DP IB Maths: AI SL

Topic Questions

5.1 Differentiation

1a
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2 marks

The equation of a curve is  y equals x minus 9 over x plus 8 for x greater than 0 .

Find fraction numerator straight d y over denominator straight d x end fraction.

1b
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3 marks

The gradient of the tangent to the curve at point straight A is 2.

Find the coordinates of point straight A.

1c
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3 marks

Find the equation of the normal to the curve at point straight A spaceGive your answer in the form  a x plus b y plus d equals 0 .

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2a
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1 mark

The volume of a sphere of radius r is given by the formula  V equals 4 over 3 straight pi straight r r cubed .

Find fraction numerator straight d V over denominator straight d r end fraction.

2b
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2 marks

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

Give your answer in terms of straight pi.

2c
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3 marks

Show that fraction numerator straight d V over denominator straight d r end fraction is an increasing function for all relevant values of r.

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3a
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3 marks

A curve has the equation

f left parenthesis x right parenthesis equals 1 third x cubed minus 2 x squared minus 4 x plus 31 over 3

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

Find the coordinates of points straight A and straight B.

3b
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5 marks

Find the equations of

(i)
the tangent to the curve at point straight A

(ii)
the normal to the curve at point straight B.
3c
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2 marks

Point straight C is the point of intersection of the two lines found in part (b).

Find the coordinates of point straight C.

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4
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5 marks

The gradient of the tangent to the curve with equation  f left parenthesis x right parenthesis equals a x squared plus 2 x plus 9  at the point left parenthesis negative 2 comma b right parenthesis is 14.

Find the values of a and b.

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5a
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2 marks

Patroclus, a would-be Olympic javelin thrower, throws a javelin during a training session.  The height of the javelin’s point can be modelled by the equation

 h left parenthesis t right parenthesis equals 1.75 plus 20.2 t minus 4.90 t squared

where t is the time, in seconds, that has passed since the javelin was released, and h left parenthesis t right parenthesis is the height of the javelin above the ground, in metres.

Find h to the power of apostrophe left parenthesis t right parenthesis.

5b
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6 marks
(i)
Find the stationary point for h left parenthesis t right parenthesis.

(ii)
Justify that the stationary point is a maximum point.
5c
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1 mark

Find the greatest vertical distance that the javelin’s point travels above the height from which it was released.

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6a
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3 marks

Check, Mate! is a company that produces luxury chess sets for discerning chess set connoisseurs.  The company’s profits P left parenthesis x right parenthesis, in thousands of UK pounds (£1000), can be modelled by the function

P left parenthesis x right parenthesis equals 0.32 x cubed minus 12.4 x squared plus 150 x minus 480

where x is the number of chess sets (in hundreds) sold per year.  Because of manufacturing constraints, the maximum number of chess sets that the company can sell in a year is 2500.

(i)
State why there is no need to consider values of x greater than 25.

(ii)
Sketch a graph of P left parenthesis x right parenthesis for 0 less or equal than x less or equal than 25 .
6b
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5 marks
(i)
Find the stationary points on the graph, and the numbers of chess sets sold and profits that correspond to those points.

(ii)
Find the maximum profit that the company can make in a year, and the number of chess sets the company must sell to make that profit.
6c
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5 marks

Calculate

(i)
the average rate of change of P left parenthesis x right parenthesis between  x equals 5  and x equals 6

(ii)
the instantaneous rate of change of P left parenthesis x right parenthesis at  x equals 5.

In each case include the units, and explain the meaning of the value you find.

6d
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3 marks

State the values of x spacefor which the instantaneous rate of change of P left parenthesis x right parenthesis is negative.  Explain the meaning of this result.

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7a
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2 marks

The diagram below shows a part of the graph of the function y equals f left parenthesis x right parenthesis ,  where

f left parenthesis x right parenthesis equals 4 over x plus x squared minus 4 comma space space space space space space space x greater than 0

ib7a-ai-sl-5-1-ib-maths-hard

Calculate the instantaneous rate of change of f left parenthesis x right parenthesis when x equals 2.

7b
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4 marks

Calculate the average rate of change of f left parenthesis x right parenthesis between x equals 2 spaceand 

(i)
x equals 3

(ii)
x equals 2.5

(iii)
x equals 2.25
7c
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2 marks

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

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8a
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4 marks

A manufacturing company is producing tins that must have a capacity of 470 cm3.  The tins are in the shape of a cylinder with a height of h cm and a base radius of r cm.

