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

The equation of a curve is $y=\frac{3}{2}{x}^2-15x+2$.

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

The gradient of the tangent to the curve at the point $\mathrm{A}$ is $-3$.

Find

(i) the coordinates of $\mathrm{A}$

(ii) the equation of the tangent to the curve at $\mathrm{A}$. Give your answer in the form $y=mx+c$.

Consider the function $f\left(x\right)=3x^7-12x$.

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

Find the gradient of the graph of $f$ at $x=0$.

Find the coordinates of the points on the graph of $f$ where the normal has a gradient of $4$.

The equation of a curve is $y=4-\frac{4}{x}$.

Find the equation of the tangent to the curve at the point where $x=2$. Give your answer in the form $y=mx+c$.

Find the coordinates of the points on the curve where the gradient is $16$.

Consider the function $f\left(x\right)=\frac{4}{x}+\frac{2x^4}{5}-\frac{2}{5}$, $x\ne 0$.

Find

(i) $f\left(2\right)$

(ii) $f\text{'}\left(2\right)$.

The line $l$ is tangent to the graph of $y=f\left(x\right)$ at the point where $x=2$.

Find the equation of $l$. Give your answer in the form $y=mx+c$.

The graph of $y=f\left(x\right)$ and the line $l$ intersect again at the point $\mathrm{A}$.

Use your graphic display calculator to find the coordinates of $\mathrm{A}$.

Consider the function $f\left(x\right)=x^2-bx+c$, where $b,c\in \mathrm{ℝ}$.

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

The tangent to the graph of $y=f\left(x\right)$ at $x=2$ has equation $y=x-1$.

Find the value of $b$.

Find the value of $c$.

The curve with equation $y=a{x}^2+bx+c$ has a gradient of $-7$ at the point $\left(-1,13\right)$ and a gradient of $-3$ at the point $\left(1,3\right)$.

By considering $\frac{\mathrm{d}y}{\mathrm{d}x}$, show that $2a+b=-3$ and $-2a+b=-7$.

Hence find the values of $a$ and $b$.

By considering a point that you know to be on the curve, find the value of $c$.

The curve $C$ has equation $y=3x^2-6+\frac{4}{x}$. The point $\mathrm{P}\left(1,1\right)$ lies on $C$.

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

Show that the equation of the normal to $C$ at $\mathrm{P}$ is $x+2y=3$.

The normal meets the $x$-axis at the point $\mathrm{Q}$.

Find $\mathrm{PQ}$, giving your answer as an exact value.

The graph of the cubic function $y=f\left(x\right)$ is shown below. Point $\mathrm{A}$, a local minimum, is located at the origin and point $\mathrm{B}$, a local maximum, sits at the point $\left(4,8\right)$.

State the equations of the horizontal tangents to the curve.

Write down the value of $x$ where the point of inflexion is located.

Find the intervals where $f$ is decreasing.

Sketch the graph of $f\text{'}\left(x\right)$, labelling clearly any intercepts and axis of symmetry.

The diagram below shows part of the curve with equation $y=x^3+11x^2+35x+25$. The curve touches the $x$-axis at $\mathrm{A}$ and cuts the $x$-axis at $\mathrm{C}$. The point $\mathrm{A}$ is a local maximum point and the point $\mathrm{B}$ is a local minimum point.

Using calculus, and showing all your working, find the coordinates of $\mathrm{A}$ and $\mathrm{B}$.

Show that $\left(-1,0\right)$ lies on the curve, and explain why it must be the point $\mathrm{C}$.

The equation of the curve $C$ is $y=\frac{1}{35}{x}^5-\frac{3}{4}{x}^3+6x$. A section of the curve $C$ is shown on the diagram below.

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

There are two points, $\mathrm{R}$ and $\mathrm{S}$, on the curve $C$ at which the gradient of the normal to $C$ is $-\frac{1}{10}$, where the $x$-coordinate of $\mathrm{R}$ is positive.

Find the $x$-coordinates of $\mathrm{R}$ and $\mathrm{S}$.

The diagram below shows 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 $f\left(x\right)$ 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 $y=x-\frac{9}{x}+8$, for $x>0$.

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

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

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

Find the equation of the normal to the curve at $\mathrm{A}$. Give your answer in the form $ax+by+d=0$, where $a$, $b$ and $d$ are integers.

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

Find the value of $a$ and the value of $b$.

A curve has equation $y=\frac{1}{6}{x}^3-\frac{3}{8}{x}^2-\frac{3}{2}x+4$.

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

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

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

Find the rate of change of $V$ with respect to $r$ when $r=5$. Give your answer in terms of $\pi$.

Show that $\frac{\mathrm{d}V}{\mathrm{d}r}$ is an increasing function for $r>0$.

A curve has equation $y=f\left(x\right)$, where $f\left(x\right)=\frac{1}{3}{x}^3-2x^2-4x+\frac{31}{3}$.

The points $\mathrm{A}$ and $\mathrm{B}$ are the two points on the curve where the gradient is $1$, and the $x$-coordinate of $\mathrm{A}$ is negative.

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

Find the equation of

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

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

The lines in part (b) meet at the point $\mathrm{C}$.

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

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 points where the gradient of the curve $y=f\left(x\right)$ is zero satisfy the equation $\left(x^2-1\right)\left(x^2+3\right)=0$.

(ii) Hence find the coordinates of these points.

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

(i) increasing

(ii) decreasing.

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

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

Show that $f$ is increasing for all values of $x$.

Let $f\left(x\right)=a{x}^2+bx+c$. The tangents to the graph of $f$ at the points $\left(-3,d\right)$ and $\left(2,-1\right)$ have gradients $25$ and $-5$ respectively.

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

The diagram below shows 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 (a) if you continued to calculate them, moving the second $x$ value closer and closer to $3$ each time.

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. The radius may vary, but the height is always 7 cm more than the radius.

Write down an expression for $V$, the volume of the cone, in terms of $r$ only.

The rate of change of $V$ with respect to $r$ is $\frac{5\pi }{3}{\text{ cm}}^3$ per cm at a particular value of $r$.

Find this exact value of $r$.

A curve has equation $y=f\left(x\right)$, where $f\left(x\right)=2x^3+\frac{3}{x}-4$.

The points $\mathrm{A}$ and $\mathrm{B}$ are the two points on the curve where the gradient is $3$, and the $x$-coordinate of $\mathrm{A}$ is negative.

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

Find the equation of

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

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

The lines in part (b) meet at the point $\mathrm{C}$.

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

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 gradient of the curve $y=f\left(x\right)$ is zero when $x=1$, and find the corresponding $y$-coordinate.

(ii) Find the $x$-coordinates of the other points on the curve where the gradient is zero.

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

(i) increasing

(ii) decreasing.

Given that $f\left(1\right)$ is greater than the value of $f$ at every other point where the gradient is zero, sketch the curve $y=f\left(x\right)$. Show the $x$-coordinates of any local maximum and local minimum points, and the coordinates of the point where the curve crosses the $y$-axis.