The linear approximation at x = 0 to
sin (5 x) is A + B x. Find A and B?
67
Which table of ordered pairs represents a proportional relationship?
mc020-1.jpg
mc020-2.jpg
mc020-3.jpg
mc020-4.jpg
To find the linear approximation of a function at a specific point, you can use the concept of the first two terms of the Taylor expansion.
The Taylor expansion of a function f(x) centered at x = a is given by:
f(x) = f(a) + f'(a) * (x-a) + f''(a)/2! * (x-a)^2 + f'''(a)/3! * (x-a)^3 + ...
For the given function f(x) = sin(5x), we need to find the derivatives of f(x) up to the second order and evaluate them at x = 0.
First, let's find the first derivative:
f'(x) = d/dx (sin(5x)) = 5cos(5x)
Now, evaluate the first derivative at x = 0:
f'(0) = 5cos(0) = 5
Next, let's find the second derivative:
f''(x) = d/dx (f'(x)) = d/dx (5cos(5x)) = -25sin(5x)
Evaluate the second derivative at x = 0:
f''(0) = -25sin(0) = 0
Now we have the values f(0) = sin(0) = 0, f'(0) = 5, and f''(0) = 0.
Using the Taylor expansion formula, the linear approximation of f(x) = sin(5x) at x = 0 is given by:
f(x) ≈ f(0) + f'(0)(x - 0)
= 0 + 5x
= 5x
So, the linear approximation of sin(5x) at x = 0 is A + Bx, where A = 0 and B = 5.