Compute the differential dy for y = (2x + sin x)^2. Use an appropriate local linear approximation to estimate the value of the given quantity. tan(-0.14) ≈

Question

Compute the differential dy for y = (2x + sin x)^2.

dy = ?

Use an appropriate local linear approximation to estimate the value of the given quantity.

tan(-0.14) ≈ ?

Answer 1

Question #1:

Note that we use the power rule and the chain rule to do the derivative.

Question #2:

Let y = tan(x)

So y(0) = tan(0) = 0

So when x = 0, y = 0

Let dx = -0.14 and we'll use the differential to find the approximate value tan(-0.14)

Calculate the differential:

dy/dx = sec²(x)

dy = (sec²(x))(dx)

For x = 0, y = 0, dx = -0.14 find dy:

dy = sec²(0)(-0.14)

dy = (1)(-0.14)

dy = -0.14

Since y = 0, the estimation for y(-0.14) = y + dy = 0 + -0.14 = -0.14

(Note: If you use your calculator, you see that tan(-0.14) = -0.140922