Differential of f

Question

Let f be the function defined as follows.
y = f(x) = √(9x+5)

(a) Find the differential of f.
Answer: dy = (9/2√(9x+5))dx


(b) Use your result from part (a) to find the approximate change in y if x changes from 4 to 4.1 (Round your answer to three decimal places.).
dy =


(c) Find the actual change in y if x changes from 4 to 4.1 and compare your result with that obtained in part (b). (Round your answer to three decimal places.)
Δy =

Answer 1

y = f(x) = (9x+5)^½

 

differential of f = dy = df = f'(x)dx = (1/2)(9x+5)^-1/2(9)dx

 

                                                   = 9/[2√(9x+5)]dx

 

b. Δf = f(4.1) - f(4) ≈ df = f'(4)(4.1-4)

 

                                    = 9/[2√41](0.1) ≈ 0.070

 

c. Δy = Δf = f(4.1) - f(4) = √41.9 - √41 ≈ 0.0698967 ≈ 0.070