1. The functions and g are twice differentiable. Selected values of f, g, and their derivatives...

  

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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). 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).