Question

(a) Find the differential dy.

y = cos(x)

dy =?


(b) Evaluate dy for the given values of x and dx. (Round your answer to three decimal places.)
x = π/3, dx = 0.1.

dy=?

Answer 1

The differential dy for y = cos(x) is evaluated by finding the derivative of y which is -sin(x), then multiplying by dx. For x = π/3 and dx = 0.1, the calculated differential dy is approximately -0.0866 when rounded to three decimal places.

Step-by-step explanation:

The differential dy of a function y with respect to x is given by the derivative of y with respect to x, multiplied by dx. For the function, y = cos(x), the derivative of y is -sin(x), hence dy = -sin(x)dx.

To evaluate dy for x = π/3 and dx = 0.1, we substitute x into -sin(x) and multiply by dx. This results in dy = -sin(π/3) × 0.1, which simplifies to dy = -0.1 √3/2. Rounding to three decimal places, dy ≈ -0.0866.

Answer 2

a. Practically speaking, you compute the differential in much the same way you compute a derivative via implicit differentiation, but you omit the variable with respect to which you are differentiating.

`y=\cos x\implies\boxed{\mathrm dy=-\sin x\,\mathrm dx}`

Aside: Compare this to what happens when you differentiate both sides with respect to some other independent parameter, say
`t`:

`(\mathrm dy)/(\mathrm dt)=-\sin x(\mathrm dx)/(\mathrm dt)`

b. This is just a matter of plugging in
`x=\frac\pi3` and
`\mathrm dx=0.1`.

`\boxed{\mathrm dy\approx-0.087}`