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

4 hours32 questions
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

          the equation of the tangent to the curve at point straight A

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

2a
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1 mark

Consider the functionspace f open parentheses x close parentheses space equals space 3 x to the power of 7 minus 12 x.

Findspace space f space apostrophe space open parentheses x close parentheses.

2b
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2 marks

Find the gradient of the graph ofspace 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 ofspace f has a gradient of 4.

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.

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) space f left parenthesis 2 right parenthesis

(ii) space f space 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.

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 functionspace f left parenthesis x right parenthesis.

6a
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2 marks

The curve with equation y equals a x squared plus b x plus c has a gradient of minus7 at the point open parentheses negative 1 comma space 13 close parentheses comma and a gradient of minus3 at the point open parentheses 1 comma space 3 close parentheses.

By considering fraction numerator d y over denominator d x end fraction show that 2 a plus b equals negative 3 and negative 2 a plus b equals negative 7.

6b
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1 mark

Hence find the values of a and b.

6c
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2 marks

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

7a
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2 marks

The curve C has equation y equals 3 x squared minus 6 plus 4 over x. The point P open parentheses 1 comma space 1 close parentheses lies on C.

Find an expression for fraction numerator d y over denominator d x end fraction.

7b
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3 marks

Show that an equation of the normal to C at point P is x plus 2 y equals 3.

7c
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2 marks

This normal cuts the x-axis at the point Q.

Find the length of P Q, giving your answer as an exact value.

8
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3 marks

Find the values of x for whichspace f left parenthesis x right parenthesis equals negative 9 x squared plus 5 x minus 3 is an increasing function.

9
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3 marks

Show that the functionspace f left parenthesis x right parenthesis equals x cubed minus 3 x squared plus 6 x minus 7 is increasing for all x element of straight real numbers.

10a
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2 marks

The graph of the cubic function y equals f open parentheses x close parentheses is shown below. Point A comma a local minimum, is located at the origin and point B, a local maximum, sits at the point open parentheses 4 comma space 8 close parentheses.

q10-5-1-medium-ib-aa-sl

State the equations of the horizontal tangent to the curve.

10b
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1 mark

Write down the value of x where the point of inflection is located.

10c
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2 marks

Find the intervals wherespace f is decreasing.

10d
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3 marks

Sketch the graph of space f apostrophe open parentheses x close parentheses comma labelling clearly any intercepts and axis of symmetry.

11a
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5 marks

The diagram below shows part of the curve with equation y equals x cubed plus 11 x squared plus 35 x plus 25. The curve touches the x-axis at C and cuts the x-axis at C. The points A and B are stationary points on the curve.

q11-5-1-medium-ib-aa-sl

Using calculus, and showing all your working, find the coordinates of A and B.

11b
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2 marks

Show that open parentheses negative 1 comma space 0 close parentheses is a point on the curve and explain why those must be the coordinates of point C.

12a
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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.

12b
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4 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.

13a
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4 marks

Find the x-coordinates of the stationary points on the graph with equation y equals x cubed minus 6 x squared plus 9 x minus 1.

13b
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3 marks

Find the nature of the stationary points found in part (a).

13c
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3 marks

Determine the x-coordinate of the point of inflection on the graph with equation y equals x cubed minus 6 x squared plus 9 x minus 1.

13d
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1 mark

Explain why, in this case, the point of inflection is not a stationary point.

14
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4 marks

The graph of a continuous function has the following properties:

The function is concave down in the interval  open parentheses negative infinity comma space a close parentheses.

The function is concave up in the interval open parentheses a comma space infinity close parentheses.

The graph of the function intercepts the -axis at the points open parentheses b comma space 0 close parentheses comma space open parentheses c comma space 0 close parenthesesand open parentheses d comma space 0 close parentheses comma where b comma space c and d are such that d greater than c greater than b greater than 0.

The x minuscoordinates of the turning points of the function are e and f, which are such thatspace f space greater than space e.

The graph of the function intercepts the y-axis at open parentheses 0 comma space g close parentheses

Given that the value of the function is positive when x equals a, sketch a graph of the function. Be sure to label the x-axis with the x-coordinates of the stationary points and the point of inflection, and also to label the points where the graph crosses the coordinate axes.

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 .

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 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.

3a
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3 marks

A curve has the equation

space 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.

4
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5 marks

The gradient of the tangent to the curve with equationspace 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.

5a
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2 marks

The diagram below shows a part of the graph of the function y equals f open parentheses x close parentheses comma where

space f open parentheses x close parentheses equals 4 over x plus x squared space minus space 4 comma space space space space space space space space space space space space space space space space x greater than 0

q5-5-1-hard-ib-aa-sl

Calculate the instantaneous rate of change of whenspace f open parentheses x close parentheses when x equals 2.

5b
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4 marks

Calculate the average rate of change of space f open parentheses x close parentheses between x equals 2 and

(i) x equals 3

(ii) x equals 2.5

(iii) x equals 2.25

5c
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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 value closer and closer to 2 each time.

