Solutions to Problem Set 10
I. Problems to be graded on completion. 1. Evaluate the following indefinite integrals: a. Let u = x3 + 1, so du = 3x2 dx, so x2 dx = x2 x3 + 1 dx =
1 3
du. 1 u3/2 2 + C = 9 (x3 + 1)3/2 + C 3 2/3
u1/2 1 du = 3
b. Let u = ex , s
Solutions to Problem Set 6
I. Problems to be graded on completion. 25. We have 180 feet of fence, so 2x + y + (y - 100) = 180, so y = 140 - x. The area of the pen is A = xy = 140x - x2 , which is maximized when 0 = dA = 140 - 2x, so x = 70, so y = 70. But
Solutions to Problem Set 5
I. Problems to be graded on completion. 14. 22 12 + 4(2)(1) = 12(1), so (2, 1) lies on the curve. Now 2xy 2 + 2x2 yy + 4y + 4xy = 12y 4 + 8y + 4 + 8y = 12y y = -2 y-1 = -2 or y = -2x + 5. x-2 16. 0 + cos(1 0) + 3 12 = 4, so (1,
Solutions to Problem Set 2
I. Problems to be graded on completion. 1. Substitute u = 4x and v = 2x. As x 0, u 0 and v 0. sin 4x 4x sin 4x 4x = = lim lim sin 2x x0 x0 sin 2x 2x 2x 4x lim x0 2x sin 4x 4x sin 2x lim x0 2x
x0
lim
=
4 lim x0 2
u0
sin u u sin v
Solutions to Problem Set 3
I. Problems to be graded on completion. 1. b = x. c = v. e = u. g = t. h = a. j = 0. l = q. p = h. r = n. s = f . t = d. u = m. x = k. 2. u1/3 - x1/3 (u1/3 - x1/3 )(u2/3 + u1/3 x1/3 + x2/3 ) = lim ux ux u-x (u - x)(u2/3 + u1/3 x
Solutions to Problem Set 7
I. Problems to be graded on completion. 1. y = x4 - 4x3 + 1, so y = 4x3 - 12x2 = 4x2 (x - 3), so y = 12x2 - 24x = 12x(x - 2). We cannot solve the equation y = 0. When y = 0, x = 0 or x = 3. When y = 0, x = 0 or x = 2. The signs
Solutions to Problem Set 8
1. (a) To go from the second to the third lines, observe that for any numbers m and n, the inequality |m| n is equivalent to the inequality -n m n. If -n m n, then -n m, so n -m, so -m n, and also m n, so |m| n.
b b
(b) If f (x)
Solutions to Problem Set 13
I. Problems to be graded on completion. 1. a. dy = y sin(2x + 3) dx dy = sin(2x + 3) dx y dy = sin(2x + 3) dx y 1 log y = - cos(2x + 3) + C 2 1 1 - 2 cos(2x+3)+C = eC e- 2 cos(2x+3) . y=e Using the initial condition, 5 = eC e-
Solutions to Problem Set 12
I. Problems to be graded on completion. 1. a. The average value is 1 1 - (-1)
1
x2 dx =
-1
1 x3 2 3
1
=
-1
1 . 3
1 The function takes this value when x = .577. 3 b. The average value is 1 1 x4 1 x3 dx = 1 - (-1) -1 2 4 The func
Solutions to Problem Set 11
I. Problems to be graded on completion. 1. a. Consider the line 2x + y = 4. If x = 0 then y = 4, so the y-intercept is 4. If y = 0 then 2x = 4, so x = 2, so the x-intercept is 2.
Around the x-axis, the radius of a slice is 4 -
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LINEAR ALGEBRA
A Geometric Approach
second edition
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LINEAR ALGEBRA
A Geometric Approach
second edition
Theodore Shifrin
Malcolm R. Adams
University of Georgia
W. H. Freeman and Company
N