Question

Please explain b!

2. Let z = f(x, y) = ln(4x2 + y2) (a) Use a linear approximation of the function z = f(x,y) at (0,1) to estimate f(0.1, 1.2) (b) Find a point P(a,b,c) on the graph of z = f(x, y) such that the tangent plane to the graph of z = f(x,y) at the point P is parallel to the plane 2x + 2y – 2=3

Answer 1

passes Eqn of plane = 2x+2y = 2 = 3 Therefore the ear of plane which through the point (a, b, c) will be - 2x+2y-Z = 2 =P where a= 2a+2b-c - Since both planes are parallel Now Now, we to the function know this plane is tangent f(x,y) = ln (4x²ty") Now with we take the partial derivation of the f(x,y) respect to a keeping y constant. af (x,y) = OPI - ax ly= b ox ly=b = 2 x8x1 4x² + y² 'y=b 8a = 2 4a²+6² We do the same take keeping & constant of(x, y) dy x=a partial derivative wity = OP I dy 1xza - 2 X=a Lyl ty? 2b ua²+6²

2b = 2 fin & 8 a = 2 (ii) 4a²+6² - 4a²+6² Divide both equations : 2b 4a² tb² = 1 = 20 = 1 8a 4a²+6² =) a = b. - putting in (i) = 2b 4xb² +6² 16 1 = b + b b=4 =) a = 1

Solving both ears give a = 1 & b = 4 which gives c = ln (4x I + 4 = -0.223

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passes Eqn of plane = 2x+2y = 2 = 3 Therefore the ear of plane which through the point (a, b, c) will be - 2x+2y-Z = 2 =P where a= 2a+2b-c - Since both planes are parallel Now Now, we to the function know this plane is tangent f(x,y) = ln (4x²ty") Now with we take the partial derivation of the f(x,y) respect to a keeping y constant. af (x,y) = OPI - ax ly= b ox ly=b = 2 x8x1 4x² + y² 'y=b 8a = 2 4a²+6² We do the same take keeping & constant of(x, y) dy x=a partial derivative wity = OP I dy 1xza - 2 X=a Lyl ty? 2b ua²+6²

Solving both ears give a = 1 & b = 4 which gives c = ln (4x I + 4 = -0.223