If f(x) = 6 cos2(x)

Question

If f(x) = 6 cos²(x), compute its differential df.

df = -12cos(x)sin(x)dx

Approximate the change in f when x changes from x = pi/ 6 to x = pi/6 +.1

deltaf = -0.520

Approximate the relative change in f as x undergoes this change. (Round your answer to three decimal places.)

i have tried and Ican get this question right.

Heres what Ive done

=6[cos²((π/6)+.1)-cos²(π/6)] which equals -0.546 which was not the c correct answer then I trie using this method that she wanted us to use and didn't get the right answer. the method she wants use to use is deltaf/f(c). WHen I used that method I got -0.047

Answer 1

The function derivative is

df(x)/dx=-12cos(x)sin(x)

df(x)=-12cos(x)sin(x)dx

this is true, when dx is infinitely small.

if Δx=.1 that is not infinitely small

so df(x)≈-12scos(x)sin(x)Δx,,,,,,,,,,,df(x) now is approximated

x=π/6,,,Δx=.1,,,,,in the approximation of df(x) the "x+Δx" is broken apart

df(x)≈-12(((√3)/2)(1/2)(.1)= -.520 is approximate change in f(x)=6cos²x when x=π/6 and Δx=.1