Question

Find a vector function r(t) that represents the curve of intersection of the two surfaces below.

z = 4x² + y²
y = 4x²

OR

z = 4x² + y²
y = 4x²"

Answer 1

The vector function representing the curve of intersection is r(t) = ti + 4t²j + (4t² + 16t´)k.

Step-by-step explanation:

The student is asking for a vector function r(t) that represents the curve of intersection of the two surfaces z = 4x² + y² and y = 4x². To find r(t), we can parameterize the curve using a single variable t. Since y is given as a function of x, we can set x = t, which gives y = 4t². Consequently, z can then be expressed as z = 4t² + (4t²)² = 4t² + 16t´.

The resultant vector function is therefore:

r(t) = t i + 4t² j + (4t² + 16t´) k.