DP IB Maths: AA SL

Topic Questions

5.1 Differentiation

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.

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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.

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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)

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.

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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 functionspace f left parenthesis x right parenthesis.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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