Question

The function f(x) is differentiable as dy/dx =2x² +x. If f(2)=-3 use differentials to estimate f(2.07).

A. −2.255
B. −3.700
C. −2.300
D. −3.745

Answer 1

To estimate f(2.07), we can use differentials. The estimated value of f(2.07) is approximately -2.3.

Step-by-step explanation:

To estimate f(2.07), we can use differentials. The differential of a function is given by dy = f'(x)dx. In this case, the derivative of f(x) is f'(x) = 2x^2 + x. To find f(2.07), we need to calculate the change in x (dx) from 2 to 2.07. This is given by (2.07 - 2) = 0.07.

Now, we substitute the values into the differential equation:

dy = f'(x)dx

dy = (2x^2 + x)dx

Using the value of x = 2, we have:

dy = (2(2)^2 + 2)dx

dy = (8 + 2)dx

dy = 10dx

Now, we substitute the value of dx = 0.07:

dy = 10(0.07)

dy = 0.7

Finally, we add the change in y (dy) to the initial value of f(2):

f(2.07) ≈ f(2) + dy

f(2.07) ≈ -3 + 0.7

f(2.07) ≈ -2.3

Therefore, the estimated value of f(2.07) is approximately -2.3.