Question

full workings required

Let f: R^2 → be a differentiable function and let CCR be a curve in R^2 described by the cartesian equation f(x,y) = Letla.b) R be a point that lies on the curve Cck and assume that the partial derivatives off evaluated at (a,b) satisfy: fr(a,b) 0 and fy(a,b) +0. Also assume that there exists an expression y-g(x) that solves the equation f(xxx)=0 fory in terms of x in a neighbourhood of the point (8.b). This means that; la) = b And flx, 8(x)) =0 is an identity in x in a neighbourhood of a a) Show that: and show each step of your derivation a)!(0,6) f(a,b) b) Letic R be the line tangent to the curve CCR at the point (a,b) Write down in terms of '(a) a vector to the CCR at the point (a,b) c) How is vector d € R? related to the gradient vector/(a,b) € R explain your answed d) Find the cartesian equation for the line R e) Find a cartesian equation that describes the tangent plane IL, CR to the graph of /:RRat the point (1.4.-) = (2,6,0). f) Explain why 11, is not a horizontal plane

Answer 1

As per HOMEWORKLIB POLICY we have solve only four subparts in one question. So,

Please comment if you need any clarification

If you find my answer helpful please put thumbs up thank you

Given fog(x)= 0 By chain rule decay) af 2x (4,4)+ Of .gy dy an dx of 8x (1.4) 2 let hal= (1,907) hilR lR² f(2)= (poby?) - 14,9(2)) = 0 11 F: RR so df da - 0 dla (fon)(x) = 0 By chain rule d ay (fa6 )) - (D+) (+ ()), Đà dy Here Df = of gf & Dh dy da of 90. ЭР oy 2x dy (roh.cz) 1 600 Of 69,4gen) ay (4,4) df 20 dy at 10 4. 9(a) so of (a,b) + glea) ay (a,b) =0 = (a) = fula,b) fy(0, b) i The slope of the tangent line is dy laib) which calculated by differentiating de f(x,y)=0 implicitety. Since y= 9(2) then dy (a,hl= gla). Thus the vector der? . dy

can be obtained from slope as follows i. with Slope me intercept c is written as A line 2:1 yo mt+C i,e <ti mi+c)=<i,m) + 0.01 so d: (1.m) Thew in our case d= (1.g'ea) © of (a,b) - df 10,6), Co. (4.b) ] D by observe that tf(0,5).d = 0 (0,6) - ca,b) per don) at the SO Tf (a,b) is nolunal to d. D Equation of line is 4-6) - g'c0) (2-a ) gicas 7-y-a-b.