Practice Problems On Differentiation And Integration

Question 1

The work that you submit should include your working as well as your final answers. Your solutions should not involve the use of Maxima, except in those parts of questions where this is explicitly required or suggested. Your solutions should not involve the use of any other mathematical software. Your work should be written in a good mathematical style, as described in Section 6 of Unit 1, and as demonstrated by the example and activity solutions in the study units. Five marks (referred to as good mathematical communication, or GMC, marks) on this TMA are allocated for how well you do this. Your score out of 5 for GMC will be recorded against Question 11.

You do not have to submit any work for Question 11. Copyright c 2017 The Open University WEB 05385 5 8.1 TMA 03 Cut-off date 14 March 2018 Question 1 – 10 marks You should be able to answer this question after studying Unit 7. Differentiate the following functions. (a) f(x) = 6x 2 ln(4x) [3] (b) g(x) = e −2x − 1 cos(2x) [3] (c) h(t) = p (t 3 + sin (2t)) [4] Question 2 – 10 marks You should be able to answer this question after studying Unit 7. A closed box is to be made in the shape of a cuboid of height h (in cm) with a rectangular base that has sides of length x (in cm) and 2x (in cm). Its volume V is required to be 1000 cm3 .

(a) Write down an expression for the volume V in terms of x and h. Hence find an expression for h in terms of x. [2] (b) Write down an expression for the surface area A of the box in terms of h and x, and use the result from part (a) to write A in terms of x only. [2] (c) Use differentiation to determine the value of x for which A is a minimum. What is the corresponding value of h? Give your answers to three decimal places. [6] Question 3 – 5 marks You should be able to answer this question after studying Unit 7. Find the indefinite integral of the function f(x) = 1 − sin2 x sin2 x .

[5] Question 4 – 10 marks You should be able to answer this question after studying Unit 8. Use integration by substitution to find the indefinite integral in part (a) and to evaluate, to four significant figures, the definite integral in part (b). (a) Z sin x (3 cos x − 4)4 dx [4] (b) Z 1 0 2x 3 e 2x 4 dx [6] page 2 of 9 Question 5 – 5 marks You should be able to answer this question after studying Unit 8. Use integration by parts to find the indefinite integral Z (x + 3)2 sin x dx. [5] Question 6 – 10 marks You should be able to answer this question after studying Unit 8.

Question 2

This question is about the function f(x) = −(x + 1) sin(5x). (a) Explain why the graph of f lies on or above the x-axis for all values of x in the interval [7π/5, 8π/5]. [3] (b) Write down an expression, involving a definite integral, that gives the area between the graph of f and the x-axis, from x = 7π/5 to x = 8π/5. [1] (c) Use integration by parts to find the area described in part (b), giving both the exact answer and an approximation to three decimal places.

[6] Question 7 – 10 marks You should be able to answer this question after studying Unit 8. Include a printout or screenshot of your Maxima worksheet for this question. You are not expected to annotate your Maxima worksheet with explanation. However, remember that for good mathematical communication you should present your answers clearly. This question is about the function f(x) = x 3 + 6x 2 − 3x x 2 + 1 . Use Maxima to do each of parts (a)–(d). (In parts (c) and (d) you should round your answers to three decimal places yourself, instead of instructing Maxima to do it.) (a) Plot the graph of f, choosing ranges of values on the x- and y-axes to make its stationary points clearly visible.

[3] (b) Find the derivative of f. [1] (c) Calculate the x- and y-coordinates of the local maximum of f, to three decimal places. [3] (d) The graph of f crosses the x-axis at x = 0. Calculate the value of x where the graph crosses the x-axis to the left of x = 0. Find the area enclosed by the graph of f and the x-axis, between this value and x = 0, to three decimal places. [3] page 3 of 9 Question 8 – 15 marks You should be able to answer this question after studying Unit 9. Give sufficient details of your working to make it clear that you have not needed to use any software or calculator that can perform matrix manipulation. For example, in part (a), show explicitly how you calculated at least one element of each matrix answer.

(a) Let A = ? ?? 1 3 −2 1 0 0 −3 ? ?? and B = 9 1 −1 1 2 ! . Evaluate each of the following expressions, if possible. Where evaluation is not possible, explain why not. (i) AB (ii) BA (iii) A2 (iv) AB − A (v) −3A [8] (b) Determine whether the following matrices are invertible, and in each case find the inverse if it exists. (i) 2 3 4 5 (ii) 4 −2 6 −3 [3] (c) Use an answer from part (b) to solve the following system of linear equations: 2x + 3y = 4, 4x + 5y = 2. [4] page 4 of 9 Question 9 – 10 marks You should be able to answer this question after studying Unit 9. You can use Maxima to do the matrix arithmetic in this question, or you can do it by hand. If you use Maxima, then you do not need to include a printout, but you must include sufficient details in your solution to make it clear what calculations Maxima has done.

