How do you use #f(x) = sin(x^2-2)# to evaluate #(f(3.0002)-f(3))/0.0002#?

Question

Evelyn Agard · Accepted Answer

Recall the derivative of a function is given by  #f'(x) = lim_(h-> 0) (f(x + h) - f(x))/h# If we let #x = 3# and #h = 0.0002# (very close to #0#), we get that the given expression is equal to #f'(3)#.  We can find the derivative using the chain rule  #f'(x)= 2xcos(x^2 - 2)#
#f'(3) = 2(3)cos(3^2 - 2) = 6cos(7) = 4.523# If we plug the given expression into our calculator we get #(f(3.0002) - f(3))/0.0002 = 4.521# So our approximation is pretty good.   Hopefully this helps!

William Abdalla · Answer

undefined To evaluate (f(3.0002)-f(3))/0.0002 using the function f(x) = sin(x^2-2), first calculate f(3.0002) and f(3), and then find their difference divided by 0.0002.

f(3.0002) = sin((3.0002)^2 - 2)
          ≈ sin(9.0006 - 2)
          ≈ sin(7.0006)
          ≈ 0.6572

f(3) = sin(3^2 - 2)
     = sin(9 - 2)
     = sin(7)
     ≈ 0.6570

Now, calculate the expression:

(f(3.0002) - f(3)) / 0.0002
= (0.6572 - 0.6570) / 0.0002
≈ 0.0002 / 0.0002
= 1

Christopher Allers · Answer

undefined To evaluate (f(3.0002)-f(3))/0.0002 using the function f(x) = sin(x^2-2), first substitute x = 3.0002 into the function to find f(3.0002), then substitute x = 3 to find f(3). Finally, subtract f(3) from f(3.0002) and divide the result by 0.0002.