Calculus: 196 IB Maths AA HL practice questions

Consider two functions $f$ and $g$ and their derivatives $f'$ and $g'$. The following table shows the values for the two functions and their derivatives at $x = 1, 2$ and $3$.

Given that $p(x) = f(x) g(x)$ and $h(x) = g \circ f(x)$, find

(a) $p'(3)$;[2]

(b) $h'(2)$.[4]

Find the equation of the tangent to the curve $y = e^{2x} - 3x$ at the point where $x = 0$.[5]

Given that the graph of $y = x^3 - 6x^2 + kx - 4$ has exactly one point at which the gradient is zero, find the value of $k$.[5]

(a) Show that $3\cos 2x + 11\sin x = 3 + 11\sin x - 6\sin^2 x$.[2]

(b) Hence, or otherwise, solve the equation $3\cos 2x + 11\sin x - 6 = 0$ for $0° \le x \le 180°$.[3]

Solve the equations

$\ln \frac{x}{y} = 1$[2]

$\ln x^3 + \ln y^2 = 5$[3]

Given that $\frac{dy}{dx} = \cos\left(x - \frac{\pi}{4}\right)$ and $y = 2$ when $x = \frac{3\pi}{4}$, find $y$ in terms of $x$.[4]

Find the exact value of $\int_1^2 \left((x-2)^2 + \frac{1}{x} + \sin \pi x\right) dx$.[6]

(a) Expand $(x + h)^3$.[2]

(b) Hence find the derivative of $f(x) = x^3$ from first principles.[3]

Find the area completely enclosed by the curves $y = e^x$, $y = -e^x$, and the lines $x = -1$ and $x = 1$.[4]

Let $y = \sin^2 \theta$, $0 \le \theta \le \pi$.

(a) Find $\frac{dy}{d\theta}$.[2]

(b) Hence find the values of $\theta$ for which $\frac{dy}{d\theta} = 2y$.[5]

(a) Find $\int (1 + \tan^2 x) dx$[2]

(b) Find $\int \sin^2 x dx$[3]

Use l'Hôpital's rule to find $$\lim_{x \to 0}\left(\frac{\arctan 2x}{\tan 3x}\right)$$[5]

The loudness of a sound, $L$, measured in decibels, is related to its intensity, $I$ units, by $L = 10\log_{10}(I \times 10^{12})$.

Consider two sounds, $S_1$ and $S_2$.

$S_1$ has an intensity of $10^{-6}$ units and a loudness of 60 decibels.

$S_2$ has an intensity that is twice that of $S_1$.

(a) State the intensity of $S_2$.[1]

(b) Determine the loudness of $S_2$.[2]

The maximum loudness of thunder in a thunderstorm was measured to be 115 decibels.

(c) Find the corresponding intensity, $I$, of the thunder.[3]

Consider the function $f(x) = \frac{4x^3}{3} - 16x$, where $x \in \mathbb{R}$.

The graph of $y = f(x)$ has a local minimum point at $(p, q)$ where $p > 0$.

Find the value of $p$ and the value of $q$.[5]

Let $l$ be the tangent to the curve $y = xe^{2x}$ at the point $(1, e^2)$.

Find the coordinates of the point where $l$ meets the $x$-axis.[4]