Calculus Examination Questions HL

On this page you can find examination questions from the topic of calculus

Introducing Derivatives

Question 1

Find the value(s) of x for which the graph \(y=x^3-8x+2\) has gradient 4

Hint

Full Solution

Question 2

Find \(f'(4)\) for the function \(f(x)=2x+\frac { 8 }{ \sqrt { x } } +\frac { 32 }{ x } \)

Hint

Full Solution

Question 3

The gradient of \(y=x^2+ax+b\) at the point (1,-3) is -4.

Find a and b

Hint

Full Solution

Graphs and Derivatives

Question 1

The graph of y = f(x) is shown below, where B is a local maximum and C is a local minimum

Sketch a graph of y = f'(x), clearly showing the images of the points B and C labellling them B' and C' respectively

Hint

Full Solution

 

Question 2

A function is given by \(f(x)=-x^3+6x^2+4\)

a) Find the coordinates of any stationary points and describe their nature

b) Determine the values of x such that f(x) is a increasing function

c) Find the coordinates of the point of inflexion

Hint

Full Solution

 

Question 3

The following diagram shows the graph of \(f'\), the derivative of f

On the graph below, sketch the graph of y = f(x) given that f(0) = 0. Mark the images of A , B and C labelling them A' , B' and C'.

Hint

Full Solution

 

Question 4

Consider the function \(f(x)=-x^3-3x^2+9x\)

a) Find the coordinates of any stationary points and determine their nature

b) Find the equation of the straight line that passes through both the local maximum and the local minimum points.

c) Show that the point of inflexion lies on this line.

Hint

Full Solution

 

Product and Quotient Rule

Question 1

Let \(y=xe^x\)

a) Find \(\frac{dy}{dx}\)

b) Show that \(\frac{d^2y}{dx^2}=e^x(2+x)\)

c) Find the coordinates of the stationary point and show that it is a local minimum.

Hint

Full Solution

 

Question 2

Let \(f(x)={ x }^{ 2 }{ (2x-3) }^{ 3 }\)

a) Find \(f'(x)\)

b) The graph of y = f(x) has stationary points at x = 0, x= \(\frac{3}{2}\) and x = a. Find the value of a

Hint

Full Solution

 

Question 3

Let \(f(x)=\frac{lnx}{x},x>0\)

a) Show that \(f'(x)=\frac{1-lnx}{x^2}\)

b) Find \(f''(x)\)

c) The graph of f has a point of inflexion at A. Find the x-coordinate of A.

Hint

Full Solution

 

Question 4

Let \(f(x)=tanx\). A\((\frac{\pi}{3},\sqrt{3})\) is a point that lies on the graph of \(y=f(x)\)

a) Given that \(tanx=\frac{sinx}{cosx}\) find \(f'(x)\)

b) Show that \(f'(\frac{\pi}{3})=4\)

c) Find the equation of the normal to the curve y = f(x) at the point A

d) Show that the normal crosses the y axis at \(\sqrt{3}+\frac{\pi}{12}\)

Hint

Full Solution

 

Question 5

Let \(f(x)=e^{2x}cosx\)

a) Find \(f'(x)\)

b) Show that \(f''(x)=4f'(x)-5f(x)\)

Hint

Full Solution

 

Equation of Tangent and Normal

Question 1

Let f(x) = (x - 1)(x - 4)(x + 2). The diagram below shows the graph of f and the point P where the graph crosses the x axis.

The line L is the tangent to the graph of f at the point P.

The line L intersects the graph of f at another point Q, as shown below

a) Find the coordinates of P

b) Show that \(f(x)=x^3-3x^2-6x+8\)

c) Find the equation of L in the form y = ax + b

d) Find the x coordinate of Q.

Hint

Full Solution

Question 2

Let \(f(x)=\frac{x^4-4x^2}{4}\) .

C(2 , 0) lies on the graph of y = f(x)

a) The tangent to the graph of y = f(x) at C cuts the y axis at A. Find the coordinates of A.

b) The normal to the graph of y = f(x) at C cuts the y axis at B. Find the area of the triangle ABC.

