DP Mathematics: Analysis and Approaches Questionbank

Sketch the graph of $f$, showing clearly the value of the $x$-intercept and the approximate position of point $\text{P}$.

$c=1$.

$c=2$.

Use your graphic display calculator to explore the graph of $y=f_n\left(x\right)$ for

•   the odd values $n=3$ and $n=5$;

•   the even values $n=2$ and $n=4$.

Hence, copy and complete the following table.

Sketch the graph of $y=f_1\left(x\right)$, stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.

Find the values of $x$ for which $f\left(x\right)=0$.

Sketch the graph of $f$ on the following grid.

Show that ${\left(f\left(t\right)\right)}^2+{\left(g\left(t\right)\right)}^2=f\left(2t\right)$.

$f\left(\text{i}u\right)$.

$g\left(\text{i}u\right)$.

Verify that $u=f\left(t\right)$ satisfies the differential equation $\frac{d^{2}u}{d{t}^2}=u$.

The hyperbola with equation $x^2-y^2=1$ can be rotated to coincide with the curve defined by $xy=k, k\in \mathrm{ℝ}$.

Find the possible values of $k$.

Show that ${\left(f\left(t\right)\right)}^2-{\left(g\left(t\right)\right)}^2={\left(f\left(\text{i}u\right)\right)}^2-{\left(g\left(\text{i}u\right)\right)}^2$.

Hence find, and simplify, an expression for ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.

Sketch the graph of $x^2-y^2=1$, stating the coordinates of any axis intercepts and the equation of each asymptote.

For $a=r$ and $b>0$, state in terms of $r$, the coordinates of points $\text{P}$ and $\text{A}$.

$y^2=x^3, x\ge 0$

$y^2=x^3+1, x\ge -1$

Sketch the curve $y=\left(x-r\right)\left(x^2-2ax+a^2+b^2\right)$ for $a=r=1$ and $b=2$.

Sketch the curve $y=f\left(x\right)$, clearly indicating the coordinates of the endpoints.

Sketch the graph of $f″$, the second derivative of $f$. Indicate clearly the $x$-intercept and the $y$-intercept.

Hence, solve the inequality $0<\frac{2x-1}{x+1}<2$.

Write down the equation of the horizontal asymptote.

On the set of axes below, sketch the graph of $y=f\left(x\right)$.

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

On the set of axes below, sketch the graph of $y=f\left(x\right)$.

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

Write down the equation of the vertical asymptote.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

Plant $A$ correct to three significant figures.

Plant $B$.

Expand the expression for $f(x)$.

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

Draw the graph of $f$ for $-3⩽x⩽3$ and $-40⩽y⩽20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

Write down the coordinates of the point of intersection.

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.

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 $p$ ≤ $x$ ≤ $q$.

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 the value of $b$.

The graph of $f\left(x\right)=x^2$ is transformed to obtain the graph of $g$.

Describe this transformation.

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

Sketch the graphs of $y=\frac{x}{2}+1$ and $y=\left|x-2\right|$ on the following axes.

Sketch the graphs $y=\text{si}{\text{n}}^{3}x+\text{ln}x$ and $y=1+\text{cos}x$ on the following axes for 0 < $x$ ≤ 9.

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y=f\left(x\right)$.

Hence, or otherwise, solve the inequality $f\left(x\right)>f^{-1}\left(x\right)$.

sketch the graph of $y=f\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

Sketch the graph of $y=f\left(x\right)$ for $-\frac{5\pi }{8}⩽x⩽\frac{\pi }{8}$.

sketch the graph of $y=f^{-1}\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.