IBMathsDPAnalysis & Approaches (AA)SLExam Questions5. CalculusDifferentiation

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

3 hours24 questions
Easy
Medium
Hard
Very Hard
1
3 marks

Let f\left(x\right) = -9x^{2} + 5x - 3.

Find the values of x for which f is increasing.

How did you do?

1a
2 marks

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

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

How did you do?

1b
4 marks

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.

How did you do?

2a
1 mark

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

Find f'\left(x\right).

How did you do?

2b
2 marks

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

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2c
3 marks

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

How did you do?

3a
3 marks

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.

How did you do?

3b
3 marks

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

How did you do?

4a
3 marks

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

Find

(i) f\left(2\right)

(ii) f'\left(2\right).

How did you do?

4b
3 marks

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.

How did you do?

4c
2 marks

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}.

How did you do?

5a
1 mark

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

Find f'\left(x\right).

How did you do?

5b
2 marks

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

Find the value of b.

How did you do?

5c
3 marks

Find the value of c.

How did you do?

6a
2 marks

The curve with equation y = ax^{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.

How did you do?

6b
1 mark

Hence find the values of a and b.

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6c
2 marks

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

How did you do?

7a
2 marks

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}.

How did you do?

7b
3 marks

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

How did you do?

7c
2 marks

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

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

How did you do?

8a
2 marks

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).

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

State the equations of the horizontal tangents to the curve.

How did you do?

8b
1 mark

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

How did you do?

8c
2 marks

Find the intervals where f is decreasing.

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8d
3 marks

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

How did you do?

9a
5 marks

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.

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

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

How did you do?

9b
2 marks

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

How did you do?

10a
2 marks

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.

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

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

How did you do?

10b
4 marks

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}.

How did you do?

11a
2 marks

The equation of a curve is y = x - \frac{9}{x} + 8, for x > 0.

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

How did you do?

11b
3 marks

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

Find the coordinates of \mathrm{A}.

How did you do?

11c
3 marks

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.

How did you do?

12
5 marks

Let f\left(x\right) = ax^{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.

How did you do?

13a
2 marks

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.

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

Calculate the instantaneous rate of change of f\left(x\right) when x = 2.

How did you do?

13b
4 marks

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

How did you do?

13c
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.

How did you do?

14a
6 marks

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.

How did you do?

14b
3 marks

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

How did you do?

1a
1 mark

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}.

How did you do?

1b
2 marks

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

How did you do?

1c
3 marks

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

How did you do?

2a
3 marks

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}.

How did you do?

2b
5 marks

Find the equation of

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

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

How did you do?

2c
2 marks

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

Find the coordinates of \mathrm{C}.

How did you do?

3a
6 marks

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.

How did you do?

3b
3 marks

Find the values of x for which f is

(i) increasing

(ii) decreasing.

How did you do?

3c
3 marks

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.

How did you do?

4
3 marks

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

Show that f is increasing for all values of x.

How did you do?

5a
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. 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.

How did you do?

5b
5 marks

The rate of change of V with respect to r is fraction numerator 5 pi over denominator 3 end fraction text  cm end text cubed per cm at a particular value of r.

Find this exact value of r.

How did you do?

6
7 marks

Let f\left(x\right) = ax^{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.

How did you do?

7a
4 marks

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.

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

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.

How did you do?

7b
3 marks

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.

How did you do?

1a
3 marks

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}.

How did you do?

1b
5 marks

Find the equation of

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

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

How did you do?

1c
3 marks

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

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

How did you do?

2a
6 marks

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.

How did you do?

2b
4 marks

Find the values of x for which f is

(i) increasing

(ii) decreasing.

How did you do?

2c
3 marks

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.

How did you do?