DP Mathematics: Analysis and Approaches Questionbank

Show that $\frac{\text{d}y}{\text{d}x}=\frac{y \cos \left(xy\right)}{2y-x \cos \left(xy\right)}$.

Prove that, when $\frac{\text{d}y}{\text{d}x}=0 , y=\pm 1$.

Hence find the coordinates of all points on $C$, for $0<x<4\mathrm{\pi }$, where $\frac{\text{d}y}{\text{d}x}=0$.

Find the equation of the tangent to the curve $y={\text{e}}^{2x}–3x$ at the point where $x=0$.

Find $f\text{'}\left(p\right)$ in terms of $k$ and $p$.

Show that the equation of $L_1$ is $kx+p^{2}y-2pk=0$.

Find the area of triangle $\text{AOB}$ in terms of $k$.

The graph of $f$ is translated by $\left(\begin{array}{c}4\\ 3\end{array}\right)$ to give the graph of $g$.
In the following diagram:

• point $\text{Q}$ lies on the graph of $g$
• points $\text{C}$, $\text{D}$ and $\text{E}$ lie on the vertical asymptote of $g$
• points $\text{D}$ and $\text{F}$ lie on the horizontal asymptote of $g$
• point $\text{G}$ lies on the $x$-axis such that $\text{FG}$ is parallel to $\text{DC}$.

Line $L_2$ is the tangent to the graph of $g$ at $\text{Q}$, and passes through $\text{E}$ and $\text{F}$.

Given that triangle $\text{EDF}$ and rectangle $\text{CDFG}$ have equal areas, find the gradient of $L_2$ in terms of $p$.

Write down $f\prime \left(x\right)$.

Write down the gradient of this tangent.

Find the value of $k$.

Show that $\frac{dy}{dx}=\frac{3x^2}{2{\text{e}}^{2y}-1}$.

The tangent to $C$ at the point Ρ is parallel to the $y$-axis.

Find the $x$-coordinate of Ρ.

Hence, show that $k=\text{e}+\frac{{\text{e}}^2}{4}$.

Show that the equation of $L$ is $y=-x+2 \ln 45+2$.

Show that $h=\frac{\text{e}+6}{2}$.

Find the exact coordinates of $\text{Q}$.

Find $f\text{'}\left(x\right)$.

Given that the gradient of $L$ is $\frac{1}{3}$, find the $x$-coordinate of $\text{B}$.

Write down the value of $f\prime \left(4\right)$.

Find $f\left(4\right)$.

Find $h\left(4\right)$.

Hence find the equation of the tangent to the graph of $h$ at $x=4$.

Hence find the equation of the tangent to $C$ at the point where $x=1$.

Show that $\frac{dy}{dx}+\left(x\frac{{\displaystyle dy}}{{\displaystyle dx}}+y\right)\left(1+\ln \left(xy\right)\right)=1$.

Show that the line $y=x-1$ is tangent to the curve $y=f\left(x\right)$ at the point $\text{A}\left(4, 3\right)$.

Hence, or otherwise, prove that the tangent to the curve $y=g\left(x\right)$ at the point $\text{A}\left(a, g\left(a\right)\right)$ intersects the $x$-axis at the point $\text{R}\left(r, 0\right)$.

Sketch the curve $y=f\left(x\right)$ and the tangent to the curve at point $\text{A}$, clearly showing where the tangent crosses the $x$-axis.

Find the equation of the tangent to $C$ at $\text{P}$.

Consider the curve with equation $y=(2x-1){\text{e}}^{kx}$, where $x\in \mathrm{ℝ}$ and $k\in \mathrm{\mathbb{Q}}$.

The tangent to the curve at the point where $x=1$ is parallel to the line $y=5{\text{e}}^{k}x$.

Find the value of $k$.

Find the equation of $L$.

Hence explain why the graph of $f$ has a local maximum point at $x=a$.

Find $f^{″}\left(b\right)$.

Sketch the graph of $y=f^{\prime }\left(x\right)$.

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

The normal to the graph of $f$ at $x=a$ and the tangent to the graph of $f$ at $x=b$ intersect at the point ($p$, $q$) .

Find the value of $p$ and the value of $q$.

Find the value of $a$ and the value of $b$.

Hence, use your answer to part (d)(i) to show that the graph of $f$ has a local minimum point at $x=b$.

The line $y=kx-5$ is a tangent to the curve of $f$. Find the values of $k$.

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.

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 $\left(\frac{2}{\text{e}},\text{ e}\right)$

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.

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

The graph of $f$ has a horizontal tangent line at $x=0$ and at $x=a$. Find $a$.

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

Find the coordinates of P.

Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.

Write down the value of $f(1)$.

Find the equation of $N$. Give your answer in the form $ax+by+d=0$ where $a$, $b$, $d\in \mathbb{Z}$.

Draw the line $N$ on the diagram above.

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 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.

Find the equation of this tangent. Give your answer in the form y = mx + c.

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

Sketch the graph of the function g (x) = 10x + 40 on the same axes.

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

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

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

Using implicit differentiation, find an expression for $\frac{\text{d}y}{\text{d}x}$.

Find the equation of the tangent to the curve at the point $\left(\frac{1}{4}\text{, }\frac{5\pi }{6}\right)$.

Write down the coordinates of $\text{B}$.

Given that $f^{\prime }\left(a\right)=\frac{1}{\text{ln}p}$, find the equation of $L_1$ in terms of $x$, $p$ and $q$.

The line $L_2$ is tangent to the graph of $g$ at $\text{A}$ and has equation $y=\left(\text{ln}p\right)x+q+1$.

The line $L_2$ passes through the point $\left(-2\text{, }-2\right)$.

The gradient of the normal to $g$ at $\text{A}$ is $\frac{1}{\text{ln}\left(\frac{1}{3}\right)}$.

Find the equation of $L_1$ in terms of $x$.

Find $h\left(x\right)$.

Let $\text{C}$ be a point on the graph of $h$. The tangent to the graph of $h$ at $\text{C}$ is parallel to the graph of $f$.

Find the $x$-coordinate of $\text{C}$.

Line $L$ passes through the origin and has a gradient of $\text{tan}\theta$. Find the equation of $L$.

At the point (1, 1) , show that $\frac{\text{d}y}{\text{d}x}=\frac{2+\pi }{2-\pi }$.

Hence, write down $f^{-1}\left(x\right)$.

Find $\left(f\circ f\right)\left(x\right)$.

Find the acute angle between $y=x$ and $L$.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

Find the value of $\text{tan}\theta$.

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.

Find the gradient of $L$.

Hence find the equation of the normal to $C$ at the point (1, 1).

Find u.

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

Determine whether $f_n$ is an odd or even function, justifying your answer.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

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