Please show all working For a vector-valued function in the plane, r : r(t) = (g

Question

Please show all working

For a vector-valued function in the plane, r : r(t) = (g(t), h(t)), write the tangent vector as a column vector, or 2 x 1-matrix. by defining the Jacobi matrix Similarly, for a function of two variables , define the Jacobi matrix Df as the 1x2 matrix of partial derivatives of z= f(x,y): For the curve r(t) in R2. and function / on R2 defined by calculate an expression for the composite function p(t) = f o r(t) as a function of 1-variable t. Calculate Df. Dr. and Hence verify directly that, as a 1 x 1 matrix, obtained by multiplication of Jacobi matrices.

Explanation / Answer

(df/dx) = 2xe^{2y} and at (2,0) it equals 4. (=A)

(df/dy) = 2(x^2)e^{2y} and at (2,0) it equals 8. (=B)

(dx,dy) = (1.9,0.2) - (2,0) = (-0.1, 0.2)

Thus, dz = (-0.1)(4) + (0.2)(8) = 1.2

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