For a function f, let f*(x) = lim as h→0 be [f(x+h) - f(x-h)]/h
a) Determine f*(x) for f(x) = cosx
b) write an equation that expresses the relationship between the functions f* and f` where f` denotes the usual derivative of f
Use the trigonometric identity
cos (x + h) = cosx cosh - sinx sinh
This means that
cos(x + h) - cosx
= (cosh -1) cosx - sin x sinh
For very small h, sin h = h and cos h = 1, so [cos(x + h) - cosx]/h -> -sin x
-sin x is the exact derivative of cos x.
a) To determine f*(x) for f(x) = cos(x), we need to evaluate the limit:
f*(x) = lim(h→0) [f(x+h) - f(x-h)]/h
Substituting f(x) = cos(x):
f*(x) = lim(h→0) [(cos(x+h) - cos(x-h))/h]
We can then simplify this expression using trigonometric identities. Applying the cosine angle sum and difference formulas:
cos(x+h) = cos(x)cos(h) - sin(x)sin(h)
cos(x-h) = cos(x)cos(h) + sin(x)sin(h)
Substituting these values back into the formula, we get:
f*(x) = lim(h→0) [cos(x)cos(h) - sin(x)sin(h) - cos(x)cos(h) - sin(x)sin(h)]/h
Combining like terms, we have:
f*(x) = lim(h→0) [-2sin(x)sin(h)]/h
Now, we can simplify this expression further by factoring out sin(x) from the numerator:
f*(x) = lim(h→0) -2sin(x) (sin(h)/h)
At this point, we recognize that the term (sin(h)/h) approaches 1 when h approaches 0. This is a well-known result:
lim(h→0) (sin(h)/h) = 1
Therefore, we can substitute this value into our expression:
f*(x) = -2sin(x) (lim(h→0) (sin(h)/h))
And finally, we obtain the derivative of the function f(x) = cos(x):
f*(x) = -2sin(x)
b) To write an equation that expresses the relationship between the functions f* and f', where f' denotes the usual derivative of f, we can simply write:
f'(x) = f*(x)
This equation states that the usual derivative of f(x) is equal to f*(x), which we determined to be -2sin(x) for the function f(x) = cos(x).