Question

Find the linear approximation, L(x), of f(x) = sin(x) at x = pi/3.

L(x) =__________
Use the linear approximation to approximate sin(5pi/12) (Round your answer to four decimal places.)
L(5pi/12) = _______

Answer 1

The linear approximation,
`\(L(x)\), of \(f(x) = \sin(x)\) at \(x = (\pi)/(3)\) is \(L(x) = (1)/(2) + (√(3))/(2)(x - (\pi)/(3))\).` Using this approximation to estimate
`\(\sin((5\pi)/(12))\),` we find
`\(L((5\pi)/(12)) \approx 0.9659\)` (rounded to four decimal places).

Step-by-step explanation:

To find the linear approximation,
`\(L(x)\), of \(f(x) = \sin(x)\) at \(x = (\pi)/(3)\),` we use the formula for linear approximation:

`\[L(x) = f(a) + f'(a)(x - a).\]`

First, we evaluate
`\(f((\pi)/(3))\) and \(f'((\pi)/(3))\),`where
`\(f'(x)\)`is the derivative of
`\(f(x)\).`For
`\(f(x) = \sin(x)\), \(f'((\pi)/(3)) = (√(3))/(2)\).`Plugging these values into the formula, we get
`\(L(x) = (1)/(2) + (√(3))/(2)(x - (\pi)/(3))\).`

To estimate
`\(\sin((5\pi)/(12))\)` using the linear approximation, we substitute
`\((5\pi)/(12)\) into \(L(x)\).`Therefore,
`\(L((5\pi)/(12)) = (1)/(2) + (√(3))/(2)((5\pi)/(12) - (\pi)/(3))\).` Simplifying this expression yields
`\(L((5\pi)/(12)) \approx 0.9659\)` (rounded to four decimal places).

In summary, the linear approximation provides an estimate of a function near a specific point by using the first-degree Taylor polynomial. In this case,
`\(L(x)\)`approximates
`\(\sin(x)\) near \(x = (\pi)/(3)\),` and the estimated value for
`\(\sin((5\pi)/(12))\)` is calculated accordingly.

Answer 2

The linear approximation of f(x) = sin(x) at x = pi/3 is L(x) = √(3)/2 + (1/2)(x - pi/3). Using the linear approximation, the approximation for sin(5pi/12) is 0.7124.

Step-by-step explanation:

To find the linear approximation of f(x) = sin(x) at x = pi/3, we use the formula for linear approximation: L(x) = f(a) + f'(a)(x-a). First, we find the value of f(a) = sin(a) at a = pi/3, which is sin(pi/3) = √(3)/2. Next, we find the derivative of f(x) = sin(x), which is f'(x) = cos(x). Evaluating f'(a) = cos(pi/3) = 1/2, we have L(x) = √(3)/2 + (1/2)(x - pi/3).

Now, to approximate sin(5pi/12) using the linear approximation, we substitute x = 5pi/12 into the linear approximation equation: L(5pi/12) = √(3)/2 + (1/2)(5pi/12 - pi/3). Simplifying this expression, we get L(5pi/12) = √(3)/2 + (1/12)(5pi - 4pi). Therefore, L(5pi/12) = √(3)/2 + pi/12. Rounding this to four decimal places, we have L(5pi/12) = 0.7124.