IB Maths AA SL – Topic 5: Calculus

Let $f(x) = 3x^4 – 2x^2 + 5x – 1$. Find $f'(x)$.

The curve $C$ has equation $y = x^3 – 4x$.(a) Find $\dfrac{dy}{dx}$.(b) Find the equation of the tangent to $C$ at the point where $x = 2$.

Find $\displaystyle\int \left(6x^2 – \frac{4}{x^2} + 3\right)\,dx$.

Differentiate with respect to $x$:(a) $y = (2x^2 – 3)^5$(b) $y = x^2 \ln x$

Evaluate $\displaystyle\int_1^4 \left(2x + \frac{1}{\sqrt{x}}\right)dx$.

The function $f$ is defined by $f(x) = x^3 – 6x^2 + 9x + 1$.(a) Find $f'(x)$ and hence find the coordinates of the two stationary points of $f$.(b) Use the second derivative to determine the nature of each stationary point.

A particle moves in a straight line. Its velocity at time $t$ seconds is $v(t) = 6t – t^2\,\text{m s}^{-1}$ for $0 \le t \le 8$. The particle starts at the origin.(a) Find the acceleration of the particle when $t = 2$.(b) Find the displacement of the particle from the origin when $t = 6$.(c) Find the total distance travelled by the particle in the first 8 seconds.

The curve $y = x^2$ and the line $y = 2x + 3$ intersect at two points $P$ and $Q$.(a) Find the $x$-coordinates of $P$ and $Q$.(b) Find the exact area of the region enclosed between the line and the curve.

The curve $C$ has equation $y = \dfrac{2x+1}{x-2}$, $x e 2$.(a) Show that $\dfrac{dy}{dx} = \dfrac{-5}{(x-2)^2}$.(b) Find the equation of the normal to $C$ at the point where $x = 3$. Give your answer in the form $ax + by + c = 0$ where $a, b, c \in \mathbb{Z}$.

An open box is made from a square sheet of card of side 30 cm by cutting a square of side $x$ cm from each corner and folding up the sides.(a) Show that the volume of the box is $V = 4x^3 – 120x^2 + 900x$.(b) Find $\dfrac{dV}{dx}$ and hence find the value of $x$ that gives the maximum volume.(c) Find the maximum volume.

Using the substitution $u = x^2 + 1$, or otherwise, find the exact value of $\displaystyle\int_0^{\sqrt{3}} x\sqrt{x^2+1}\,dx$.

Let $f(x) = x e^{-x}$ for $x \ge 0$.(a) Show that $f'(x) = e^{-x}(1 – x)$.(b) Find the coordinates of the maximum point of the graph of $f$.(c) Show that $f”(x) = e^{-x}(x – 2)$ and hence find the $x$-coordinate of the point of inflexion.(d) Find the equation of the tangent to the graph of $f$ at the point where $x = 2$. Give your answer in the form $y = mx + c$, with $m$ and $c$ in exact form.