Let f(x, y) = x 2 – y 2 .
(i) Find the level curve of f passing through the point (5, 3).
(ii) Find the gradient of f at that point.
(iii) Make a sketch of both the level curve and the gradient vector.
(iv) Determine the equations of the normal line and the tangent line to the curve
f(x, y) = x 2 – y 2 at the point (5, 3).

If g(x, y) = x2 + y2 - 6x, find the gradient vector Deltag(l, 4) and use it to find the tangent line to the level curve g(x, y) = 11 at the point (1, 4). gradient vector tangent line equation

3. Consider the function f(x) = 2x-x + 1 . a Write down f'(x). b Show that at x = 2,f(x) =f (s). c Find the coordinates of a second point on the curve y f for which f(x) =fG). Homework #1 There are two points on the curve y = x-12x + 5 at which the gradient of the curve is zero. Find the coordinates of these two points. Find the equation of the straight line that passes through these two points. #2 PointP(1,b)lies on the curvey=x2-4x+1. a Find the value of b b Find the gradient function of the curve. c Show that at P the gradient of the curve is also equal to b. d Q (c, d) is the point on the curve at which the gradient of the curve is equal to-2. Show that d =-2. 赫3 The gradient of the tangent to the curve y = P (3, b) is 7. Find the value of k and the value of b. kx at the point

Consider the following. rx, y) = 9x2 + 4y2 c:: : 13,從1,-1) (a) Find the gradient of the function at P. 1,-1) = (b) Find a unit vector that is normal to the level curve f(x, y) = c at P. (c) Find the tangent line to the level curve rx, y) = c at P.