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

The equation of a curve is  $y=x-\frac{9}{x}+8$ for $x>0$ .

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

The gradient of the tangent to the curve at point $\mathrm{A}$ is $2$.

Find the coordinates of point $\mathrm{A}$.

Find the equation of the normal to the curve at point $\mathrm{A}$Give your answer in the form  $ax+by+d=0$ .

The volume of a sphere of radius $r$ is given by the formula  $V=\frac{4}{3}\mathrm{\pi }{r}^3$ .

Find $\frac{\mathrm{d}V}{\mathrm{d}r}$.

Find the rate of change of the volume with respect to the radius when $r=5$.

Give your answer in terms of $\mathrm{\pi }$.

Show that $\frac{\mathrm{d}V}{\mathrm{d}r}$ is an increasing function for all relevant values of $r$.

A curve has the equation

$f\left(x\right)=\frac{1}{3}{x}^3-2x^2-4x+\frac{31}{3}$

Points $\mathrm{A}$ and $\mathrm{B}$ are the two points on the curve where the gradient is equal to 1, and the  $x$ -coordinate of $\mathrm{A}$ is less than zero.

Find the coordinates of points $\mathrm{A}$ and $\mathrm{B}$.

Find the equations of

(i)     the tangent to the curve at point $\mathrm{A}$

(ii)    the normal to the curve at point $\mathrm{B}$.

Point $\mathrm{C}$ is the point of intersection of the two lines found in part (b).

Find the coordinates of point $\mathrm{C}$.

The gradient of the tangent to the curve with equation$f\left(x\right)=a{x}^2+2x+9$  at the point $(-2,b)$ is 14.

Find the values of $a$ and $b$.

The diagram below shows a part of the graph of the function $y=f\left(x\right),$ where

$f\left(x\right)=\frac{4}{x}+x^2-4,x>0$

Calculate the instantaneous rate of change of when$f\left(x\right)$ when $x=2.$

Calculate the average rate of change of $f\left(x\right)$ between $x=2$ and

(i) $x=3$

(ii) $x=2.5$

(iii) $x=2.25$

Explain what would happen if you continued to calculate the average rates of change in part (b), moving the second $x$ value closer and closer to 2 each time.

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

Show that the curve has exactly two stationary points. Determine the coordinates and nature of each point.

Show that the curve has exactly one point of inflection and determine its coordinates.

Give an example of a curve with equation $y=a{x}^3+b{x}^2+cx+d$, where $a,b,c$ and $d$ are real numbers and $a\ne 0,$ for which there is a point of inflection that is also a stationary point. Be sure to justify your answer.

A function$f$ is defined by

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

for all real numbers $x$.

(i) Show that the $x$-coordinates of the stationary points on the curve $y=f\left(x\right)$ must satisfy the equation

$(x^2-1)(x^2+3)=0$

(ii) Hence determine the coordinates of the stationary points on the curve.

Determine the ranges of values of $x$ for which$f\left(x\right)$ is

(i) increasing

(ii) decreasing.

Sketch the curve of $y=f\left(x\right)$, showing the coordinates of any minimum and maximum points, as well as the point where the curve crosses the $y$-axis.

A function$f$ is defined for all 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 when $x>n,$ where $n$ is the least possible number that makes that inequality true.

Find the value of $n$.

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

Let$f$ be a function defined by$f\left(x\right)=-3x^3+30x^2-87x+60$ for all $x$ in the interval. The following diagram shows the graph of$f$:

The curve intercepts the $x$-axis at points $\mathrm{A}\left(\mathrm{a},0\right),\mathrm{B}\left(\mathrm{b},0\right)$, and $C\left(c,0\right)$. There is a point of inflection at point $\mathrm{X}$ and a local maximum at point $\mathrm{Y}.$

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

Find$f^{\text{'}}\left(x\right),$ and hence determine the $x$-coordinate of the local maximum at point $\mathrm{Y}$. You should give your answer as an exact value.

Find the equation of the tangent at point $\mathrm{X}$. Give your answer in the form$px+qy+r=0$ where$p,q$ and $r$ are integers.

A curve is given by the equation

$y=\frac{1}{6}{x}^3-\frac{3}{8}{x}^2-\frac{3}{2}x+4$

Determine the coordinates of the points on the curve where the gradient is 2. You must show all your working, and give your answers as exact fractions.

