MEC702_Wk_4_Tutorial

# MEC702_Wk_4_Tutorial - MEC702 Wk 4 Tutorial Explicit...

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Unformatted text preview: MEC702 Wk 4 Tutorial Explicit Dierentiation 1. Dierentiate the following functions. (a)f (x) = (4x2 − 1)(7x3 + x) 3 (b)f (x) = (x + 1)(3x − 1) 2 x2 − 1 x4 + 1 3 (f )(x + 2x)37 x + 2x − 1 x+5 2 6 (e)(x + 1) (c)y = (d)f (x) = 2. Dierentiate the following logarithmic and exponential functions. (a)f (x) = ln(x2 + 1) (c)y = (ln x)3 (e)y = ln 2x2  2 (g)y = ex (i)e (b)y = ln x 3 (d)y = x3 ln x (f )y = e5x (h)y = e . 1 x (x−3x) 3. Dierentiate the following trigonometric functions. (a)y = 10 sin x (c)y = 4x cos x + 2 sin x sec x 1 + tan x (g)y = sin x csc x (e)f (x) = (b)y = 2 cos 5t 5 − cos x (d)f (x) = 5 + sin x  (f )f (x) = sin x2 (h)y = x2 cot x 1 Integration 1. Integrate the following. ˆ a.) ˆ c.) ˆ e.) ˆ x2 dx b.) 1 dx x2 d.) x−1 dx f.) ˆ ˆ xdx 4 cos xdx ˆ  x + x2 dx i.) √ ˆ ˆ g.) x3 dx h.)  3x6 − 2x2 + 7x + 1 dx t2 − 2t4 dt t4 2. Evaluate the following integrals using the indicated u-substitutions. ˆ a.) 2x(x2 + 1)23 dx; u = x2 + 1 cos3 x sin xdx; u = cos x ˆ b.) ˆ √ 1 √ sin xdx; x ˆ 3xdx d.) √ ; 4x2 + 5 ˆ e.) sec2 (4x + 1)dx; ˆ p f.) y 1 + 2y 2 dy; ˆ √ sin πθ cos πθdθ; g.) ˆ 4 h.) (2x + 7) x2 + 7x + 3 5 dx; c.) 2 u= √ x u = 4x2 + 5 u = 4x + 1 u = 1 + 2y 2 u = sin πθ u = x2 + 7x + 3 3. Evaluate the following integrals by using the method of Integration by Parts. ˆ ˆ a.) xe−2x dx b.) x sin 3xdx d.) ˆ c.) ˆ ˆ e.) xe3x dx x2 cos 2xdx ˆ √ x ln xdx x ln xdx f.) ex sin xdx h.)e3x cos 2xdx ˆ g.) 4. Evaluate the following integrals using partial fractions. ˆ ˆ dx x2 − 6x − 7 ˆ dx d.) x(x2 − 1) ˆ 2 x +1 f.) dx x−1 ˆ x2 h.) dx 2 x − 3x + 2 dx a.) x2 − 3x − 4 ˆ 2x2 − 9x − 9 c.) dx x3 − 9x ˆ 2 x −8 e.) dx x+3 ˆ 3x2 − 10 g.) dx 2 x − 4x + 4 b.) 5. Evalulate the following integrals by using trigonometric substitutions. ˆ p a.) 4 − x2 dx ˆ x2 c.) √ dx 5 + x2 √ ˆ x2 − 9 e.) dx x ˆ x2 g.) √ dx 16 − x2 ˆ √ 1 − 4xdx ˆ √ 1 + t2 dt d.) t ˆ dx √ f.) 2 x x2 − 16 ˆ dx √ h.) 2 x 9 − x2 b.) 3 ...
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• Winter '16
• Alveen Chand
• Trigonometry, Logarithm, dx, Euler's formula

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