Calculus Examination Questions SL

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

 

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

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

 

 

 

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