Techniques & Applications of Differentiation (DP IB Analysis & Approaches (AA): HL): Exam Questions

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 $x=-1$.

Let $f$ be a function defined by $f\left(x\right)={\mathrm{e}}^{x^3}$, $x\in \mathrm{ℝ}$.

Find $f\text{'}\text{'}\left(x\right)$.

Find the values of $x$ for which the graph of $f$ is

(i) concave-up

(ii) concave-down.

Give your answers as exact values.

Hence show that the graph of $f$ has two points of inflexion, and find the exact coordinates of these points.

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 $g\left(x\right)=\sin x\cos x$, for $0\le x\le \pi$.

Solve the equation $f\text{'}\left(x\right)=g\text{'}\left(x\right)$.

The equation of a curve is $y=x^3+9x^2+24x+17$.

Show that the curve has exactly two points where the gradient is zero. Find the coordinates of each point, and determine whether it is a local maximum or a local minimum.

Show that the curve has exactly one point of inflexion, and find its coordinates.

Give an example of a curve with equation $y=a{x}^3+b{x}^2+cx+d$, where $a,b,c,d\in \mathrm{ℝ}$ and $a\ne 0$, that has a point of inflexion with zero gradient. Justify your answer.

A function $f$ is defined for $x>0$. The derivative of $f$ is given by $f\text{'}\left(x\right)=3x^2+\frac{2}{x^3}-25$.

Find $f\text{'}\text{'}\left(x\right)$.

The graph of $f$ is concave-up for $x>n$.

Find the smallest possible value of $n$.

Show that the curve $y=f\left(x\right)$ has only one point of inflexion, and find the gradient of the curve at that point.

Let $f\left(x\right)=-6x^3+7x^2+3x$, for $x\in \mathrm{ℝ}$.

The curve $y=f\left(x\right)$ meets the $x$-axis at the points $\mathrm{A}\left(a,0\right)$, $\mathrm{B}\left(b,0\right)$ and $\mathrm{C}\left(c,0\right)$, where $a<b<c$.

Find the values of $a$, $b$ and $c$.

The curve has a local minimum point at $\mathrm{D}$.

Show that the $x$-coordinate of $\mathrm{D}$ is $\frac{7-\sqrt{103}}{18}$. Justify that this point is a local minimum, and that it is the only local minimum point.

The curve has a point of inflexion at $\mathrm{E}$.

Find the gradient of the normal to the curve at $\mathrm{E}$.

A function $f$ is defined for $x\ne 0$. The derivative of $f$ is given by $f\text{'}\left(x\right)=48x^2+\frac{1}{x^3}-22$.

The graph of $f$ is concave-up for $x>n$.

Find the smallest possible value of $n$.

Show that the curve $y=f\left(x\right)$ has only one point of inflexion, and find the gradient of the normal to the curve at that point.

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

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

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

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

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

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

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

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

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.

A curve has equation $y=a{x}^3+b{x}^2+cx+d$, where $a,b,c,d\in \mathrm{ℝ}$ and $a\ne 0$.

(i) Show that the curve has points where the gradient is zero only if $b^2\ge 3ac$.

(ii) In the case where $b^2\ge 3ac$, find the $x$-coordinates of the points where the gradient is zero, giving your answer as simply as possible in terms of $a$, $b$ and $c$.

Show that the curve always has exactly one point of inflexion, and find its $x$-coordinate in terms of $a$ and $b$.

In the case where the gradient at the point of inflexion is zero, show that there are no other points on the curve where the gradient is zero.

In the case where there are two distinct points where the gradient is zero, show that the $x$-coordinate of the point of inflexion lies halfway between the $x$-coordinates of these two points.

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 function $f$ defined by $f\left(x\right)=\cos \left(kx\right){\mathrm{e}}^{\sin \left(kx\right)}$, $x\in \mathrm{ℝ}$, where $k$ is a positive integer.

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

(i) Show algebraically that in general the $x$-coordinates of the points at which the graph of $f$ 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) Show algebraically that in general the $x$-coordinates of the points at which the graph of $f$ is neither concave-up nor concave-down will be the solutions to the equation

$\sin \left(2kx\right)=0$

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

In terms of $k$, state in general how many (i) local maximum and minimum points and (ii) points of inflexion the graph of $f$ will have in the interval $-\frac{\pi }{2}\le x<\frac{3\pi }{2}$. Give a reason for your answers.

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

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

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

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

Explain why, for odd $k\ge 3$,

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

Hence show that the graph of $f$ will only have points of inflexion 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 inflexion, giving your answer in terms of $k$ where appropriate.