Further Differentiation (DP IB Applications & Interpretation (AI): HL): Exam Questions

A particle P moves in a straight line. Its velocity, $v$ m s^-1, at time $t$ seconds satisfies

$\frac{\mathrm{d}v}{\mathrm{d}t}=\sin t\cos 4t$

for $0\le t\le 2$.

Sketch the graph of $\frac{\mathrm{d}v}{\mathrm{d}t}$ against $t$ on the following grid.

Find the values of $t$ at which the displacement-time graph of P has a point of inflexion, and explain the significance of these points for the motion of P.

Hence find the values of $t$ for which the displacement-time graph of P is concave down.

Use the product rule to find the derivative of $f\left(x\right)=(3x-7)(4-2x^2)$

Use the quotient rule to find the derivative of $g\left(x\right)=\frac{-7x}{x^3-1}$

Use the chain rule to find the derivative of $h\left(x\right)=(5-3x{)}^5$

Find the derivative of   $j\left(x\right)=\frac{3x^{{\displaystyle \frac{1}{5}}}}{1-x^{{\displaystyle \frac{2}{3}}}}$

Find an expression for the derivative of each of the following functions:

$f\left(x\right)=e^{3x}\tan x$

$g\left(x\right)=sin(3x^2+5)$

$h\left(x\right)=\frac{-{\cos }^{2}x}{\ln x}$

$j\left(x\right)=x\sqrt{x}-\frac{1}{\sqrt[3]{x}}$

Consider the function $f$ defined by $f\left(x\right)=2x+{\cos }^{3}x$, $x\in \mathrm{ℝ}$.

By considering the derivative of the function, show that $f$ is increasing for all $x$.

Let $f\left(x\right)=g\left(x\right)h\left(x\right)$, where $g$ and $h$ are functions such that $g\left(x\right)=3x^{2}h\left(x\right)$ for all $x\in \mathrm{ℝ}$.

Given that $h\left(-1\right)=2$ and $h\text{'}\left(-1\right)=-2$, find the equation of the tangent to the graph of $f$ at $x=-1$.

Consider the curve with equation $y=e^{x^3}$ ,  defined for all values of $x\in \mathrm{ℝ}$ .

Find an expression for $\frac{d^{2}y}{d{x}^2}$.

Hence determine the values of $x$for which the curve is

(i)     concave up

(ii)    concave down.

Your answers should be given as exact values.

Use your answer to part (b) to show that the curve has two points of inflection, and determine the exact values of their coordinates.

Consider the function $f$ defined by $f\left(x\right)=x{\mathrm{e}}^{3\cos x}$, for $-\pi \le x\le \pi$.

Find the number of points on the graph of $f$ at which the tangent is horizontal.

The point $\mathrm{A}$ is the point on the graph of $f$ for which the $x$-coordinate is $\frac{\pi }{2}$.

Show that the gradient of the tangent to the graph of $f$ at $\mathrm{A}$ is $\frac{2-3\pi }{2}$.

Hence find the equation of the normal to the graph of $f$ at $\mathrm{A}$, and find the coordinates of the point where the normal meets the $x$-axis.

Show algebraically that the graph of $f$ meets the line $y=x$ at exactly three points, and find the coordinates of these points.

Let   $f\left(x\right)=\frac{\sqrt{3}}{2}\cos 2x$  and  $g\left(x\right)=\frac{1}{2}\sin 2x$ ,  for $0\le x\le \pi .$

Solve the equation $f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right)$  ,  giving your answers as exact values.

An ice sculptor has created an abstract minimalist ice sculpture in the shape of a cylinder with radius $\mathrm{r}\mathrm{m}$ and height $8r\mathrm{m}$.  The sculpture is of solid ice throughout.

After a power cut that shuts off the sculptor’s freezer, the sculpture begins melting such that the volume of ice is decreasing at a constant rate of $0.4{\mathrm{m}}^3$  per hour.

Assuming that while it melts the sculpture remains at all times in the shape of a cylinder which is mathematically similar to the original cylinder, find the rate at which the sculpture’s surface area is changing at the point when its radius is $0.3\mathrm{m}$. 

A hemispherical bowl is supported with its curved surface on the bottom,  such that the plane defined by the open top of the bowl is at all times horizontal.  The bowl contains liquid, with the volume of liquid in the bowl being given by the formula

$V=\frac{1}{3}\pi h^2(3r-h)$

where $r$ is the radius of the bowl and $h$ is the depth of liquid (i.e., the height between the bottom of the bowl and the surface level of the liquid). 

The bowl is leaking liquid through a small hole in its bottom at a rate directly proportional to the depth of liquid.

Show that the rate of change of the depth of liquid in the bowl is

$-\frac{k}{\mathrm{\pi }\left(2\mathrm{r}-\mathrm{h}\right)}$

where $k$ is a positive constant.

Consider the function $g$ defined by $g\left(x\right)={\mathrm{e}}^{2x}-2x$, $x\in \mathrm{ℝ}$.

Point $A$ is the point on the graph of $g$ for which the $x$-coordinate is $\ln \sqrt{3}$.

Find the equation of the tangent to the graph of $g$ at point $A$.

Point $B$ is the point on the graph of $g$ at which the normal to the graph is vertical.

Show that the coordinates of the point of intersection between the tangent to the graph of $g$ at point $A$ and the tangent to the graph of $g$ at point $B$ are

$\left(\frac{3\ln 3-2}{4},1\right)$

Let $f\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}$, where $g$ and $h$ are differentiable functions with $h\left(x\right)\ne 0$ anywhere on the common domain of $g$ and $h$.

