f*(x) = lim as h -> 0 [f(x+h)-f(x-h)]/ h
write an equation that expresses the relationship between the functions f*(x) and f`
I know that f` is [f(x+h)-f(x)]/h but i have no clue how to write an equation to relate the two!!!
f(x+h)-f(x-h) =
f(x+h) - f(x) + f(x) - f(x-h)
lim as h -> 0 [f(x+h)-f(x-h)]/ h =
lim as h -> 0 [f(x+h)-f(x)]/ h +
lim as h -> 0 [f(x)-f(x-h)]/ h
lim as h -> 0 [f(x)-f(x-h)]/ h
also equals the derivative of f if the functon is differentiable.
To express the relationship between the functions f*(x) and f'(x), you can start by substituting the expression for f'(x) into the equation for f*(x):
f*(x) = lim as h -> 0 [f(x+h)-f(x-h)]/ h
Replace f'(x) into the equation:
f*(x) = lim as h -> 0 [f(x+h)-f(x)]/ h
Notice that the expression in the numerator of f*(x) is f(x+h) - f(x), which matches f'(x). So, you can rewrite the equation as:
f*(x) = lim as h -> 0 f'(x)
This equation expresses the relationship between the functions f*(x) and f'(x), indicating that f*(x) is equal to the limit of f'(x) as h approaches 0.