Find the range of values for $x$ for which the curve is increasing.

An engineer is designing a right cone that is to be produced on a 3D printer. The cone has a base radius of $r$ cm and a height of $h$ cm, and while the radius may vary freely the height must always be 7 cm more than the radius.

Write down, in terms of $r$ only, the formula for the volume of the cone.

Find the exact value of the radius at the point where the instantaneous rate of change of the volume with respect to the radius is$\frac{5\pi }{3}$${\mathrm{cm}}^3/\mathrm{cm}$ . 

A curve has the equation

$f\left(x\right)=2x^3+\frac{3}{x}-4$

Points $\mathrm{A}$ and $\mathrm{B}$ are the two points on the curve where the gradient is equal to 3, and the  $x$ -coordinate of $\mathrm{A}$ is less than zero.

Find the coordinates of points $\mathrm{A}$ and $\mathrm{B}$.

Find the equations of

(i)     the tangent to the curve at point $\mathrm{A}$.

(ii)    the normal to the curve at point $\mathrm{B}$.

Point $\mathrm{C}$ is the point of intersection of the two lines found in part (b).

Find the coordinates of point $\mathrm{C}$. Give your answers as exact fractions.

A curve has equation  $f\left(x\right)=a{x}^2+bx+c$.

The gradient of the tangent to the curve at the point $(-3,d)$ is 25.

The gradient of the tangent to the curve at the point $(2,-1)$ is $-5$.

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

The diagram below shows a part of the graph of the function  $y=f\left(x\right)$,  where

$f\left(x\right)=9-\frac{1}{18}{x}^3-\frac{6}{x},x>0$

Calculate the average rate of change of $f\left(x\right)$ between $x=3$and

(i)     $x=4$

(ii)    $x=3.5$

(iii)   $x=3.25$

Explain what would happen to the values of the average rates of change in part (b) if you continued to calculate them, moving the second $x$ value closer and closer to 3 each time.

Let$f$ be a function defined by $f\left(x\right)=-6x^3+7x^2+3x$ for all $x\in \mathrm{ℝ}.$

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

Find the values of $a,b$ and $c.$

The curve $y=f\left(x\right)$ has a local minimum at point $D$.

Show that the $x$-coordinate of point $D$ is equal to $\frac{7-\sqrt{103}}{18}.$ Be sure to justify that the point corresponding to that $x$-coordinate is indeed a local minimum, and that it is the only local minimum.

The curve $y=f\left(x\right)$ has a point of inflection at point $E.$

Find the gradient of the normal to the curve at point $E.$

A function $f$ is defined by

$f\left(x\right)=\frac{1}{5}{x}^5-\frac{5}{3}{x}^3+4x-\frac{38}{15}$

for all real numbers $x$.

(i) Show that the curve $y=f\left(x\right)$ has a stationary point when $x=1$ and determine the corresponding $y$-coordinate.

(ii) Find the $x$-coordinates of any other stationary points on the curve.

Determine the ranges of values of $x$ for which $f\left(x\right)$ is

(i) increasing

(ii) decreasing.

Given that the value of the function when $x=1$ is greater than the value of the function at any other stationary point, sketch the curve of $y=f\left(x\right).$ Be sure to show clearly the $x$-coordinates of any minimum and maximum points, as well as the coordinates of the point where the curve crosses the -axis.

A function $f$ is defined for all 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 when $x>n$,  where $n$ is the least possible number that makes that inequality true.

Find the value of $n$.

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

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 will only have stationary points if $a,b$ and $c$ satisfy the inequality $b^2\ge 3ac.$

(ii) In the case where the inequality in part (a)(i) is satisfied, determine the $x$-coordinate(s) of the stationary point(s), giving your answer as simply as possible in terms of $a,b$ and $c.$

Show that the curve will always have exactly one point of inflection, and determine its $x$-coordinate in terms of $a$ and b.

In the case where the point of inflection is also a stationary point, show that the curve will have no other stationary points.

In the case where the curve has two distinct stationary points, show that the $x$-coordinate of the point of inflection will lie halfway between the $x$-coordinates of the two stationary points.