Let f be the function defined as follows.
$ y = f(x) = \sqrt{{\color{red}3} x +{\color{red}10}}$
(a) Find the differential of f.
dy =
(b) Use your result from part (a) to find the approximate change in y if x changes from 4 to 4.1 (Round your answer to three decimal places.).
dy =
(c) Find the actual change in y if x changes from 4 to 4.1 and compare your result with that obtained in part (b). (Round your answer to three decimal places.)
Δy =
To find the differential of the function f(x), we can use the concept of differentials in calculus.
(a) The differential of a function can be found using the formula: dy = f'(x) * dx, where f'(x) represents the derivative of the function with respect to x, and dx represents the change in x.
Let's find the derivative of f(x) first.
Given: f(x) = sqrt(3x + 10)
To find f'(x), we can use the chain rule.
Apply the chain rule for the square root function:
f'(x) = (1/2) * (3x + 10)^(-1/2) * (d(3x + 10)/dx)
Differentiating 3x + 10 with respect to x gives us 3.
So, f'(x) = (1/2) * (3x + 10)^(-1/2) * 3 = 3/(2√(3x + 10))
Now, we can find the differential dy:
dy = f'(x) * dx
Substituting f'(x) = 3/(2√(3x + 10)) and dx = (change in x) gives:
dy = (3/(2√(3x + 10))) * (change in x)
Hence, the differential dy is given by dy = (3/(2√(3x + 10))) * (change in x).
(b) To find the approximate change in y if x changes from 4 to 4.1, we can use the differential dy.
Substituting x = 4 and (change in x) = 0.1 into the differential dy, we get:
dy = (3/(2√(3(4) + 10))) * 0.1
Calculating this value will give us the approximate change in y.
(c) To find the actual change in y if x changes from 4 to 4.1, we can substitute x = 4.1 into the original function f(x) and subtract the value of f(4).
Δy = f(4.1) - f(4)
Calculate this value to find the actual change in y.
Comparing the results obtained in parts (b) and (c) will allow us to compare the approximate change with the actual change in y.