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...