Zadanie 12c
Define for −0.2≤x≤0.2. Find its Maclaurin expansion through x².
Maclaurin expansions and series operations
Construct Maclaurin expansions about zero from derivatives or differential equations, and manipulate standard expansions to approximate functions and evaluate limits.
Tę umiejętność sprawdzają 34 zadania wzorowane na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 34 Otwórz w wyszukiwarce z filtrami
Zadanie 12e
The first two nonzero Maclaurin terms of x/(a + bx) match those in 12c, where a,b are rational. Find a,b.
Zadanie 9
Evaluate .
Zadanie 12e
For positive integers n, define . (i) Find its Maclaurin expansion through the term in x2. (ii) Hence or otherwise find in terms of n.
Zadanie 2b
Hence use integration to establish for |x|<1.
Zadanie 2c
Use the first three nonzero terms of the arctan series at x=a to approximate π, to three decimal places.
Zadanie 2d
Find the smallest number N of nonzero arctan terms giving an absolute error less than 10⁻⁵ when approximating arctan a. Use the alternating-series bound or directly compare the actual errors.
Zadanie 2f
Evaluate to six decimal places.
Zadanie 2g
Using I = (1 + a2)π/8 − a and the approximation , estimate π to four decimal places.
Zadanie 2h
The approximation is a sum of four alternating integrated terms. Verify numerically that its actual absolute error in approximating I is no greater than the next integrated term.
Zadanie 2i
Find the smallest N for which . Use the alternating-series bound or directly compare the actual errors.
Zadanie 5a
(i) Show that a=3. (ii) Find b.
Zadanie 5b
Use the displayed cubic approximation to estimate the signal strength before storage.
Zadanie 10a
(i) Show that the first four terms of (1 + 3x2)−1 are 1 − 3x2 + 9x4 − 27x6, for . (ii) Hence approximate its indefinite integral through the x⁷ term.
Zadanie 10b
(i) Expand (1 − 2x2)−1/2 through x⁴. (ii) Hence give the polynomial approximation to H(x) through x⁵.
Zadanie 10c
Using the polynomial for H, approximate in the form 13/24 + k/320, where k is a positive integer. Find k.
Zadanie 10d
Hence find an approximation to π in the form , where p is a positive integer.
Zadanie 1c
For |r|<1, a + ar + ar2 + ⋯ = a/(1 − r). (i) Differentiate both sides with respect to r and express in terms of a,r. (ii) Hence show that E(X)=1/p.
Zadanie 8a
Find the first two nonzero terms of (i) ; (ii) .
Zadanie 8b
Hence or otherwise find the first two nonzero terms of .
Zadanie 12b
Use derivatives at zero or another valid method to show that the Maclaurin expansion begins .
Zadanie 12c
Hence, by multiplying the relevant expansions, show that when terms beyond degree 2 are discarded.
Zadanie 12d
Let P(x) = (1 − x)−3/2(1 + x)−3/2. The two Maclaurin series in this product are those of f1(x) and f−1(x). State the range of x for which both Maclaurin series used in the product converge.
Zadanie 12e
Let P(x) = (1 − x)−3/2(1 + x)−3/2. The quadratic approximation to P is P(x) ≈ 1 + 3x2/2. Use x=1/5 in the quadratic approximation 1 + 3x2/2 to obtain a rational approximation to .
Zadanie 11d
Use the derivative relations y′ = (1 − y2)3/2, y′′ = −3y(1 − y2)2 and y′′′ = −3(1 − 5y2)(1 − y2)5/2 to find the Maclaurin series through the cubic term.
Zadanie 12c
Given y‴(0)=−34, (i) find the first four terms of the Maclaurin expansion; (ii) use them to estimate y(0.20) to six significant figures.
Zadanie 11d
Find, through x⁴, the Maclaurin expansions of (i) cos4x; (ii), using (i); (iii) F(x), using (ii).
Zadanie 11e
Use the first two nonzero terms F(x) ≈ (e + 1) − 8(e + 1)x2 to show that .
Zadanie 11d
Find, through x⁴, the Maclaurin expansions of (i) cos4x; (ii), using (i); (iii) F(x), using (ii).
Zadanie 11e
Use the first two nonzero terms F(x) ≈ (e + 1) − 8(e + 1)x2 to show that .
Zadanie 12a
Find the Maclaurin expansion of f up to the x³ term.
Zadanie 12b
Hence approximate using terms up to degree 3 in the Maclaurin expansion of f. For standalone use, f(u) = u + 2u2 + 11u3/6 + ⋯.
Zadanie 12d
Using g′′ = 4g′ − 5g and its differentiated recurrences, find the Maclaurin expansion of g through x⁴.
Zadanie 12e
Hence or otherwise evaluate .