Hint

Full Solution

Question 3

The function \(f(x)=x^3-x^2-9x+9\) intersects the x axis at A, B and C.

The x coordinate of the point D is the mean of the x coordinates of B and C.

a) Find the coordinates of A, B and C.

b) Find the equation of the tangent to the curve at D.

c) Find the point of the intersection of the tangent with the curve. Interpret your result.

Hint

Full Solution

Optimisation

Question 1


Hint

Full Solution

Question 2

A container is made from a cylinder and a hemisphere. The radius of the cylinder is r m and the height is h. The volume of the container is \(45\pi\)

a) Find an expression for the height of the cylinder in terms of r

b) Show that the surface area of the container, \(A=\frac{5 \pi r^2}{3}+\frac{90 \pi}{r}\)

c) Hence, find the values of r and h that give the minimum surface area of the container

* Volume of a sphere = \(\frac{4}{3}\pi r^3\)

** Surface area of a sphere = \(4\pi r^2\)


Hint

Full Solution

 

Question 3

The diagram below shows the graph of the functions f(x) = sinx and g(x) = 2sinx

A rectangle ABCD is placed in between the two functions as shown so that B and C lie on g , BC is parallel to the x axis and the local minima of the function f lies on AD.

Let NA = x

a) Find an expression for the height of the rectangle AB

b) Show that the area of the rectangle, A can be given by A = 4xcosx - 2x

c) Find \(\frac{dA}{dx}\)

d) Find the maximum value of the area of the rectangle.

Hint

Full Solution

 

Implicit Differentiation

Question 1

  

Find the equation of the tangent to the curve \(x^3 + y^3 -6xy = 0\) at the point (3,3)

Hint

Full Solution

 

Question 2

Find the gradient of the curve \(x^2+2e^{(x+2y)}=3\) at the point when x=-1

Hint

Full Solution

 

Question 3

A curve is defined by \(x\arcsin y=e^{2y}\)

Show that \(\frac{dy}{dx}=\frac{\sqrt{1-y^{2}}\arcsin y}{2e^{2y}\sqrt{1-y^{2}}-x}\)

Hint

Full Solution

Question 4

  1. Find the coordinates of the stationary point on the graph \(x^2y+2x=1\)
  2. Show that \(2y + 4x\frac{dy}{dx}+ x^2\frac{d^2y}{dx^2}=0\) and hence justify that the stationary point is a local minima.

Hint

Full Solution

Related Rates of Change

Question 1

The radius of a sphere is increasing at 2.5 cms-1

Find the rate at which the volume of the sphere is increasing when the radius is 8 cm.

Give your answer in terms of \(\large \pi\)

Hint

Full Solution

Question 2

The diagram shows a container in the form of a right circular cone. The height of the cone is equal to its diameter. Initially the cone is empty, then water is added at a rate of \(\large 18\pi\) cm3 per minute. the depth of water in the container at the time is given by h cm.

a) Show that the volume, V cm3 , of water in the container when the depth is x cm is given by

\(\large V=\frac{1}{12} \pi x^3\)

b) Find the rate at which the depth of the water is increasing at the instant when the depth is 12 cm


Hint

Full Solution

Question 3

A searchlight rotates at 2 revolutions per minute. The beam hits a wall 30m away and produces a spot of light that moves horizontally along the wall. How fast is the spot moving along the wall when the angle, \(\large \theta\) between the beam and the line through the spotlight perpendicular to the wall is 45°?


Hint

Full Solution

Question 4

A ladder AB of length 8m has one endA on horizontal ground and the other end B resting against a vertical wall.

The end A slips away from the wall at a constant speed of 0.5 ms-1 and the end B slips down the wall.

Determine the speed of the end B is slipping down the wall when the top of the ladder is 5 m above the ground.


Hint

Full Solution

Question 5

 

 

A solid is made up of a cylinder and a hemisphere as shown in the diagram.

a) Write down a formula for the volume of the solid.

b) At the time when the radius is 6cm, the volume of the solid is \(\large 684\pi\) cm3 , the radius is changing at a rate of 1.5 cm/minute and the volume is changing at a rate of \(\large 1800\pi\) cm3 /min. Find the rate of change of the height at this time.