Question 3

Two nearby holiday resorts, A and B, receive visitors from the three countries X, Y and Z. The visitors are distributed as follows. 25% of the visitors from country X chose resort A, and 75% chose resort B; 85% of the visitors from country Y chose resort A, and 15% chose resort B; 30% of the visitors from country Z chose resort A, and 70% chose resort B.

(a) (i) Draw a network diagram, with input nodes X, Y and Z, and output nodes A and B, that represents the flow of visitors from the three countries to the two resorts. [2] (ii) Write down the matrix that represents this network. [2] (iii) Use the matrix that you found in part (a)(ii) to determine how many visitors visit resort A in a certain year if the total numbers of visitors from the three countries X, Y and Z in that year are 12 000, 15 000 and 7500, respectively. [2]

(b) The two resorts share three nearby beaches, E, F and G. The network diagram below represents the proportions of visitors at resorts A and B that use the three beaches E, F and G.

The matrix that represents this network is ? ? 0.6 0.1 0.1 0.5 0.3 0.4 ? ? . (i) Find the single matrix that represents the network obtained by combining the network in part (a)(i) with the network above. (That is, the combined network represents the flow of visitors from countries X, Y and Z to beaches E, F and G.) [2] (ii) Find the numbers of visitors at the beaches E, F and G if the total number of travellers from the three countries X, Y and Z in that year are 12 000, 15 000 and 7500, respectively. [2] page 5 of 9 Question 10 – 10 marks You should be able to answer this question after studying Units 7, 8 and 9. In this question, you are asked to attempt the mini examination paper on pages 8–9 and enter your answers on an answer form of the type that you will use in the real examination (see page 7).

This will give you practice in completing the form. You must submit your completed answer form as part of this TMA. If you are posting the TMA, then print and complete the form, and include it with your TMA. If you are submitting electronically, then either print and complete the form and scan or photograph it to produce a PDF, or complete it electronically by using annotation software. In either case you must then incorporate it into your TMA.

Question 4

There is a copy of the form as a single-page PDF in the ‘Assessment’ area of the MST124 website. You are not required to submit any working for the mini examination paper. Of the 10 marks for this question, 8 marks are for correct answers to the questions in the mini examination paper, and 2 marks are for completing the form correctly. Print or download a copy of the computer-marked examination form on page 7, read the instructions for completing it on page 8, and complete Part 1 of the form.

Then work through the mini examination paper on pages 8–9, and mark your answers on Part 2 of the form. [10] Question 11 – 5 marks Your score out of 5 marks for good mathematical communication in Questions 1 to 9 will be recorded under Question 11. You do not need to submit any work for this question. [5] page 6 of 9 page 7 of 9 MST124 Mini examination paper This paper has TWO sections. You should attempt ALL questions in each section. Section A has 2 questions, each worth 2 marks. Section B has 1 question, worth 4 marks. Each question is multiple-choice, with ONE correct answer from five options. No marks will be deducted for incorrectly answered questions. Mark your answers on the computer-marked examination (CME) form provided. Instructions for filling in the CME form are given below.

The instructions below are almost identical to those that will be provided in the real examination. You should follow the instructions carefully, just as you would in the real examination, except that the instruction to use an HB pencil is irrelevant if you are annotating the form electronically for this TMA, and of course the instruction that if you need a new form then you should ask an invigilator does not apply for this TMA. Instructions for completing the CME form If you do not follow these instructions, then the examiners may not be able to award you a score for the examination.

Writing on the form • Use an HB pencil.

• To mark a cell, pencil across it, as demonstrated on the form.

• To cancel a mark, pencil in the coloured part of the cell, as demonstrated on the form.

• If you make any unwanted marks on the form that you cannot cancel clearly, then ask the invigilator for a new form, and transfer your entries to it. Completing Part 1

• Enter your personal identifier (NOT your examination number) and the ‘assignment number’ for this examination, which is MST124 81, in the boxes provided.

• In the blocks headed ‘personal identifier’ and ‘module and assignment number’, pencil across the cells corresponding to your personal identifier and the assignment number given above. Completing Part 2

• For each question (numbered 1 to 3), mark your answer by pencilling across ONE of cells A, B, C, D or E.