By first writing $f\left(x\right)=g\left(x\right){\left[h\left(x\right)\right]}^{-1}$, use the product and chain rules to show that

$f\text{'}\left(x\right)=\frac{h\left(x\right)g\text{'}\left(x\right)-g\left(x\right)h\text{'}\left(x\right)}{{\left[h\left(x\right)\right]}^2}$

Consider the function $h$ defined by $h\left(x\right)=\sin 3x+{\mathrm{e}}^{3\sqrt{3}x}\cos 3x$, $x\in \mathrm{ℝ}$.

Show that the normal to the graph of $h$ at $x=\frac{\pi }{9}$ meets the $y$-axis at the point

$\left(0,\frac{2\pi }{27}+\frac{\sqrt{3}+{\mathrm{e}}^{\frac{\pi \sqrt{3}}{3}}}{2}\right)$

Find an expression for the derivative of each of the following functions:

$f\left(x\right)=\left(12x^2-7\right)e^{-2x}$

$g\left(x\right)=\frac{\tan 3x}{4-5x^3}$

$h\left(x\right)={\left(\ln \left(2x^2-x-2\right)\right)}^5$

$j\left(x\right)=\frac{2x^{-\frac{3}{4}}}{1-x^{\frac{3}{5}}}$

Find an expression for the derivative of each of the following functions:

$f\left(x\right)=\left(3x-1\right)e^{\sin x}D$

$g\left(x\right)=\ln \left(\cos \left(x^2-1\right)\right)$

$h\left(x\right)=\frac{-\sin \left(e^{-x}\right)}{e^{x\cos x}}$

$j\left(x\right)=\tan \left(\frac{1}{x^2\sqrt[3]{x}}\right)$

Consider the function $f$ defined by $f\left(x\right)=-x+\frac{2}{3}{\sin }^{3}x$, $x\in \mathrm{ℝ}$.

Show that $f$ is decreasing for all $x$.

Let $f\left(x\right)=g\left(x\right)h\left(x\right)$, where $g$ and $h$ are real-valued functions such that

$g\left(x\right)=\ln \left(\frac{x}{3}\right)h\left(x\right)$

for all $x>0$.

Given that $h\left(3\right)=a$ and $h\text{'}\left(3\right)=b$, where $a\ne 0$, find the distance between the $y$-intercept of the tangent to the graph of $f$ at $x=3$ and the $y$-intercept of the normal to the graph of $f$ at $x=3$. Give your answer in terms of $a$ and/or $b$ as appropriate.

Consider the curve with equation $y=\cos \left(kx\right)e^{\sin \left(kx\right)}$ defined for all  $x\in \mathrm{ℝ}$, where  $k\ne 0$  is a positive integer.

For the case where $k=1$, find the number of points in the interval $-\frac{\mathrm{\pi }}{2}\le x<\frac{3\mathrm{\pi }}{2}$ at which the curve has a horizontal tangent.

(i) Show algebraically that in general the $x$-coordinates of the points at which the curve has horizontal tangents will be the solutions to the equation

${\sin }^2\left(kx\right)+\sin \left(kx\right)-1=0$

(ii) Hence, for the case where $k=1$, find the $x$-coordinates of the points identified in part (a).

(i) By considering $\frac{{\mathrm{d}}^{2}y}{d{x}^2}$, show algebraically that in general the $x$-coordinates of the points at which the curve is neither concave up nor concave down will be the solutions to the equation

 $\sin \left(kx\right)\cos \left(kx\right)=0$ 

(ii) Hence, for the case where $k=1$, find the -coordinates of the points of inflection on the curve in the interval $-\frac{\mathrm{\pi }}{2}\le x<\frac{3\mathrm{\pi }}{2}.$

In terms of $k$, state in general how many (i) turning points and (ii) points of inflection the curve will have in the interval $-\frac{\mathrm{\pi }}{2}\le x<\frac{3\mathrm{\pi }}{2}$.  Give a reason for your answers.

Consider the function $f$ defined by $f\left(x\right)=e^{x^k}$, $x\in \mathrm{ℝ}$,  where $k\ge 1$ is a positive integer.

Show that the graph of $f$ will have no points of inflection in the case where $k=1$.

Show that, for $k\ge 2$, the second derivative of is given by

            $f"\left(x\right)=k{x}^{k-2}\left(k{x}^k+k-1\right)e^{x^k}$

Hence show that the graph of $f$ will only have points of inflection in the case where $k$ is an odd integer greater than or equal to 3.  In that case, give the exact coordinates of the points of inflection, giving your answer in terms of $k$  where appropriate.  In your work you may use without proof the fact that for odd integers $k$ with $k\ge 3$

$-1<\sqrt[k]{-\frac{k-1}{k}}<-\frac{1}{2}$

A small conical flask, in the shape of a right cone stood on its flat base, is being filled with perfume via a small hole at its vertex.  The cone has a height of 6 cm and a radius of 2 cm. 

Perfume is being poured into the flask at a constant rate of 0.3 cm³s^-1.

Find the rate of change of the depth of the perfume in the flask at the instant when the flask is half full by volume.

A large block of ice is being prepared for use by a team of ice sculptors.  The block is in the shape of a cuboid with the ratio of its length to width to height being equal to  1 : 2 : 5. The block melts uniformly such that its surface area decreases at a constant rate, losing $k{m}^2$ of surface area every hour.  You may assume that as the block melts, its shape remains a cuboid with the dimensions in the same ratio to each other as in the original cuboid.

The block of ice is considered stable enough to be sculpted so long as the loss of volume due to melting does not exceed a rate $0.05{\mathrm{m}}^3$ per hour. 

Find, in terms of $k$, the volume of the largest block of ice that can be used for ice sculpting under such conditions.