Advertisements
Advertisements
प्रश्न
Expand sin x in ascending powers `x - pi/4` upto three non-zero terms
उत्तर
Let f(x) = sin x
f(x) = sin x, `"f"(pi/4) = 1/sqrt(2)`
f(x) = cos x, `"f'"(pi/4) = 1/sqrt(2)`
f(x) = – sin x, `"f''"(pi/4) = - 1/sqrt(2)`
f(x) = – cos x, `"f'''"(pi/4) = - 1/sqrt(2)`
Taylor series of f(x) is
f(x) = `sum_("n" = 0)^("n" = oo) "a"_"n" (x - pi/4)^pi`
Where an = `("f"^(("n"))(pi/4))/("n"!)`
∴ The required expansion is
sin x = `1/sqrt(2) + 1/sqrt(2) ((x - pi/4))/(1!) - 1/sqrt(2) ((x - pi/4)^2)/(2!) - 1/sqrt(2) (x - pi/4)^3/(3!) + ...`
= `1/sqrt(2) [1 + ((x - pi/4))/(1!) - (x - pi/4)^2/(2!) - (x - pi/4)^3/(3!) + ...]`
= `sqrt(2)/2 [1 + ((x - pi/4))/(1!) - (x - pi/4)^2/(2!) - (x - pi/4)^2/(3!) + ...]`
APPEARS IN
संबंधित प्रश्न
Write the Maclaurin series expansion of the following functions:
ex
Write the Maclaurin series expansion of the following functions:
sin x
Write the Maclaurin series expansion of the following functions:
cos x
Write the Maclaurin series expansion of the following functions:
log(1 – x); – 1 ≤ x ≤ 1
Write the Maclaurin series expansion of the following functions:
tan–1 (x); – 1 ≤ x ≤ 1
Write the Maclaurin series expansion of the following functions:
cos2x
Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0
Expand the polynomial f(x) = x2 – 3x + 2 in power of x – 1