• If you think that a question is unsound in any way, pencil across the ‘unsound’ cell (‘U’), as well as pencilling across an answer cell. • Do not pencil across any other cell. page 8 of 9 SECTION A Question 1 For a certain function g, it is known that Z 5 −4 g(x) dx = 2, Z 0 −2 g(x) dx = −2 and Z 5 0 g(x) dx = 1. What is the value of Z −2 −4 g(x) dx? A 1 B 3 C 5 D −1 E −5 Question 2 Given that y = 1 3 x 9 − 2 x 3 − 1, which of the following is equal to dy dx ? A 3x 8 + 2 3x 4 + 5x B 1 27 x 10 − 2 3x 4 C 1 30 x 10 − 1 2x 4 + 5x D 1 3 x 8 − 2 x 4 E 3x 8 + 6 x 4 SECTION B Question 3 Using the method of integration by substitution, which of the following is equal to Z (x + 6) (3x + 1)4 dx ? A 1 2 x 6 + 18 5 x 5 + 1 2 x 2 + 6x + c B 5 3 (3x + 1)4 + 68 3 (3x + 1)3 + c C 1 54 (3x + 1)6 + 17 45 (3x + 1)5 + c D 2(3x + 1)6 + 85 3 (3x + 1)5 + c E (3x + 1)6 + 3 5 (x + 6)(3x + 1)5 + c

Question 1

Differentiating 

(a).  f(x)=6x^2ln(4x)

f’(x)=d/dx(6x^2ln(4x))

First we remove the constant

f’(x)=6 d/dx(x^2ln(4x))

Second the product rule

(f*g)’=f’*g+f*g’

f=x^2 , g=ln(4x)

f’=d/dx(x^2)=2x ,  g’=d/dx(ln(4x))=1/x

Third substitute the values into the product rule

f’(x)=6 (2xln(4x)+(1/x)*x^2)

Final Result

f’(x)=6(2xln(4x)+x) 

(b).  f(x)=

f’(x)=d/dx()

First use the quotient rule

(f/g)’=(f’*g-f*g’)/g^2

f=e^-2x-1 ,  g=cos(2x)

f’=2e^-2x , g’=-2sin(2x)

Second substitute the values in the quotient rule

f’(x)=

Final Result

f’(x)= 

(c). h(t)=

h’(t)=

First use the chain rule

dfh(u)/dt=dh/du*du/dt

h= , u=)

d/du(=1/2

d/dt())=

Second substitute the values in the chain rule

h’(t)=1/2 *)

h’(t)=1/(2)*)

Final Result

h’(t)= 

Question 2

Closed box

(a). V=L*W*H

V=2x*x*h

V=

Expression of h in terms of x if V=1000

h=1000/

(b).  A=2LW+2LH+2WH

A=2(2x*x)+2(2x*h)+2(x*h)

Final result

A=

A=

A=

A= 

(c).  A’=

A’=

0=

x=7.211248

A”=8+6000/x^3

A”=24

h=1000/2x^2

h=9.615 

Question 3 

Expansion of the expression

 = 

Using the Sum rule

= - 

==-cos(x)

==x

Final result

=-cos(x)-x+C 

Question 4

(a).

First using substitution

U=cos(x)

=

By taking out the constant

=

Applying substitution with V=3u-4

=

Taking the constant out

=

=

Applying the power rule

=

Substituting back all the values

V=3u-4 and u=cos(x)

=

Final result

=

(b). 

Taking the constant out

=

Making u=2x^4 and substituting

=

=

=

We end up with

(

 =1.59726

Final Answer

1.597 

Question 5

Using integration by parts

Let u= and V=sin(x)

V’=-cos(x) , u’=2(x+3)

Hence

=

Further integrating the second part

 =

Hence

Simplifying

=

Final result

=+C

Question 6

Area under a curve

(a). The value of f(x) for both x’s

f(x)=

f(x)=2.021

f(x)=

f(x)=2.56

Hence, the region has positive value meaning it is above the x-axis

(b)  Upper limit (-Lower limit (

Where the limits are the given values for upper and lower bounds

(c).

Integration by part

Where

U=(x+1),  V=sin (5x)

U’=1,  V’=-1/5cos(5x)

Hence

=

Integrating the second part

 =

Finally

=+C

After integration

) for  =

) for  =

=

=

Area=-2.285 

Question 8

(a)

(i) AB(ii).   BA

Not possible, the number of columns in the first matrix must be equivalent to the number of row of the second matrix

(iii). A^2

No possible, a matrix can only be raised to a given power if the matrix has the same number of rows and columns

(iv).  AB-A

AB=

A=

Hence 

 

 

 Question 10

The form is missing from page 8-9 

Question 11

This Part is supposed to be BLANK.