1. Let f(x) = sin (x) and a = a) Find the linear approximation L(x) = f(a) + f'(a)(x-a). Give exact values for a, f(a), and f'(a). b) Rewrite L(x) given decimal approximations for a, f (a), and f'(a). Round to three decimal places. c) In your calculator find sin (0.9). Round to three decimal places. d) Use the linear approximation and find L(0.9). e) What is the error [sin(0.9) - L(0.9)|? f) Graph y = f(x) and y = L(x) in the same set of axes. Observe how close the graphs are for values of x close to 1. 2. Let f(x) = tan (x) and a = 1. 4 a) Find the linear approximation L(x) = f(a) + f'(a)(x-a). Give exact values for a, f(a), and f'(a). b) Rewrite L(x) given decimal approximations for a, f(a), and f'(a). Round to three decimal places. c) In your calculator find tan (0.9). Round to three decimal places. d) Use the linear approximation and find L(0.9). e) What is the error |tan(0.9) - L(0.9)|? f) Graph y= f(x) and y = L(x) in the same set of axes. Observe how close the graphs are for values of x close to 1. TT 3. Let f(x)=tan (x) and a = 3 a) Find the linear approximation L(x) = f(a) + f'(a)(x-a). Give exact values for a, f(a), and f'(a). b) Rewrite L(x) given decimal approximations for a, f(a), and f'(a). Round to three decimal places. c) In your calculator find tan (1.162). Round to three decimal places. d) Use the linear approximation and find L(1.162). e) What is the error |tan(1.162) - L(1.162)|? f) Graph y= f(x) and y = L(x) in the same set of axes. Observe how close the graphs are for values of x close to 1. 4. Why do you think the error for problems 2 and 3 is different even though the function is the same?