Show that the surface area of the cylinder in cm2, including the two circular ends, may be written as

A equals 2 pi r squared plus 940 over r

8b
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2 marks

Sketch the graph of A equals 2 pi r squared plus 940 over r.

8c
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5 marks

The company would like to minimise the amount of metal used to make the tins.

(i)
Find the stationary point on the graph of  A equals 2 pi r squared plus 940 over r ,  and justify that it is a minimum point.

(ii)
Hence find the minimum possible surface area for the tin, and the base radius that corresponds to that minimum area.
8d
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3 marks

A commercially available tin of chopped tomatoes on sale in the UK has a capacity of 470 cm cubed and a base radius of 3.7 cm.

Determine the percentage difference between the surface area of that tin of chopped tomatoes and the minimum possible surface area for a tin with the same capacity.

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9
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6 marks

Two numbers, x and y, are such that  x greater than y  and the difference between the two numbers is 7.

Find the minimum possible value of the product x y, and the values of x and y that correspond to that minimum value.

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1a
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2 marks

The equation of a curve is y equals 3 over 2 x squared minus 15 x plus 2

Find fraction numerator dy over denominator straight d x end fraction.

1b
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4 marks

The gradient of the tangent to the curve at point A is negative 3.

Find

(i)
the coordinates of straight A

         

(ii)
the equation of the tangent to the curve at point straight A
Give your answer in the form y equals m x plus c.

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2a
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1 mark

Consider the function f left parenthesis x right parenthesis equals 3 x to the power of 7 minus 12 x.

Find f to the power of apostrophe left parenthesis x right parenthesis.

2b
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2 marks

Find the gradient of the graph of f at x equals 0.

2c
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3 marks

Find the coordinates of the points at which the normal to the graph of f spacehas a gradient of 4.

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3a
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3 marks

The equation of a curve is y equals 4 minus 4 over x.

Find the equation of the tangent to the curve at x equals 2.

Give your answer in the form y equals m x plus c.

3b
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3 marks

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

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4a
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3 marks

Consider the function f left parenthesis x right parenthesis equals 4 over x plus fraction numerator 2 x to the power of 4 over denominator 5 end fraction minus 2 over 5 comma space space space space space space space x not equal to 0.

Calculate

(i)

f left parenthesis 2 right parenthesis

(ii)
f apostrophe left parenthesis 2 right parenthesis.
4b
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3 marks

A line, l comma is tangent to the graph of y equals f left parenthesis x right parenthesis at the point x equals 2 .

Find the equation of l. Give your answer in the form y equals m x plus c.

4c
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2 marks

The graph of y equals f left parenthesis x right parenthesis and l have a second intersection at point straight A.

Use your graphic display calculator to find the coordinates of straight A.

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5a
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1 mark

Consider the function f left parenthesis x right parenthesis equals x squared minus b x plus c.

Find f to the power of apostrophe left parenthesis x right parenthesis.

5b
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2 marks

The equation of the tangent line to the graph y equals f left parenthesis x right parenthesis at x equals 2 is y equals x minus 1.

Calculate the value of b.

5c
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3 marks

Calculate the value of c and write down the function f left parenthesis x right parenthesis.

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6a
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2 marks

The equation of the curve C is y equals 1 over 35 x to the power of 5 minus 3 over 4 x cubed plus 6 x. A section of the curve C is shown on the diagram below.

ib6-ai-sl-5-1-ib-maths-medium

Find fraction numerator straight d y over denominator straight d x end fraction.

6b
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4 marks

Points A and B represent the local maximums on the diagram above.

Write down the coordinates of

(i)

A

(ii)
B
6c
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2 marks

There are two points, straight R and straight S, along the curve C at which the gradient of the normal to the curve C is equal to negative 1 over 10.

Calculate the x-coordinates of points straight R and straight S.

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7a
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1 mark

The daily cost function of a company producing pairs of running shoes is modelled by the cubic function

C left parenthesis x right parenthesis equals 1225 plus 11 x minus 0.009 x squared minus 0.0001 x cubed comma space space space space space space space space space space space space 0 less or equal than x less than 160

where x is the number of pairs of running shoes produced and C the cost in USD.