Hint

Full Solution

L'Hôpital's Rule

Question 1

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}( x \ln x)\)

Hint

Full Solution

Question 2

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}\frac{e^{x^2}-1}{\sin x^2}\)

Hint

Full Solution

Question 3

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}\frac{1-\cos x^2}{x^4}\)

Hint

Full Solution

Question 4

Use l’Hôpital’s rule to determine the value of \(\large \lim\limits_{x\rightarrow 0}\frac{6\tan x-6x}{x^3}\)

Hint

Full Solution

Definite Integration

Question 1

Given that \(\int _{ 4 }^{ 8 }{ \frac { 1 }{ 2x-4 } dx= } ln\sqrt { a } \) , find the value of a

Hint

Full Solution

Question 2

Consider a function f(x) such that \(\int _{ 0 }^{ 4 }{ f(x)dx } \) = 6

Find

a) \(\int _{ 0 }^{ 4 }{3 f(x)dx } \)

b) \(\int _{ 0 }^{ 4 }{[ f(x)+3]dx } \)

c) \(\int _{ -3 }^{ 1 }{\frac{1}{3} f(x+3)dx } \)

d) \(\int _{ 0 }^{ 4 }{ [f(x)+x]dx } \)

Hint

Full Solution

Question 3

Given that \(\int _{ 2 }^{ 5 }{ ln(sinx)dx=A } \)

show that \(\int _{ 2 }^{ 5 }{ ln({ e }^{ x }sinx)dx=A } +\frac { 21 }{ 2 } \)

Hint

Full Solution

Area between Graphs

Question 1

Let f(x)=sinx, for \(0\le x\le 2\pi \)

The following diagram shows the graph of f

The shaded region R is enclosed by the graph of f, the line x=a , where a<\(\pi\) and the x-axis.

The area of R is \(\left( 1-\frac { \sqrt { 3 } }{ 2 } \right) \). Find the value of b.

 

Hint

 

Full Solution

 

Question 2

Show that the area bounded by the graphs of y = f(x) and y = g(x) in the interval \(0\le x\le \pi \) is given by 2 - \(\frac { \pi }{ 2} \)

f(x) = sinx

g(x) = sin²x

Hint

Full Solution

 

Question 3

Let \(f(x)=ln\left( \frac { x }{ x-1 } \right) \) for x>1

a) Find f ' (x)

b) Hence, show that the area bounded by g(x) = \(\frac { 1 }{ x(x-1) } \) , the x axis, x = 2 and x = e is given by \(ln\left( \frac { 2e-2 }{ e } \right) \)

Hint

Full Solution

Question 4

The graph below shows the function f(x) = sinx where a < b \(\le \frac { \pi }{ 2 } \)

a) Write an expression for the area A, bounded by the curve y = f(x), the x axis, x = a and x = b

b) Hence, without using any further integration, show that the area B = \(bsinb - asina - cosb + cosa\)

c) Area B can also be found by finding the area bounded by the curve x = f(y), the y axis, y = f(a) and y = f(b). Find the area B using this method and show that your answer is the same as the one you found in part b)

 

Hint

 

Full Solution

Volume of Revolution

Question 1

a) Show that \(\large \int {x^2e^{2x}\,dx}=\frac{1}{4}e^{2x}(2x^2-2x+1)+c\)

The following graph shows the function \(\large f(x)=xe^x\)

b) Show that the equation of the tangent to the graph at x = 1 has the equation \(\large y=(2e)x-e\)

The region bounded by \(\large f\), the tangent \(\large y=(2e)x-e\) and y = 0 is rotated \(\large 2\pi\) around the x axis

c) Find the volume of this solid in terms of \(\large \pi\)

Hint

Full Solution

Question 2

The following graph shows the curve defined by the equation \(\large (x-1)^2+y^2=4\).

The region bounded by the curve and the lines y = 0 and x = 0 is rotated \(\large 2\pi\) around the x axis.