6a
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6 marks

The equation of a curve is y equals x cubed plus 9 x squared plus 24 x plus 17.

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

6b
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3 marks

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

6c
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3 marks

Give an example of a curve with equation y equals a x cubed plus b x squared plus c x plus d, where a comma space b comma space c and d are real numbers and a space not equal to space 0 comma for which there is a point of inflection that is also a stationary point. Be sure to justify your answer.

7a
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6 marks

A functionspace f is defined by

space f open parentheses x close parentheses space equals space 1 fifth x to the power of 5 plus 2 over 3 x cubed minus 3 x plus 3

for all real numbers x.

(i) Show that the x-coordinates of the stationary points on the curve y equals f open parentheses x close parentheses must satisfy the equation

left parenthesis x squared minus 1 right parenthesis left parenthesis x squared plus 3 right parenthesis equals 0

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

7b
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3 marks

Determine the ranges of values of x for whichspace f open parentheses x close parentheses is

(i) increasing

(ii) decreasing.

7c
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3 marks

Sketch the curve of y equals f open parentheses x close parentheses, showing the coordinates of any minimum and maximum points, as well as the point where the curve crosses the y-axis.

8a
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2 marks

A functionspace f is defined for all for x greater than 0. The derivative ofspace f is given by

space f apostrophe space open parentheses x close parentheses equals 3 x squared plus 2 over x cubed minus 25

Findspace space f " space open parentheses x close parentheses.

8b
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4 marks

The graph ofspace f is concave up when x greater than n comma where n is the least possible number that makes that inequality true.

Find the value of n.

8c
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3 marks

Show that the curve y equals f left parenthesis x right parenthesis has only one point of inflection, and find the gradient of the curve at that point of inflection.

9a
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2 marks

Letspace f be a function defined byspace f left parenthesis x right parenthesis equals negative 3 x cubed plus 30 x squared minus 87 x plus 60 for all x in the interval. The following diagram shows the graph ofspace f:

q9-5-1-hard-ib-aa-sl

The curve intercepts the x-axis at points straight A open parentheses straight a comma space 0 close parentheses comma space straight B open parentheses straight b comma space 0 close parentheses, and C open parentheses c comma space 0 close parentheses. There is a point of inflection at point straight X and a local maximum at point straight Y.

Find the values of a comma space b and c.

9b
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4 marks

Findspace f space apostrophe open parentheses x close parentheses comma and hence determine the x-coordinate of the local maximum at point straight Y. You should give your answer as an exact value.

9c
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6 marks

Find the equation of the tangent at point straight X. Give your answer in the formspace p x plus q y plus r equals 0 wherespace p comma space q and r are integers.

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.

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 . 

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.

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.

5a
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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

5b
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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.

6a
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3 marks

Letspace f be a function defined by f left parenthesis x right parenthesis equals negative 6 x cubed plus 7 x squared plus 3 x for all x element of straight real numbers.

The curve y equals f left parenthesis x right parenthesis intercepts the x-axis at points A left parenthesis a comma space 0 right parenthesis comma space B open parentheses b comma space 0 close parentheses and C open parentheses c comma space 0 close parentheses comma where a less than b less than c.

Find the values of a comma space b and c.

6b
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6 marks

The curve y equals f left parenthesis x right parenthesis has a local minimum at point D.

Show that the x-coordinate of point D is equal to fraction numerator 7 minus square root of 113 over denominator 18 end fraction. 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.

6c
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3 marks

The curve y equals f open parentheses x close parentheses has a point of inflection at point E.

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

7a
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6 marks

A function f is defined by

f open parentheses x close parentheses equals 1 fifth x to the power of 5 minus 5 over 3 x cubed plus 4 x minus 38 over 15

for all real numbers x.

(i) Show that the curve y equals f open parentheses x close parentheses has a stationary point when x equals 1 and determine the corresponding y-coordinate.

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

7b
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4 marks

Determine the ranges of values of x for which f open parentheses x close parentheses is

(i) increasing

(ii) decreasing.

7c
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3 marks

Given that the value of the function when x equals 1 is greater than the value of the function at any other stationary point, sketch the curve of y equals f open parentheses x close parentheses. 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.

8a
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6 marks

A function f is defined for all for x not equal to 0. The derivative of f is given by

 f apostrophe open parentheses x close parentheses equals 48 x squared plus 1 over x cubed minus 22

The graph of f is concave up when x greater than n,  where n is the least possible number that makes that inequality true.

Find the value of n.

8b
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5 marks

Show that the curve y equals f open parentheses x close parentheses has only one point of inflection, and find the gradient of the normal line to the curve at that point.

9a
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7 marks

A curve has equation  y equals a x cubed plus b x squared plus c x plus d comma where a comma space b comma space c comma space d space element of space straight real numbers and a not equal to 0.

(i) Show that the curve will only have stationary points if a comma space b and c satisfy the inequality b squared greater or equal than 3 a c.

(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 comma space b and c.

9b
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4 marks

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

9c
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3 marks

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

9d
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1 mark

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.