1 The functions and g are twice differentiable Selected values of f, g, and their derivatives f' and g' are given in the table below X 0 2 3 4 f(x) 6 1 1 0 f'(x) 1 2 g(x) 3 3 5 1 2 1 3 g'(x) 4 3 2 2 (a) The function kis defined by k (x) f'(x)g(x), x 2 2f(x)g'(x), x2 Use the definition of continuity to show that k(x) is continuous at x 2 (b) Evaluate lim f(2x 1) x or show that it does not exist Justify your answer 2g (sin(x) 6 x 1 (c) The function h is defined by h(x) f(g(3x)) The slope of the line tangent to h(x) at x 1 is 15 Find f'( 1) (d) Evaluate g'(x) f'(g(x))dz Show the work that leads to your answer (e) Let y p(x) be a particular solution to the differential equation dy dx f'(x)g(x) with p(0) 4 Use Euler's method, starting at x 0 with two steps of equal size to approximate p(4) p'(x) can be approximated using a right Riemann sum with three subintervals indicated in the above table Find this approximation and use it to approximate p(4)

Question: 1. The functions and g are twice differentiable. Selected values of f, g, and their derivatives f' and g' are given in the table

1. The functions and g are twice differentiable. Selected values of f, g, and their derivatives f' and g' are given in the table below. X 0 2 3 4 f(x) 6 1 1 0 f'(x) -1/2 g(x) 3 -3 5 1 2 -1 3 g'(x) 4 3 2 -2 (a) The function kis defined by k (x) = [ f'(x)g(x), x

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a To show that kx is continuous at x 2 we need to evaluate the lefthand limit and the righthand limit of kx as x approaches 2 and show that they are e... View full answer

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