What is the linear approximation of the square root of 8.9? Must be accurate to the fifth decimal place.
I have tried entering 2.98328 and 2.98328 but that does not appear to be correct. Please help!!
Did you try 2.98329 -- as the Google calculator showed?
Reiny did it the long way and came up with a number that was close to that.
If the computer doesn't accept the Google answer -- I'd ask the teacher about it.
Did you enter
2.98333 , which was my answer using linear approximation method ? ,
(we can't use a calculator for this)
To find the linear approximation of a function, we use its tangent line at a certain point. In this case, we want to approximate the square root function near the point x = 9.
The tangent line of the square root function near x = 9 can be found using calculus techniques. The derivative of the square root function f(x) = √x is given by f'(x) = 1 / (2√x).
We can plug in x = 9 into the derivative to find the slope of the tangent line at that point:
m = f'(9) = 1 / (2√9) = 1 / (2 * 3) = 1/6.
Using the point-slope form of a line, we can write the equation of the tangent line as:
y - f(9) = m(x - 9).
Since we want to approximate the square root of 8.9, which is slightly smaller than 9, let's use x = 8.9 as the input. Plugging in x = 8.9 and solving for y, we can find our linear approximation.
y - f(9) = (1/6)(x - 9)
y - √9 = (1/6)(8.9 - 9)
y - 3 = (1/6)(-0.1)
y - 3 = -0.1/6
y - 3 = -0.0166667.
Adding 3 to both sides, we get:
y = -0.0166667 + 3
y ≈ 2.9833333.
Therefore, the linear approximation of the square root of 8.9 accurate to the fifth decimal place is approximately 2.98333.