Find the volume of this solid in terms of \(\large \pi\)

Hint

Full Solution

 

Question 3

The graph shows \(\large f(x) = -\arctan(1-x^2)\), the tangent to the curve at (1 , 0) and the tangent to the curve at the point \(\large(-\frac{\pi}{4},0)\).

The shaded region is bounded by the curve and the two tangents. This region is rotated \(\large 2\pi\) around the y axis to forma solid.

Find the volume of this solid correct to 3 significant figures.

Hint

Full Solution

 

Question 4

The following diagram shows the graph of \(\large x^2=\cos^3y\) for \(\large -\frac{\pi}{2}\le y\le\frac{\pi}{2}\)

The shaded region R is the area bounded by the curve, the y axis and the lines \(\large y=-\frac{\pi}{2}\) and \(\large y=\frac{\pi}{2}\).

The rgion is rotated about the y axis through \(\large 2\pi\) to form a solid.

Show that the volume of the solid is \(\large \frac{4\pi}{3}\)

Hint

Full Solution

Question 5

The following diagram shows the graph of the function \(\large f(x)=\frac{\sqrt{x}}{\cos x}\) for \(\large 0\le x \le\frac{\pi}{2}\)

The shaded region is the area bounded by f, the x axis, x= 0 and x = \(\large \frac{\pi}{3}\)

The region is rotated about the x axis through \(\large 2\pi\) to form a solid.

Find the volume of the solid.

Hint

Full Solution

 

Integration with Partial Fractions

Question 1

Let \(\large f(x)=\frac{-x-7}{x^2-x-6} \quad ,x\neq-2,x\neq3\)

a) Express f(x) in partial fractions

b) Hence, find the exact value of \(\large \int_{-1}^2f(x)dx\) giving your answer in the form \(\large lnq \quad,q\in\mathbb{Q}\)

Hint

Full Solution

 

Question 2

The graph shows the function \(\large f(x)=\frac{x+10}{4-x^2} \quad ,x\neq-2,x\neq2\)

By writing f(x) as the sum of two partial fractions, show that the area bounded by the curve, the x axis and the lines x = -1 and x = 1 is equal to \(\large 5ln3\)

Hint

Full Solution

 

Question 3

Consider the finction \(\large f(x)=\frac{x^2-7x+12}{x-1} \quad ,x\neq1\)

Find the coordinates where the graph of f crosses the

a i) x axis

ii) y axis

b) Write down the equation of the vertical asymptote of f

c) Write down the equation of the oblique asymptote of f

d) Sketch the graph of f for \(\large -15\le x \le15\) clearly indicating the points of intersection with each axis and any asymptotes

e) Express \(\large \frac{1}{f(x)}\) in partial fractions

f) Hence, find the exact value of \(\large \int_1^2\frac{1}{f(x)}dx\) in the form \(\large lnq \quad,q\in\mathbb{Q}\)

Hint

Full Solution

 

 

 

 

Integration by Substitution

Question 1

a) Find \(\int { \frac { { e }^{ x } }{ 1+{ e }^{ x } } dx } \)

b) Evaluate \(\int _{ \frac { { \pi }^{ 2 } }{ 4 } }^{ \pi ^{ 2 } }{ \frac { cos\sqrt { x } }{ \sqrt { x } } } dx\)

Hint

Full Solution

 

Video Solution

Question 2

The following diagram shows the graph of f(x) = \(\frac { 4x }{ \sqrt { { x }^{ 2 }+1 } }\)

Let R be the region bounded by f, the x-axis, x = 1 and x = 2

Find R

Hint

Full Solution

 

Video Solution

Question 3

a) Using the fact that \(tanx = \frac{sinx}{cosx} \), show that \(\frac{d}{dx}(tanx) = \frac{1}{cos^2x}\)

b) Hence, find \(\int { \frac { \sqrt { tanx } }{ cos^{ 2 }x } } dx\)

Hint

Full Solution

 

Video Solution

Question 4

Find \(\int { \frac { arcsinx+x }{ \sqrt { 1-x^{ 2 } } } } dx\)

Hint

Full Solution

 

