Question

A. Find the region bounded by the curves y = (x−3)^2 and y = 12−4x. Show all of your work.

B. Find the equation of the tangent line to the curve 5x^2 −6xy + 5y^2 = 4 at the point (1,1) Show all of your work. Thanks

Answer 1

Solution:

A.

Given

Finding intersection points of two curves

Hence intersection points are (-1, 16) and (3, 0).

Thus integral must be evaluate from x=-1 to x=3, the top one to bottom curve as

B.

The tangent is a straight line so it will be of the form y=mx+c

We can get m by finding the 1st derivative dy/dx as this is the gradient of the line. We can then get c, the intercept, by using the values of x and y which are given. To find the 1st derivative we can use implicit differentiation.

Given

Differentiating with respect to x

Point at tangent line x=1 and y=1

This corresponds to the gradient m. The tangent line is of the form y = mx + c

Putting in the values (x=1, y=1, m=-1):       

So the equation of the tangent line becomes: