Zadanie 12b
Use derivatives at zero or another valid method to show that the Maclaurin expansion begins .
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Use derivatives at zero or another valid method to show that the Maclaurin expansion begins .
Hence, by multiplying the relevant expansions, show that when terms beyond degree 2 are discarded.
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.
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 .
Use Euler’s method with step length π/12 to estimate y at x=π/3, giving three significant figures.
Show that .
Show that cot x is an integrating factor.
Hence solve the initial-value problem and show that y=x tan x.
The exact solution is y=x tan x. (i) Find y at x=π/3 to three significant figures. (ii) Considering the gradient, explain why the forward Euler estimate from π/6 using steps of π/12 is inaccurate here. (iii) Explain why that estimate is less than the exact…
Using the differential equation and its solution y=x tan x, deduce that the curve has positive gradient throughout 0<x<π/2.