Problem 4. (. Let C be the curve given by parametric equations x(t) = t², y(t) = 2t, z(t) = lnt, t€ (-∞0, +∞0). (c) Find an equation of the tangent line to the curve C when t = 3. (d) Find the arc length of the curve C when 1 ≤ t ≤e.

Q: Prove each of the following. (a) Given a real number a > 1 with a N, show that there is some natural…

A: a) As a is a real number, then it can be written as sum of a natural number and a fraction.  Hence,…

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A: Here ds is arc length.

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A: Explanation of the answer is as follows

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A: Since multiple questions are asked, I am answering first question

Q: дz For z = x² + xy³ ; x = uv² + w², y = u + vew, find when u = 2, v = 1, and w=0 Əw

A: z=x2xy3, x=uv2+w2, y=u+ve2at u=2, v=1 and w=0, x=2×12+0=2y=2+1e0=2

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Q: 149. Determine the real number a such that vectors a = 2i +3j and b=9i+aj are orthogonal.

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Q: [x²√√x + 17 dx

A: ∫ x2x+17dxLet u=x+17.=∫u-172udu=∫ u52-34u32+289udu=∫ u52du-∫ 34u32du+∫…

Q: e dx + (2y + xe") dy = 0

A: Given ey dx+2y+xey dy=0 Here M=ey and N=2y+xey ∂M∂y=ey  , ∂N∂x=ey Therefore,   ∂M∂y=∂N∂x is exact…

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A: Given: y=2x-123x+4

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A: Break the tanx into Sinx/Cosx

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A: We are allow to solve one question at a time. If you need rest of answer then upload separately.…

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A: Basic identities of integration are used

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Q: Evaluate the definite integral. I's 5 cos(nt/2) dt

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Q: 9. J et dx 10. 4лr² dr

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A: Use first derivative to sketch curve of function g(theta)

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A: To find out the marginal revenue at 2785 items.

Q: 3. Use a known Maclaurin series to obtain the Maclaurin series for the following. f(x) = x cos (₁)

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A: We need to find critical numbers and classify them.

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A: We know that, Newton’s law of cooling is given by, dT/dt = k(Tt – Ts) Where, Tt = temperature of…

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A: If value of y increases as x increases then function is increasing in that interval. If value of y…

Q: 20. (x-2x³+1)dx (x² - 2x+5)

A: I am attaching image so that you understand each and every step.