Exercise 11 11 Consider The Circles C X Y 1 C X 1 Y 1 With Radius 1 And Respective Centers 0 0 1 (70.47 KiB) Viewed 25 times
Exercise 11 11 Consider The Circles C X Y 1 C X 1 Y 1 With Radius 1 And Respective Centers 0 0 2 (66.73 KiB) Viewed 25 times
Exercise 11.11. Consider the circles C = {x² + y² = 1}, C'= {(x-1)² + y² = 1} with radius 1 and respective centers (0,0) and (1,0). (a) Use algebra to compute the two points where these meet, and draw a picture to show why your answer is reasonable. (b) Use calculus to compute the (acute) angle at which the tangent vectors to C and C" meet at both of these points. (Informally, one may regard this as the angle at which the curves meet at P.) Hint: explain why it is the same as to find the acute angle between the gradient vectors at those points. The problem in (b) can be done directly via Euclidean geometry without recourse to calculus because of the special angles involved. The point of the exercise is to work out a special case of a general method (applicable in settings which Euclidean geometry cannot handle).
linger Exercise 10.1. Let f(x, y) = xy² - 4xy + (1/2)x² + 1. (a) Show that this function f has exactly 3 critical points: (0, 0), (0,4), and (4,2). (b) By examining the behavior of f on the lines y = 0 and y = x that pass through (0, 0), explain why (0, 0) is a saddle point. (It turns out that (0, 4) is also a saddle point, whereas (4, 2) is a local minimum.) (c) For the square S = {(x, y) = R² -1 ≤ x ≤ 1,-1 ≤ y ≤ 1}, analyze f on each side of S and compare with f(0, 0) to determine where f as a function on S attains its maximal value and where f as a function on S attains its minimal value. (Hint: to save work, check that f on each edge of the square has non-vanishing derivative and so is strictly increasing or strictly decreasing along the edge.)
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