linearization calculus

Question

Find the linearization of the given function at the given point, then use it to approximate the given value. (a) f(x) = 3^√ x at x = 343 to approximate 3^√350 (b) f(x) = 2^x at x = 4 to approximate 2^3.99

Answer 1

If you "linearize" a function, you will be considering it a "line" at the point in question. The slope of that line is the derivative of the function at that point.

For f(x) = 3^√x, take the derivative using the exponential rule and the chain rule:

f '(x) = ln(3)•3^√x/(2√x)

To find the derivative at x = 343:

f '(343) = ln(3)•3^√343/(2√343) ≈ 20350575.41

Now we have the slope (aka "m", now if we had a point, we could write the equation of the line using the point-slope form. To find that point, we can plug in x = 343 into the function. y = 3^√343 = 686134562.5 so the point is (343, 686134562.5)

So the linear equation is y - y₁ = m(x - x₁) or y - 686134562.5 = 20350575.41(x - 343).

To approximate 3^√350, we just plug in x = 350 into this equation:

y - 686134562.5 = 20350575.41(350 - 343)

y - 686134562.5 = 20350575.41(7)

y = 142454027.9 + 686134562.5 = 828588590.4

so 3^√350 ≈ 828588590.4

Repeat the same process for the 2nd problem.