IB Maths AA HL – Topic 5: Calculus

The curve $C$ has equation $x^2 + xy + y^2 = 7$. Find the gradient of $C$ at the point $(1, 2)$.

Find $\displaystyle\int x\,e^{2x}\,dx$.

Using the substitution $u = 1 + x^2$, or otherwise, find the exact value of $\displaystyle\int_0^1 x\sqrt{1 + x^2}\,dx$.

Find $\displaystyle\int \frac{3x + 1}{(x – 1)(x + 2)}\,dx$.

Find the Maclaurin series of $e^x \sin x$ up to and including the term in $x^3$.

Solve the differential equation $\dfrac{dy}{dx} = xy$, given that $y = 2$ when $x = 0$. Give your answer in the form $y = f(x)$.

A spherical balloon is inflated so that its radius increases at a constant rate of $0.5\,\text{cm s}^{-1}$.(a) Find the rate of increase of the volume when the radius is $4$ cm.(b) Find the rate of increase of the surface area at the same instant.

A ladder of length $5$ m rests against a vertical wall. The foot of the ladder slides away from the wall at $0.3\,\text{m s}^{-1}$. Find the rate at which the top of the ladder is sliding down the wall when the foot is $3$ m from the wall.

Consider the differential equation $\dfrac{dy}{dx} + \dfrac{2y}{x} = x$, $x > 0$, with $y = 1$ when $x = 1$.(a) Find the integrating factor.(b) Hence find $y$ in terms of $x$.

Consider $\dfrac{dy}{dx} = x + y$ with $y(0) = 1$.(a) Use Euler’s method with step length $h = 0.1$ to estimate $y(0.2)$.(b) The exact solution is $y = 2e^x – x – 1$. Find the percentage error in your estimate.

(a) Find the Maclaurin series of $\ln(1 + 2x)$ up to and including the term in $x^3$.(b) Use it to find an approximation for $\ln 1.2$, and compare with the calculator value.(c) Use l’Hôpital’s rule to find $\displaystyle\lim_{x \to 0}\frac{e^x – 1 – x}{x^2}$.

Consider $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, $x > 0$, $y > 0$, with $y = 2$ when $x = 1$.(a) Use the substitution $y = vx$ to show that $v\dfrac{dv}{dx} = \dfrac{1}{x}$.(b) Hence find $y^2$ in terms of $x$.