MTH405_Wk_8_Tutorial - MTH405 Wk 8 Tutorial Cross Products...

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Unformatted text preview: MTH405 Wk 8 Tutorial Cross Products 1. Find the determinant of the following matrices. 4 −2 2 0 4 3 1 a.) 3 −1 2 c.) 0 0 4 3 0 0.1 e.) −0.3 0.5 2 b.) 1 1 −1 4 0 6 1 −5 −3 d.) 7 1 0.2 0.3 0.2 0.2 0.4 0.4 −0.4 f.) 0.2 0.3 3 4 2 0 0 11 0 2 2 0.4 0.2 0.2 0.3 0.2 0.2 2. Find the cross product ~a ×~b and verify that it is orthogonal to both ~a and ~b. ~b = h0, 8, 0i ~b = h2, 4, 6i a.)~a = h6, 0, −2i b.)~a = h1, 1, −1i c.)~a = ˆi + 3ˆj − 2kˆ d.)~a = ˆj + 7kˆ e.)~a = ˆi − ˆj − kˆ ~b = −ˆi + 5kˆ ~b = 2ˆi − ˆj + 4kˆ ~b = 1 ˆi + ˆj + 1 kˆ 2 2 ~b = h1, 2t, 3t2 i f.)~a = ht, t2 , t3 i   3. Show that ~a × ~b · ~b = 0 for all vectors ~a and ~b in R3 .   4. Show that ~a × ~b = − ~b × ~a for the vectors ~a and ~b given in 2(a) above. 5. Find the scalar triple product ~u · (~v × w) ~ for the vectors 3ˆi − 2ˆj − 5kˆ = ˆi + 4ˆj − 4kˆ ˆ = ˆi + 3ˆj + 2k. ~u = ~v w ~ 1 Limits of Functions 1. Use the graphical method to nd the following limits. (a) (b) (c) (d) (e) limx→0 sin x. limx→π cos x. limx→0 ex . limx→1 ln x. x+1 limx→2 x−2 2. Use the tabular method to nd the following limits. x−1 x3 −1 . √ limx→0 x+1−1 . x sin 3x limx→0 x . limx→0− |x| x . tan(x+1) limx→−1 x+1 . (a) limx→1 (b) (c) (d) (e) 3. Find the limits of the following functions. (a) limx→5 (x2 − 4x + 3). (b) limx→−25 3. x (c) limx→3 x−3 (d) limx→2 (e) (f) 5x3 +4 x−3 . 2 limx→3 x −6x+9 x−3 . limx→−4 x22x+8 +x−12 . x2 −3x−10 limx→5 x2 −10x+25 . (g) (h) limx→2 x(x − 1)(x + 1) 2 −2x (i) limx→3 xx+1 . (j) limx→1 (k) x4 −1 x−1 . 2 +x−1 limx→−1 2x x+1 4. Use the Denition of the Derviatives to nd the derivative of the following functions. (a) (b) (c) (d) (e) (f) f (x) = 4x − 3. f (x) = −2x + 3. f (x) = x2 + 1. f (x) = 3x2 . f (x) = 2x2 + 4x. f (x) = x2 + 3x + 5. 2...
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