Please answer all 6 parts! When y is an algebraic function of x described by the

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Please answer all 6 parts!

When y is an algebraic function of x described by the equation f(x,y) = 0, we have a simpler process to find the tangent line to the curve f(x,y) = 0 at a chosen point.

Consider the curve C which is described by the equation 2x2 + y2 ? xy ? 4x + y = 0. Thus C is the set of all points that satisfy the equation 2x2+y2?xy?4x+y=0. Using implicit differentiation, calculate the equation of the tangent line to this curve at the origin (0, 0).

Let (a,b) be any point on the same curve C. Using implicit differentiation, show that the equation of the tangent line to the curve C at (a, b) is given by:

(4a ? b ? 4) (x ? a) + (2b ? a + 1) (y ? b) = 0.

Using the formula from the previous part, determine the equation of the tangent line at the point (2, 1). Verify that the point (2, 1) is on the curve and verify that you get the same equation by the usual calculation using implicit differentiation.

Suppose we consider a new curve C? which is the graph of the equation 2x2 + y2 + pxy ? 4x + y = 0, wherep is any constant. Explain why the new curve C? still has the same tangent line at (0,0). Suggestion: It is a good idea to recognize the pattern in your calculations.

Suppose that a curve C is the graph of an equation f(x,y) = 0 where f (x, y) = ax + by + pxy and p is a constant and b ?= 0.

Show that the tangent line to this curve at (0, 0) has equation ax + by = 0.

Give an example of a term beside pxy that we can add to f without changing the tangent line at (0, 0).

6. In case a ?= 0 but b = 0, explain why ax = 0 is still the equation of the tangent line. In this case dy/dx will be undefined, but you can find dx/dy. Make a sketch of a sample curve of this type to illustrate your conclusion.

Explanation / Answer

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