Question

​​​​​​​The linear approximation of a function f(x) given by the tangent line at x=a is L(x)=f(a)+ f'(a)(x-a). The error E(x) is the difference between the value given by the linear approximation and the actual function value, i.e. E(x)=∫(x)−L(x). (a) What is the L(x) for the function f(x)= √1−x at x=a ?

Answer 1

The linear approximation of the function f(x) = √1−x at x=a is found by substituting a into the function and taking the derivative of the function at x=a. The linear approximation equation is L(x) = f(a) + f'(a)(x-a).

Step-by-step explanation:

The linear approximation of a function f(x) given by the tangent line at x=a is L(x)=f(a)+ f'(a)(x-a).

To find the linear approximation of the function f(x)= √1−x at x=a, we need to find the value of f(a) and f'(a).

1. First, find f(a) by substituting a into the function: f(a) = √1−a.
2. Next, find f'(a) by taking the derivative of the function: f'(a) = d/dx (√1−x) evaluated at x=a.
3. Finally, substitute f(a) and f'(a) into the linear approximation equation L(x) = f(a) + f'(a)(x-a).