Write down the daily cost to the company if no pairs of running shoes are produced.

7b
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2 marks

The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.

Find an expression for the marginal cost,C to the power of apostrophe left parenthesis x right parenthesis , of producing pairs of running shoes.

7c
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2 marks

Find the marginal cost of producing

(i)

40 pairs of running shoes

(ii)
90 spacepairs of running shoes.
7d
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3 marks

The optimum level of production is when marginal revenue,R to the power of apostrophe left parenthesis x right parenthesis , equals marginal cost, C to the power of apostrophe left parenthesis x right parenthesis. The marginal revenue,R to the power of apostrophe left parenthesis x right parenthesis , is equal to 4.5.

Find the optimum level of production.

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8a
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2 marks

A cyclist riding over a hill can be modelled by the function

h left parenthesis t right parenthesis equals negative 1 over 24 t squared plus 3 t plus 12 comma space space space space space space 0 less or equal than t less or equal than 70

where h is the cyclist’s altitude above mean sea level, in metres, and t spaceis the elapsed time, in seconds.

Calculate the cyclist’s altitude after a minute.

8b
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2 marks

Find h to the power of apostrophe left parenthesis t right parenthesis.

8c
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3 marks

Calculate the cyclist’s maximum altitude and the time it takes to reach this altitude.

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9a
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1 mark

A company produces and sells cricket bats. The company’s daily cost, C, in hundreds of Australian dollars open parentheses AUD close parentheses, changes based on the number of cricket bats they produce per day. The daily cost function of the company can be modelled by

C left parenthesis x right parenthesis equals 6 x cubed minus 10 x squared plus 10 x plus 4

where x hundred cricket bats is the number of cricket bats produced on a particular day.

Find the cost to the company for any day zero cricket bats are produced.

9b
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2 marks

The company’s daily revenue, of AUD, from selling x hundred cricket bats is given by the function R left parenthesis x right parenthesis equals 42 x.

Given that profitequals revenue minuscost, determine a function for the profit, P left parenthesis x right parenthesis comma in hundreds of AUD from selling x hundred cricket bats.

9c
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2 marks

Find P to the power of apostrophe left parenthesis x right parenthesis.

9d
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3 marks

The derivative of P left parenthesis x right parenthesis gives the marginal profit. The production of bats will reach its profit maximising level when the marginal profit equals zero and P left parenthesis x right parenthesis is positive.

Find the profit maximising production level and the expected profit.

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10a
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2 marks

Dora decides to build a cardboard container for when she goes strawberry picking from a rectangular piece of cardboard, 55 space cm space cross times 28 space cm. She cuts squares with side length x cm from each corner as shown in the diagram below.

ib10-ai-sl-5-1-ib-maths-medium

Show that the volume, V cm cubed, of the container is given by

V equals 4 x cubed minus 166 x squared plus 1540 x

10b
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2 marks

Find begin inline style fraction numerator straight d v over denominator straight d x end fraction end style.

10c
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4 marks

Find

(i)

the value of x spacethat maximises the volume of the container

(ii)

the maximum volume of the container. Give your answer in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 spaceand k element of straight integer numbers.

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1a
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6 marks

A curve is given by the equation

y equals 1 over 6 x cubed minus 3 over 8 x squared minus 3 over 2 x plus 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.

1b
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3 marks

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

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2a
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2 marks

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.

2b
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5 marks

Find the exact value of the radius at the point where the instantaneous rate of change of the volume with respect to the radius isfraction numerator 5 pi over denominator 3 end fractioncm cubed divided by cm . 

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3a
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3 marks

A curve has the equation

f left parenthesis x right parenthesis equals 2 x cubed plus 3 over x minus 4

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

Find the coordinates of points straight A and straight B.

3b
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5 marks

Find the equations of

(i)
the tangent to the curve at point straight A.

(ii)
the normal to the curve at point straight B.
3c
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3 marks

Point straight C is the point of intersection of the two lines found in part (b).

Find the coordinates of point straight C. Give your answers as exact fractions.

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4
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7 marks

A curve has equation  f left parenthesis x right parenthesis equals a x squared plus b x plus c.

The gradient of the tangent to the curve at the point left parenthesis negative 3 comma d right parenthesis is 25.

The gradient of the tangent to the curve at the point left parenthesis 2 comma negative 1 right parenthesis is negative 5.

Find the values of a,b ,c  and d.

