Partial derivatives (a) For the functions below, evaluate all first and second derivatives, and simplify as...

 

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Partial derivatives (a) For the functions below, evaluate all first and second derivatives, and simplify as much as possible. m (²7/2+1), ƒ(x, y, 2) = €²V²/2 ii. f(z,g)=sinh(ry), f(2,3)=tan'(ry), f(r,y)=sin (ry) i. f(x, y) = In iii. f(x, y, z): 4/9 (z+ y sin r) (x² + y² +24)¹/3 1 af 8ºf ƏrƏ0* Ər (b) Let f(r, y) =r²y. Use the chain rule to find the polar derivatives and -Da't sin(Tr) satisfies the diffusion equation u₂ = Dur, where D (e) Show that u(x, t) = e is a constant. (d) Find the tangent plane to f(x, y) = ln(r2/y) at the point (3, 4). 2.024 (e) Use a linear approximation to estimate Q= 1.973 +4 (f) Find the rate of change of the function f(x, y, 2) = r²y +2³ at the point (1,2,3) in the direction given by v = (-1,4,5). (g) Find the upward pointing normal vector to the curve y = e in R². (h) Find the downward pointing normal vector to the surface z = 2² + y². (i) Find and classify the critical points of f(r, y)=r³-3r²y + y². (j) Find the extreme values of f(r, y) =r²+2ry² on the domain r² + y² ≤ 2. (k) Find the maximum and minimum values of f(x, y) = yer along the constraint curve 2² + 2y² = 4. (1) Find the maximum value of f(x, y) = 2²-6y²+2 along the constraint curve 2r+3y = 5. Partial derivatives (a) For the functions below, evaluate all first and second derivatives, and simplify as much as possible. m (²7/2+1), ƒ(x, y, 2) = €²V²/2 ii. f(z,g)=sinh(ry), f(2,3)=tan'(ry), f(r,y)=sin (ry) i. f(x, y) = In iii. f(x, y, z): 4/9 (z+ y sin r) (x² + y² +24)¹/3 1 af 8ºf ƏrƏ0* Ər (b) Let f(r, y) =r²y. Use the chain rule to find the polar derivatives and -Da't sin(Tr) satisfies the diffusion equation u₂ = Dur, where D (e) Show that u(x, t) = e is a constant. (d) Find the tangent plane to f(x, y) = ln(r2/y) at the point (3, 4). 2.024 (e) Use a linear approximation to estimate Q= 1.973 +4 (f) Find the rate of change of the function f(x, y, 2) = r²y +2³ at the point (1,2,3) in the direction given by v = (-1,4,5). (g) Find the upward pointing normal vector to the curve y = e in R². (h) Find the downward pointing normal vector to the surface z = 2² + y². (i) Find and classify the critical points of f(r, y)=r³-3r²y + y². (j) Find the extreme values of f(r, y) =r²+2ry² on the domain r² + y² ≤ 2. (k) Find the maximum and minimum values of f(x, y) = yer along the constraint curve 2² + 2y² = 4. (1) Find the maximum value of f(x, y) = 2²-6y²+2 along the constraint curve 2r+3y = 5.

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a Lets evaluate the derivatives for each given function i fx y lnx 1 First derivatives fx 1 x 1 fy 0 ... View the full answer