Video Solution

Question 5

Find \(\int { sin^{ 5 }x\ dx } \)

Hint

Full Solution

 

Video Solution

Question 6

Find \(\int { \sqrt { 16-9x^{ 2 } } dx } \) using the substitution 3x = 4 sin\(\theta\)

Hint

Full Solution

 

Video Solution

Integration by Parts

Question 1

\(\int { { e }^{ \frac { x }{ 2 } }sin(x)\quad dx } \)

Hint

Full Solution

 

Video Solution

Question 2

\(\int { arctanx \ dx } \)

Hint

Full Solution

 

Video Solution

Question 3

\(\int { 2x\cdot arctanx \ dx } \)

Hint

Full Solution

 

Video Solution

Differential Equations - Separable Variables

Question 1

Solve the differential equation given that y(0) = 2

\({\large \frac{dy}{dx}=\frac{e^x}{y}, \quad y>0}\)

Hint

Full Solution

 

Question 2

a) Write \({\huge \frac{1}{4-x^2}}\)as the sum of two partial fractions

b) Hence, given that y(0) = 0, find the particular solution of the differential equation in the form

Hint

Full Solution

 

Question 3

a) Express \(\frac{3}{x(x-3)}\) as the sum of two partial fractions

The population of a species of fish can be modelled by the differential equation \(\large \frac{\text{d}N}{\text{d}t}=\frac{2}{3}N(N-3)cos2t\)

where N = population in thousands, t = time in years

b) Given that initially the population of fish is 4000, show that \(\large N=\frac{12}{4-e^{sin{2t}}}\)

c) How many days during the first year is the population of fish above 8000?

Hint

Full Solution

 

Differential Equations - Integrating Factor

Question 1

Consider the following first order differential equation

\(\large x\frac{\text{d}y}{\text{d}x}+3y=\frac{lnx}{x}\)

a) Show that x3 is the integrating factor for this differential equation

b) Hence, find the general solution of this differential equation in the form y = f(x)

Hint

Full Solution

 

Question 2

Consider the following differential equation \(\large \frac{\text{d}y}{\text{d}x}+ytanx=secx\)

a) Using a suitable integrating factor show that the differential equation can be written as \(\large \frac{y}{cosx}=\int sec^²x {dx}\)

b) Given that (0 , 2) lies on the curve, show that the particular solution of the differential equation is \(\large y=sinx+2cosx\)

Hint

Full Solution

 

Question 3

Consider the following differential equation \(\large \sin x\frac{\text{d}y}{\text{d}x}+y\cos x=2\sin^2 x\)

a) Show that the integrating factor of this differential equation is \(\large \sin x\)

b) Solve the differential equation, giving your answer in the form \(y=f(x)\)

Hint

Full Solution

 

Differential Equations - Homogeneous

Question 1

Consider the first order differential equation

\( \large xy\frac{\text{d}y}{\text{d}x}+4x^2+y^2=0 \quad,\quad y\geq0\)

a) Use the substitution y = vx to show that \( \large \frac{\text{d}v}{\text{d}x}=-\frac{4+2v^2}{vx}\)

b) Show that the solution to the differential equation for which y(1) = 0 is \(\large y=\frac{\sqrt{2-2x^4}}{x}\)

Hint

Full Solution

 

Question 2

Consider the homogeneous differential equation \(\large \frac{\text{d}y}{\text{d}x}=\frac{x^2+y^2}{2xy}\)

a) Using the substitution y = vx , show that \(\large \frac{\text{d}v}{\text{d}x}=\frac{1-v^2}{2vx}\)

b) Solve the differential equation and show that y² = x² - cx

Hint

Full Solution

 

Question 3

Consider the differential equation \(\large xy\frac{\text{d}y}{\text{d}x}=x^2\text{cosec}\frac{y}{x}+y^2\)

a) Show that this equation can be written in the form \(\large \frac{\text{d}y}{\text{d}x}=f(\frac{y}{x})\)

b) Using the substitution y = vx , show that the particular solution to the equation, given that y(1)= 0 is

Hint

Full Solution

 

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