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.

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

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

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

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

Let $g(x)=\frac{\sin(2x)}{2}\left(\frac{1}{\tan(x)}+2\right)-\sin^2(x)$ . The domain is $x\in\mathbb R\setminus\{n\pi/2:n\in\mathbb Z\}$ .

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

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

This question explores integration using substitution and integration by parts.

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

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

(d) Evaluate the integral $\int x \sin(x) \, dx$ 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 $t$ is time in seconds (radians are assumed for trigonometric functions).

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

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

Let $f(x)=x^{3}-3x^{2}+kx+2$ , where $k$ is a constant. The function has a point of inflection at $x=1$ and this point lies on the $x$ -axis.

(a) Find the value of $k$ .

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

(b) the nature of the stationary points of $f(x)$ for this value of $k$ .

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