DP Mathematics: Analysis and Approaches Questionbank

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

Calculate the area of cloth, in cm², needed to make Haruka’s bag.

Write down the $y$-intercept of the graph of $y=f\left(x\right)$.

Write down the number of possible solutions to the equation $f(x)=5$.

Sketch the graph of $y=f\left(x\right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.

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

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

Find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$.

Differentiate from first principles the function $f\left(x\right)=3x^3-x$.

In the context of the population model, interpret the meaning of $\frac{dP}{dt}$.

Write down the $x$-coordinates of these two points;

Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d are integers.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Determine the equation of the tangent to $C$ at the point $\left(\frac{2}{\text{e}},\text{ e}\right)$

Find the rate of change of the ball’s height above the ground when $t=\frac{1}{3}$. Give your answer in the form $p\mathrm{\pi }\sqrt{q} {\mathrm{ms}}^{-1}$ where $p\in \mathrm{\mathbb{Q}}$ and $q\in {\mathrm{\mathbb{Z}}}^{+}$.

Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.

Write down the coordinates of C, the midpoint of line segment AB.

Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.

Find the gradient of the line DC.

Write down the $y$-intercept of the graph.

Show that $a=8$.

Find $f^{\prime }(x)$.

Determine the range of $f\left(x\right)$ for $p$ ≤ $x$ ≤ $q$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find the value of this minimum area.

Use your answers to parts (c) and (d) to show that

$A=3x^2+\frac{10368}{x}$.

Write down a formula for $A$, the surface area to be coated.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Find $f(2)$.

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the range of $f(x)$.

Calculate the volume, in cm³, of the bag.

Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.

Find $\frac{\text{d}A}{\text{d}x}$.