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 $x^4$ .

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 .

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

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).

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

Determine the first three terms of the Maclaurin series expansion for $(1-2x)^{-\frac{1}{2}}$ in ascending powers of .

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.

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

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

Use the substitution $x=\frac{2}{25}$ to find a rational approximation for $\frac{5}{\sqrt{21}}$ .

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