Question

For the following exercises, find the linear approximation L(x) to y = f (x) near x = a for the function. 50. [T] f (x) = x + x4, a = 0 51. [T] f (x) =1x , a = 2 52. [T] f (x) = tanx, a = π 4 53. [T] f (x) = sin x, a = π 2 54. [T] f (x) = xsin x, a = 2π 55. [T] f (x) = sin2 x, a = 0

Answer 1

For the given functions and values of a, we can find the linear approximations L(x) using the formula L(x) = f(a) + f'(a)(x - a). The linear approximations are: 1) L(x) = x for f(x) = x + x^4 and a = 0, 2) L(x) = 1 - (1/4)x for f(x) = 1/x and a = 2, and 3) L(x) = x - (π/4) for f(x) = tan(x) and a = π/4.

Step-by-step explanation:

1. For the function f(x) = x + x^4 and a = 0, we can find the linear approximation L(x) using the formula L(x) = f(a) + f'(a)(x - a).
   Plugging in the values, we have L(x) = f(0) + f'(0)(x - 0) = 0 + 1( x - 0) = x. Therefore, the linear approximation is L(x) = x.
2. For the function f(x) = 1/x and a = 2, we can find the linear approximation L(x) using the formula L(x) = f(a) + f'(a)(x - a).
   Plugging in the values, we have L(x) = f(2) + f'(2)(x - 2) = 1/2 + (-1/4)(x - 2) = 1/2 - (1/4)x + 1/2 = 1 - (1/4)x. Therefore, the linear approximation is L(x) = 1 - (1/4)x.
3. For the function f(x) = tan(x) and a = π/4, we can find the linear approximation L(x) using the formula L(x) = f(a) + f'(a)(x - a).
   Plugging in the values, we have L(x) = f(π/4) + f'(π/4)(x - π/4) = 1 + 1(x - π/4) = 1 + x - π/4. Therefore, the linear approximation is L(x) = x - (π/4).