DP IB Maths: AA SL

Topic Questions

5.2 Further Differentiation

1
Sme Calculator
4 marks

Differentiate fraction numerator 5 x to the power of 7 over denominator sin space 2 x end fraction with respect to x.

Did this page help you?

2a
Sme Calculator
4 marks

Find fraction numerator straight d y over denominator straight d x end fraction for each of the following:

y space equals space cos open parentheses x squared minus 3 x plus 7 close parentheses plus sin open parentheses straight e to the power of x close parentheses

2b
Sme Calculator
3 marks

y space equals space ln space open parentheses 2 x cubed close parentheses

Did this page help you?

3a
Sme Calculator
3 marks

Differentiate with respect to x, simplifying your answers as far as possible:

open parentheses 4 space cos space x minus 3 space sin space x close parentheses straight e to the power of 3 x minus 5 end exponent

3b
Sme Calculator
3 marks

left parenthesis x cubed minus 4 x squared plus 7 right parenthesis space ln space x

Did this page help you?

4
Sme Calculator
4 marks

A curve has the equation  y equals straight e to the power of negative 3 x end exponent plus space ln space x comma space space x space greater than space 0.

Find the gradient of the normal to the curve at the point open parentheses 1 comma space straight e to the power of negative 3 end exponent close parentheses comma giving your answer correct to 3 decimal places.

Did this page help you?

5
Sme Calculator
4 marks

Find the equation of the tangent to the curve y space equals space e to the power of 3 x squared plus 5 x minus 2 end exponent at the point open parentheses negative 2 comma space 1 close parentheses comma giving your answer in the form a x plus b y plus c equals 0, where a comma space b and c are integers.

Did this page help you?

6
Sme Calculator
6 marks

Let f open parentheses x close parentheses space equals space fraction numerator g open parentheses x close parentheses over denominator h open parentheses x close parentheses end fraction comma where g open parentheses 2 close parentheses equals 4 comma space h open parentheses 2 close parentheses equals negative 1 comma space g apostrophe open parentheses 2 close parentheses equals 0 and h apostrophe open parentheses 2 close parentheses equals 2

Find the equation of the tangent of f at x equals 2.

Did this page help you?

7a
Sme Calculator
3 marks

A curve has the equation y equals x cubed minus 12 x plus 7.

Find expressions for fraction numerator straight d y over denominator straight d x end fraction and fraction numerator straight d squared y over denominator straight d x squared end fraction.

7b
Sme Calculator
3 marks

Determine the coordinates of the local minimum of the curve.

Did this page help you?

8a
Sme Calculator
4 marks

The diagram below shows part of the graph of y space equals space f open parentheses x close parentheses comma where f open parentheses x close parentheses is the function defined by

f left parenthesis x right parenthesis equals left parenthesis x squared minus 1 right parenthesis space ln space left parenthesis x plus 3 right parenthesis comma   x greater than 3

q8-5-2-medium-ib-aa-sl

Points A comma space B and C are the three places where the graph intercepts the  x-axis.

Find f apostrophe space open parentheses x close parentheses.

8b
Sme Calculator
2 marks

Show that the coordinates of point A are open parentheses negative 2 comma space 0 close parentheses.

8c
Sme Calculator
3 marks

Find the equation of the tangent to the curve at point A.

Did this page help you?

9a
Sme Calculator
3 marks

Let f left parenthesis x right parenthesis equals x squared straight e to the power of x.

Find f apostrophe open parentheses x close parentheses.

9b
Sme Calculator
3 marks

Find f " space open parentheses x close parentheses.

9c
Sme Calculator
4 marks

Find the exact x of the points of inflection for the graph of f.

9d
Sme Calculator
1 mark

Find limit as space space x rightwards arrow negative 2 of x squared straight e to the power of x.

Did this page help you?

10a
Sme Calculator
1 mark

Let f open parentheses x close parentheses equals 2 straight e to the power of 2 cos x end exponent comma where negative pi less or equal than x less or equal than pi.

Find the number of points containing a horizontal tangent.

10b
Sme Calculator
4 marks

Show algebraically that the gradient of the tangent at x equals straight pi over 2 is negative 4.

10c
Sme Calculator
1 mark

State the gradient of the tangent at x equals fraction numerator 3 straight pi over denominator 2 end fraction.

10d
Sme Calculator
3 marks

It can be found that as the function, f comma undergoes a transformation f open parentheses k x close parentheses comma the number of stationary points found between negative pi less or equal than x less or equal than pi  increases.

Find the number of stationary points on f after a transformation of f open parentheses 2 x close parentheses and hence, state the general rule representing the number of stationary points in terms of k where k element of Z to the power of plus.

Did this page help you?

11
Sme Calculator
5 marks

Let f left parenthesis x right parenthesis equals sin space x and g left parenthesis x right parenthesis equals sin squared x comma for 0 less or equal than x less or equal than 2 pi.

Solve f to the power of apostrophe left parenthesis x right parenthesis equals g to the power of apostrophe left parenthesis x right parenthesis.

Did this page help you?