Question

Find the local quadratic approximation of f at x=x0, and use that approximation to find the local linear approximation of f at x0. Use a graphing utility to graph f and the two approximations on the same screen. f(x)=sin(2x), x0=pi/4 Enter Approximation Formulas below. Local Quadratic Approx = Local Linear Approx =

Answer 1

The local quadratic approximation of the function f(x) at x0 = π/4 is L_2(x) = 1 - 2(x - π/4)^2. The local linear approximation derived from the quadratic approximation is simply L_1(x) = 1.

Step-by-step explanation:

To find the local quadratic approximation of the function f(x) = sin(2x) at x0 = π/4, we use Taylor series expansion around x0. The local quadratic approximation can be represented as:
L_2(x) = f(x0) + f'(x0)(x - x0) + ½ f''(x0)(x - x0)^2

Since sin'≡ cos and cos'(x) is -sin(x), at x0 = π/4, f(x0) = sin(π/2) = 1, f'(x0) = 2cos(π/2) = 0, and f''(x0) = -4sin(π/2) = -4. Plugging these values into the local quadratic approximation we get:
L_2(x) = 1 + 0(x - π/4) + ½ (-4)(x - π/4)^2

To find the local linear approximation, which is the first degree Taylor polynomial, we can derive it from the quadratic approximation by eliminating the quadratic term:
L_1(x) = f(x0) + f'(x0)(x - x0) = 1 + 0(x - π/4) = 1

The Local Quadratic Approximation is L_2(x) = 1 - 2(x - π/4)^2 and the Local Linear Approximation is L_1(x) = 1.