Let a < b and let ƒ(x) be a continuously differentiable function on the interval [a, b] with f(x) > 0 for all x = [a, b]. Let A > 0, B> 0 be constants. Show that the stationary path of the functional S[(y)] = [" is given by [° dx f(x)√1+y¹², y(a) = A, y(b) = B, rx 1 y(x) = A + 8 [² du √ √5)² - 32 dw - where is a constant satisfying · A=D [ ² de √(1-3³²-13² dw a B-A= B

a) Find dx of the function y = (x+1)²using the limit definition of the derivative. b) What is the slope at x = -1?

Let f(x) = √100x². Use a linear approximation to ƒ at (6,0) to approximate ƒ(6.1). Your answer must be simplified and in decimal form.

Consider the following R2 →R function: b) Plot the level curves for c = 0, 1, 2 and 3