IB Mathematics Analysis and Approaches Topic Calculus…

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, $\lim_{x \to 0} \frac{e^{x^2} \sin(x^2) - x^2 - x^4}{x^6}$ .

Let $g(x)=\frac{\sin(2x)}{2}\left(\frac{1}{\tan(x)}+2\right)-\sin^2(x)$ .

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

This question explores integration using substitution and integration by parts.

the integral $\int x e^{2x} \, dx$ using integration by parts.

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).

Show that $g(x)=\sin(2x)+\cos(2x)$ .

Show that $g''(x)=-4g(x)$ .

Hence, using your answer from Part 2, find $\int\!g(x)\,dx$ in terms of $g'(x)$ .

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

the integral $\int \frac{\ln(x)}{x^2} \, dx$ using integration by parts.

the integral $\int \frac{1}{x \ln(x)} \, dx$ using the substitution $u = \ln(x)$ .

the integral $\int x \sin(x) \, dx$ using integration by parts.

the integral $\int e^{x} \cos(x) \, dx$ using integration by parts twice.

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 $h(t)$ .

IB Mathematics Analysis and Approaches Calculus Questionbank

IB Mathematics Analysis and Approaches Calculus Questionbank

IB Mathematics Analysis and Approaches Calculus Questionbank

IB Mathematics Analysis and Approaches Calculus Questionbank

IB Mathematics Analysis and Approaches (AA) Topic Calculus Questionbank