DP Mathematics: Applications and Interpretation Questionbank

Identify the $x$ value of the point where $|h\prime (x\left)\right|$ has its maximum value.

Interpret this point in the given context.

Juri starts at a height of $60$ metres and finishes at $\text{F}$, where $x=f$.

Sketch a possible diagram of the hill on the following pair of coordinate axes.

Determine the value of $a$ and of $b$. Give your answers correct to one decimal place.

Given that $\underset{T\to \infty }{\lim }k=A$ and $\underset{T\to 0}{\lim }k=0$, sketch the graph of $k$ against $T$.

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

Determine the values of $x$ for which $f(x)$ is a decreasing function.

Sketch the graph of $y=f(x)$ showing clearly the minimum point and any asymptotic behaviour.

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y=-x$.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

Determine an expression for $f^{\prime }(x)$ in terms of $x$.

Sketch a graph of $y=f^{\prime }(x)$ for .

Find the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y=f(x)$, labelling these clearly on the graph of $y=f^{\prime }(x)$.

Express $\sin x$ in terms of $\mu$.

Express $\sin 2x$ in terms of $u$.

Hence show that $f(x)=0$ can be expressed as $u^3-7u^2+15u-9=0$.

Solve the equation $f(x)=0$, giving your answers in the form $\arctan k$ where $k\in \mathbb{Z}$.

Show that there are exactly two points on the curve where the gradient is zero.

Find the equation of the normal to the curve at the point P.

The normal at P cuts the curve again at the point Q. Find the $x$-coordinate of Q.

The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.

Show that $\frac{\text{d}y}{\text{d}x}=-\left(\frac{1+y\text{sin}\left(xy\right)}{1+x\text{sin}\left(xy\right)}\right)$.

Find the coordinates of P and Q.

Given that the gradients of the tangents to C at P and Q are m₁ and m₂ respectively, show that m₁ × m₂ = 1.

Find the coordinates of the three points on C, nearest the origin, where the tangent is parallel to the line $y=-x$.

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

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

A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{\text{d}\theta }{\text{d}t}$, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .

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

Find the equations of the tangents to this curve at the points where the curve intersects the line $x=1$.

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

Show that $k=6$.

Find the equation of the tangent to the graph of $y=g(x)$ at $x=2$. Give your answer in the form $y=mx+c$.

Use your answer to part (a) and the value of $k$, to find the $x$-coordinates of the stationary points of the graph of $y=g(x)$.

Find $g^{\prime }(-1)$.

Hence justify that $g$ is decreasing at $x=-1$.

Find the $y$-coordinate of the local minimum.

Find f'(x)

Find the gradient of the graph of f at $x=-\frac{1}{2}$.

Find the x-coordinate of the point at which the normal to the graph of f has gradient $-\frac{1}{8}$.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.

Find $\frac{\text{dy}}{\text{dx}}$.

Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.

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

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

Show that $a=8$.

Find $f(2)$.

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

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

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

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

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

Write down the derivative of $f$.

Find the point on the graph of $f$ at which the gradient of the tangent is equal to 6.

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

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.

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

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

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

Find the value of this minimum area.

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

Show that $k=-6$.

Find the coordinates of the local minimum.

Write down the interval where the gradient of the graph of $f\left(x\right)$ is negative.

Determine the equation of the normal at $x=-2$ in the form $y=mx+c$.

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

Find the gradient of the line DC.

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

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

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.

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

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

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

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

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

The cloth used to make Nanako’s bag costs 4 Japanese Yen (JPY) per cm².

Find the cost of the cloth used to make Nanako’s bag.

Find $f^{\prime }\left(x\right)$.

Find the gradient of the graph of $y=f\left(x\right)$ at $x=2$.

Find the equation of the tangent line to the graph of $y=f\left(x\right)$ at $x=2$. Give the equation in the form $ax+by+d=0$ where, $a$, $b$, and $d\in \mathbb{Z}$.