linearization formula

Question

Let f(x)=3sqrtx​ (a) Find a formula for the linearization L(x) of f(x) at x=8.

(b) Use your answer from (a) to estimate 3sqrt10​.

full answer and work pls

Answer 1

If f(x) = 3√x the at x = 8, f(8) = 3√8 = 6√2 so the point on the curve for which we are "linearizing" the function is (8, 6√2). If we are creating a line at that point, we now need the slope. To get the slope, we use the derivative:

f(x) = 3√x

f(x) = 3(x)^1/2

f '(x) = (3/2)(x)^-1/2 (using the power rule)

f '(8) = (3/2)(8)^-1/2 = 3/(2√8) = 3/(4√2) = (3√2)/8

Using the point-slope form of a line:

y - 6√2 = [(3√2)/8](x - 8)

y = [(3√2)/8]x - 3√2 + 6√2

y = [(3√2)/8]x + 3√2

L(x) = [(3√2)/8]x + 3√2

To estimate f(10):

L(10) = [(3√2)/8]10 + 3√2 = (15√2)/4 + 3√2 = (15√2)/4 + 12√2/4 = (27√2)/4 which is approx 9.55