Applications of Derivatives
There are two areas where derivatives are used
1. Increasing and Decreasing Functions
2. Maxima and Minima
- Relative maxima and minima
- Absolute maxima and minima
3. Taylor’s and Maclaurin’s Series Expansion of Functions
4. Slope determination of line
Taylor’s and Maclaurin’s Series Expansion of Functions
Taylor’s Series
If (i) f(x) and its first (n – 1) derivatives be continuous in [a, a + h], and (ii) fn(x) exists for every value of x in (a, a + h), then there is at least one number θ (0 < θ < 1), such that
f (a + h) = f(a) + hf′(a) + h2 / 2! f” (a)+…..+ hn / n! fn (a + θh)
which is called Taylor’s theorem with Lagrange’s form of remainder, the remainder Rn being hn / n! fn (a + θh)
Maclaurin’s Series
If f(x) can be expanded as an infinite series, then
f (x) = f(0) + xf′(0) + x2 / 2! f”(0) + x3 / 3! f”(0) + ….∞
If f (x) possesses derivatives of all orders and the remainder Rn in (3) on page 154 tends to zero as n → ∞, then the Maclaurin’s theorem becomes the Maclaurin’s series (1).
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