IB Mathematics Analysis and Approaches Topic AHL…

This question involves finding the Maclaurin series expansion of given functions and using them to approximate integrals and evaluate limits.

Find the Maclaurin series expansion of the function $f(x) = \cos(x^2)$ up to and including the term in $x^4$ .

Find the Maclaurin series for the function $f(x) = \ln(1 + x)$ up to and including the term in .

In physics, the displacement of a particle in a damped oscillatory system is modeled by $h(t)=e^{-t} \sin (2 t)$ , where is time in seconds (radians are assumed for trigonometric functions).

The function $y$ satisfies the differential equation $\frac{dy}{dx} = x + \cos y$ , with when .

Find the Maclaurin series expansion of the function $f(x)=\ln(1+x^2)$ up to and including the term in .

Using the result from part 1, find an approximate value for $\int_0^1 \cos(x^2) \, dx$ .

Find the Maclaurin series for the function $g(x) = e^{x^2} \sin(x^2)$ up to and including the term in $x^6$ .

Using the Maclaurin series found in part 3, $\lim_{x \to 0} \frac{e^{x^2} \sin(x^2) - x^2 - x^4}{x^6}$ .

Using the result from part 1, find an approximate value for $\int_0^1 \ln(1 + x) \, dx$ .

Find $h^{\prime}(t)$ using first principles.

Show that $h(t)$ satisfies the differential equation $h^{\prime \prime}(t)+2 h^{\prime}(t)+5 h(t)=0$ .

Find the Maclaurin series for $h(t)$ up to the $t^{5}$ term.

the coordinates of the first positive maximum of $h(t)$ .

Show that $\frac{d^{2}y}{dx^{2}} = 1 - \sin y\,\frac{dy}{dx}$ , and find the values of $\frac{d^{2}y}{dx^{2}}$ , $\frac{d^{3}y}{dx^{3}}$ and $\frac{d^{4}y}{dx^{4}}$ at $x = 0$ .

Hence find the Maclaurin polynomial of $y$ of degree $4$ , and use it to $y$ when $x = 0.4$ .

Use Euler's method with a step length of $0.1$ to $y$ when $x = 0.4$ , giving your answer correct to five decimal places.

Use Euler's method with a step length of $0.05$ to $y$ when $x = 0.4$ , and find the difference between each of the two Euler estimates and their answer to part (b).

The initial condition is now replaced by $y = c$ when $x = 0$ , where $0 \lt c \lt \pi$ . Find the exact value of for which takes its least value at , and find that least value.

Using the result from part 1, find an approximate value for $\int_0^{1/2} \ln(1+x^2)\,dx$ .

IB Mathematics Analysis and Approaches AHL 5.19—maclaurin Series Questionbank

IB Mathematics Analysis and Approaches AHL 5.19—maclaurin Series Questionbank

IB Mathematics Analysis and Approaches AHL 5.19—maclaurin Series Questionbank

IB Mathematics Analysis and Approaches AHL 5.19—maclaurin Series Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 5.19—maclaurin Series Questionbank