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

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

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

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

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

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 everywhere on its domain.

Consider the function $g$ defined by $g\left(x\right)={\mathrm{e}}^x-7x,x\in \mathrm{ℝ}.$

Show that the equation of the tangent to the graph of $g$ at $x=\ln 3$ may be written in the form $y=-4x-3(\ln 3-1).$

Show that there is a point on the graph of $g$ at which the normal to the graph is vertical, and determine the coordinates of that point.

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

Find an expression for $h\text{'}\left(x\right).$

Hence determine an equation for the tangent to the graph of $h$ at  $x=\mathrm{\pi }.$.

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 $h(-1)=2$ and $h^{\text{'}}(-1)=-2,$ 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 an expression for $f^{\text{'}\text{'}}\left(x\right).$

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

(i) concave up

(ii) concave down.

Your answers should be given as exact values.

Hence show that the graph of $f$ 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 at which the graph of $f$ has a horizontal tangent.

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

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

Hence find the equation of the normal line to the graph of $f$ at point A, and determine where that line intersects the $x$-axis.

Show algebraically that the graph of $f$ intersects the line $y=x$ in exactly three places, and determine the coordinates of the points of intersection.

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 $f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right).$

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

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

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

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

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

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

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

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

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 everywhere on its domain.

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

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

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

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 line to the graph of $h$ at $x=\frac{\mathrm{\pi }}{9}$intercepts the $y$-axis at the point

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

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 $f\left(x\right)=\cos \left(kx\right){\mathrm{e}}^{\sin \left(kx\right)},$ 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 graph of $f$ has a horizontal tangent.

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

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

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 inflection on the graph of $f$ 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 graph of $f$ will have in the interval $-\frac{\mathrm{\pi }}{2}\le x<\frac{3\mathrm{\pi }}{2}.$ Give a reason for your answers.

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

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 $f$ defined by $f\left(x\right)={\mathrm{e}}^{x^k},x\in \mathrm{ℝ},$ 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 $f$ 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 $k\ge 2,$

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

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.