IB Math AA HL

Consider the function f(x) = $x^3$ $\ln$(x), for x > 0.

(a) Find f'(x). [3]

(b) Find the exact x-coordinate of the stationary point of f(x). [3]

(a) Evaluate $\int$ u $\sin$ u du. [4 marks]

(b) By using an appropriate substitution and the result from part (a), evaluate $\int_0^1$ $\frac{x \arcsin x}{\sqrt{1-x^2}}$ dx. Express your answer in the form A + B$\pi$, where A, B $\in$ $\mathbb{Q}$. [6 marks]

Let f(x) = $\ln$($\cos$ x).

(a) Write down the Maclaurin series for $\cos$ x up to and including the term in $x^4$. (1 mark)

(b) Using your answer from part (a) and the Maclaurin series for $\ln$(1+u) = u - $\frac{u^2}{2} + \frac{u^3}{3} - \dots$, find the first three non-zero terms of the Maclaurin series for f(x) = $\ln$($\cos$ x). (5 marks)

(c) Let g(x) = x f'(x). Find the first two non-zero terms of the Maclaurin series for g(x). (3 marks)

A particle moves in a straight line such that its velocity, v m/s, at time t seconds, for t $\geq$ 0, is given by v(t) = (t-1)(t-3)(t-5).

(a) Find the initial velocity of the particle.

(b) Determine the exact times when the particle is momentarily at rest.

(c) Find the acceleration of the particle when t=2 seconds.

(d) Calculate the total distance traveled by the particle in the interval 0 $\leq$ t $\leq$ 3 seconds.

Consider the differential equation $\frac{dy}{dx} = \frac{y^2 \ln x}{x(y-1)}$ for x>0 and y $\ne$ 0, y $\ne$ 1.

(a) Show that the general solution of the differential equation is $\frac{1}{y} + \ln$ y = $\frac{1}{2}(\ln x)^2$ + C, where C is an arbitrary constant.

(b) Find the particular solution given that y(e) = -1.

(c) Explain why y must be negative for the particular solution found in part (b).