IB Maths AI SL – Topic 5: Calculus

Find $\dfrac{dy}{dx}$ when $y = 3x^4 – 2x^2 + 5x – 7$.

The function $f$ is defined by $f(x) = 2x^3 + \dfrac{1}{x}$, $x eq 0$.(a) Find $f'(x)$.(b) Find the gradient of the curve at $x = 2$.

The curve $y = x^2 – 4x$ passes through the point $P(3, -3)$.(a) Find the gradient of the curve at $P$.(b) Hence find the equation of the tangent to the curve at $P$ in the form $y = mx + c$.

Find $\displaystyle\int (6x^2 – 4x + 3)\,dx$.

Evaluate $\displaystyle\int_1^3 (2x + 1)\,dx$.

A function has derivative $f'(x) = 4x – 3$ and passes through the point $(2, 5)$.Find $f(x)$.

Consider the curve $y = x^3 – 6x^2 + 9x + 1$.(a) Find $\dfrac{dy}{dx}$.(b) Find the coordinates of the two stationary points of the curve.(c) Determine the nature of each stationary point, justifying your answer.

Let $f(x) = \sqrt{x^2 + 1}$.(a) Complete a table of values of $f(x)$ for $x = 0, 0.5, 1, 1.5, 2$, giving values to 4 decimal places.(b) Use the trapezoidal rule with four intervals to estimate $\displaystyle\int_0^2 \sqrt{x^2+1}\,dx$.(c) Use your GDC to find the exact integral to 4 significant figures, and hence find the percentage error in your estimate.

A company sells $x$ units of a product per week. The weekly revenue is $R(x) = 80x – 0.5x^2$ dollars and the weekly cost is $C(x) = 20x + 500$ dollars.(a) Write down an expression for the weekly profit $P(x)$.(b) Find $\dfrac{dP}{dx}$.(c) Find the number of units that maximises the weekly profit, and the maximum profit.

The curve $y = 4x – x^2$ meets the $x$-axis at the origin and at the point $A$.(a) Find the $x$-coordinate of $A$.(b) Find the area of the region enclosed by the curve and the $x$-axis.

A cylindrical can with a closed top and bottom is to hold $500\,\text{cm}^3$ of drink. The radius of the can is $r$ cm and its height is $h$ cm.(a) Show that $h = \dfrac{500}{\pi r^2}$.(b) Show that the total surface area of the can is $S = 2\pi r^2 + \dfrac{1000}{r}$.(c) Find $\dfrac{dS}{dr}$.(d) Find the value of $r$ that minimises the surface area, and the corresponding minimum surface area.

A surveyor measures the depth of a river at 2 m intervals across its width. The depths (in metres) are $0,\ 1.2,\ 2.5,\ 3.1,\ 2.8,\ 1.6,\ 0$.(a) Use the trapezoidal rule to estimate the cross-sectional area of the river.(b) The water flows at an average speed of $0.5\,\text{m s}^{-1}$. Estimate the volume of water passing a point on the river each second.(c) Explain how the surveyor could obtain a more accurate estimate of the area.