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5a
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5 marks

A newly-commissioned attack submarine is performing a series of manoeuvres to test its propulsion and steering systems.  The vertical position of the submarine relative to sea level (where sea level is represented by h equals 0) is given by the equation

h left parenthesis t right parenthesis equals 0.0125 t cubed minus 1.03 t squared plus 16.6 t minus 165 comma space space space space space space space space space space 0 less or equal than t less or equal than 60

where t is the time, in minutes, that has passed since the submarine began its manoeuvres, and h left parenthesis t right parenthesis is the vertical position of the submarine in metres.

Find the stationary points for h left parenthesis t right parenthesis.

5b
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4 marks

For each of the stationary points found in part (a), determine whether the point is a maximum point or a minimum point. Justify your answer in each case.

5c
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1 mark

Explain why, in order to find the maximum and minimum depths reached by the submarine in the interval 0 less or equal than t less or equal than 60, it is not sufficient merely to consider the stationary points found in part (a).

5d
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2 marks

Find the greatest vertical distances that the submarine travels in the interval  0 less or equal than t less or equal than 60  above and below the depth from which it started its manoeuvres.

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6a
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2 marks

Muggins! is a company that produces luxury cribbage boards for discerning collectors of pub game paraphernalia.  For sales of between 0 and 100 cribbage boards in a month, the company’s profits P left parenthesis x right parenthesis, in thousands of UK pounds (£1000), can be modelled by the function

P left parenthesis x right parenthesis equals 4.53 x squared minus 8.51

where x is the number of cribbage boards (in hundreds) sold during the month.  For sales of between 100 and 1000 cribbage boards in a month, the corresponding formula is 

P left parenthesis x right parenthesis equals 0.02 x cubed minus 9 over x plus 5

Because of manufacturing constraints, the maximum number of cribbage boards that the company can sell in a month is 1000.

(i)
Confirm that both formulae give the same profit for sales of 100 cribbage boards in a month.

(ii)
State the ranges of x values for which each formula is valid.
6b
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3 marks

On the same set of axes, sketch the two profit functions. Each function should only be sketched over the interval of x values for which it is valid.

6c
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4 marks

Show that the combined profit function sketched in part (b) is an increasing function for all valid x values greater than zero.

6d
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5 marks

Considering only values of x spacefor which P left parenthesis x right parenthesis greater than 0,  find the value of x for which the instantaneous rate of change of P left parenthesis x right parenthesis is a minimum.  Give the value of the corresponding instantaneous rate of change, and explain the meaning of that value in context.

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7a
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4 marks

The diagram below shows a part of the graph of the function  y equals f left parenthesis x right parenthesis,  where

f left parenthesis x right parenthesis equals 9 minus 1 over 18 x cubed minus 6 over x comma space space space space space space space space space space space x greater than 0

ib7a-ai-sl-5-1-ib-maths-veryhard

Calculate the average rate of change of f left parenthesis x right parenthesis between x equals 3 spaceand

(i)
x equals 4

(ii)
x equals 3.5

(iii)
x equals 3.25
7b
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3 marks

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.

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8a
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9 marks

An artist is producing large pieces of sculpture for an art installation.  Each piece is in the form of a cylinder with base radius r metres, on top of which is a hemisphere with the same radius as the cylinder’s base radius.  The hemisphere is fitted exactly to the top of the cylinder, so that the circular bottom of the hemisphere lines up exactly with the circular top of the cylinder. 

Every side of each piece of sculpture must be painted, so the artist is eager to find a design for his sculptures such that, for any given volume of a piece of sculpture, the total surface area will be the minimum possible.

Show that for a piece of sculpture with volume k pi straight m cubed, the minimum surface area occurs when

r equals cube root of fraction numerator 3 k over denominator 5 end fraction end root

8b
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2 marks

Find the minimum possible surface area for a piece of sculpture with volume fraction numerator 40 space over denominator 3 end fraction straight pi straight m cubed.  Give your answer as an exact value.

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9a
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6 marks

Two numbers, x and y, are such that  x greater than y  and the difference between the two numbers is k, where k is a positive constant.

Find the minimum possible value of the sum x squared plus 3 y squared, and the values of x and y that correspond to that minimum value.  Your answers should be given in terms of k.

9b
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4 marks

Justify that your answer in part (a) is a minimum value.

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