +1 t Hint: simplify first. = t + t-1 2 Antideriv = t2 + ln |t| + C (c) 2 cos(2x) sin(2x) + C t2 +1...

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