Let f be the function defined as follows.
$ y = f(x) = {\color{red}5} x^2 -{\color{red}9} 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 2 to 1.97. (Round your answer to two decimal places.)
dy =
(c) Find the actual change in y if x changes from 2 to 1.97 and compare your result with that obtained in part (b). (Round your answer to two decimal places.)
Δy =
To find the differential of a function, we differentiate it with respect to the independent variable. In this case, we need to find dy for the function f(x) = 5x^2 - 9x + 10.
(a) To find the differential of f, we will differentiate the function with respect to x:
dy = f'(x) dx
First, let's find f'(x):
f'(x) = d/dx (5x^2 - 9x + 10)
= 10x - 9
Therefore, dy = (10x - 9) dx.
(b) To find the approximate change in y if x changes from 2 to 1.97, we need to substitute the values of x and dx into the differential equation we found in part (a). Here, x = 2 and dx = 1.97 - 2 = -0.03 (since x is decreasing).
dy = (10x - 9) dx
= (10*2 - 9) * (-0.03)
= (-7) * (-0.03)
= 0.21
Therefore, the approximate change in y is 0.21.
(c) To find the actual change in y if x changes from 2 to 1.97, we need to substitute these values into the original function f(x) = 5x^2 - 9x + 10.
Δy = f(1.97) - f(2)
Substituting x = 1.97 into f(x):
f(1.97) = 5*(1.97)^2 - 9*(1.97) + 10
Calculating the value of f(1.97) and f(2), we get:
f(1.97) ≈ 14.6809
f(2) = 5*(2)^2 - 9*(2) + 10 = 10
Δy = f(1.97) - f(2)
≈ 14.6809 - 10
≈ 4.68
Therefore, the actual change in y is approximately 4.68. Comparing this with the approximate change we obtained in part (b), we see that there is some difference between the two values.