Use a linear approximation (or differentials) to estimate the given number.

Tan 44 degrees

Please help.

f(x)=tan(x)

f'(x)=sec²(x)
Using linear approximation:
f(xo+δ)=f(xo)+δf'(xo) (approx.)

Put xo=π/4 (in radians, = 45°)
δ=π/180 (1°)
f(44°)
=f(xo-δ)
=tan(π/4)-δ*sec²(π/4) (approx.)
=1-π/180/cos²(π/4)
=1-π/180/(1/2)
=1-π/90
=0.96509...

accurate value of tan(44°)=0.96569...

To estimate the value of tan (44 degrees) using linear approximation, we can start by considering the tangent function's behavior near 45 degrees, which has a known value.

The tangent function has an approximate linear behavior near 45 degrees due to its derivative. At 45 degrees, the tangent function has a value of 1. So, we can use this information to estimate the value of tan (44 degrees) by applying the concept of differentials.

We can first rewrite the angle of 44 degrees as a difference from 45 degrees:
44 degrees = 45 degrees - 1 degree

Next, let's find the differential of the tangent function at 45 degrees:
d(tan x) = sec^2 x * dx

Since dx = -1 degree, we can substitute these values into the differential:
d(tan 45) = sec^2(45) * -1

The secant function at 45 degrees has a value of √2, so we can substitute this value:
d(tan 45) = (√2)^2 * -1

Simplifying, we get:
d(tan 45) = -2

Now we can use the differential to estimate the difference between tan (45 degrees) and tan (44 degrees):
tan 44 ≈ tan 45 + d(tan 45)

Substituting the values:
tan 44 ≈ 1 + (-2)

Finally, we can calculate the estimate:
tan 44 ≈ -1

Therefore, using linear approximation, we estimate that tan (44 degrees) is approximately -1.

To estimate the value of tan 44 degrees using linear approximation, we can use the tangent function's derivative, which is sec^2(x).

First, we need to convert 44 degrees to radians. Since tan expects the angle to be in radians, we have:

44 degrees * (π radians / 180 degrees) = 0.76794 radians (approximately)

Now, we can use the linear approximation formula:

f(x) ≈ f(a) + f'(a)(x - a)

In this case, f(x) = tan(x), a = 0 radians, and x = 0.76794 radians.

So, the linear approximation for tan 44 degrees is:

tan(0.76794) ≈ tan(0) + sec^2(0)(0.76794 - 0)
≈ 0 + 1(0.76794 - 0)
≈ 0 + 0.76794
≈ 0.76794 (to approximately 5 decimal places)

Therefore, using the linear approximation, the estimated value of tan 44 degrees is approximately 0.76794.