Question

Problem 1: For problems 1 and 2, use f(x,y)=xy and the unit vector u=⟨u1,u2⟩. (a) Find an equation for the line r(t) (in the xy-plane) passing through the point (a,b) with direction vector u. Make sure to write r(t) as a vector-valued function. (b) Let x=x(t) and y=y(t) represent the x and y components of r(t), the line you found in part (a). Write out an explicit expression for f(x(t),y(t)). Fact 1: The function f(x(t),y(t)) gives the position (as a function of time) of someone walking on the surface f(x,y), in the direction of u, and passing through the point (a,b,f(a,b)).

Answer 1

To find the equation of a line passing through a point with a given direction vector, we use the vector form of a line equation. The explicit expression for f(x(t),y(t)) can be found by substituting x(t) and y(t) into the given function. Fact 1 explains the purpose of the function f(x(t),y(t)) in relation to the surface f(x,y).

Step-by-step explanation:

Problem 1:

(a) To find an equation for the line r(t) passing through the point (a,b) with direction vector u, we can use the equation of a line in vector form: r(t) = (a,b) + t(u1,u2)

• For the given function f(x,y) = xy, the explicit expression for f(x(t),y(t)) is f(x(t),y(t)) = x(t)y(t)

Fact 1:

The function f(x(t),y(t)) represents the position of someone walking on the surface f(x,y), in the direction of u, and passing through the point (a,b,f(a,b))

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