Solved Practice Problems - Partial Derivatives - Calculus IV | MTH 254, Study notes of Advanced Calculus

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Math 254 - Practice Problems Fri, Oct 14 Justify your reasoning. Partial Derivatives & Tangent Planes 1. Without appealing to Clairaut's Thm, find all second partials for f(x,y) = 2?y> — 2Qry> + 4y. 2. The temperature at any point (x,y) on a steel plate is given by T(z, y) = 500 — 2?y — 4y’ax kelvins (x,y in feet). What is the rate of change of temperature with distance at the point (2,3) in the direction bfa) the g-axis, and b) the y-axis? “Olle 3. Find the equation of the tangent plane at (—1,1,—2). Hay) 2 fea? gS V) Febery) = BxyF Oy? 6 Gea) = lO yt ¥y Cay) = 5x74" + layed fy (4) sa 20x"4? + laxy 2) Ti. Coy) Ab O44 Oxy —4y? TlI,= -202-9-41 T, (kau) = ~x2- Byx = ol tilipe gega -* “89 Ty (2,3) =- (af - 6 (23) ~ 52 Mt parallel be Kan 2-S2K/¢4 Cniding x cons teri) _ UBM parallel y-axis : ilding 4 cad) 3) fxg) =43—- Gxy® Lad=-F =a fy (xy) = -4.ey* fGuld= -4 note: O= Axi) Hg) - ED 20,4, 17 Gel 2