2014 MAM - ANALYSIS TASK 1

Student Name

TeacherAMADWEHSOINAJWGMMD

MATHEMATICAL METHODS (CAS) Unit 4Analysis Task 1Monday 4th August 2014Reading time: 10 minutesWriting time: 1 hour

Number of questionsNumber of questions to be answeredNumber of marks

3340

Students are permitted to bring into the test room: pens, pencils, highlighters, erasers, sharpeners, rulers, one bound reference, one approved CAS calculator or CAS software and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. Students are not permitted to bring into the examination room: blank sheets of paper and/or white out liquid/tape.Materials supplied Question and answer book of 11 pages with a sheet of formulas at the end. Working space is provided throughout the book.Instructions Write your name in the space provided above on this page. All responses must be written in English.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

Students must not disclose the contents of the task; to do so will be a breach of VCE guidelines and will be dealt with according to VCAA regulations.

InstructionsAnswer all questions in the spaces provided.Unless otherwise specified an exact answer is required to a question.In questions where more than one mark is available, appropriate working must be shown.Unless otherwise indicated, the diagrams in this book are not drawn to scale.The axes on any graphs drawn in this section must be labelled.

Question 1 (13 marks)

An isosceles triangle, ABC, can be constructed by joining the following three points on a cartesian plane: and .

-4

a. Determine the equations, in the form of y = mx + c, of the straight lines joining the points of the triangle listed above.

3 marks

b. State the base length and perpendicular height of the triangle and use these values to determine its area.

2 marks

c. State, but do not evaluate, the definite integral(s) that would allow you to find the shaded area of the isosceles triangle.

1 mark

d. Using calculus and algebra, show that the value of the definite integral(s) in part c is the same as the area found in part b.

3 marks

e. A rectangle is wholly enclosed inside the isosceles triangle so that one side of the rectangle lies along the base of the triangle. [As indicated in the diagram to the right]

(i) Write down, in terms of x, an equation for the area, A, of the rectangle.

(ii) Hence, use calculus to find the maximum possible area of the rectangle placed inside the triangle. You do not have to prove your value is a maximum.

1 + 3 = 4 marks

Question 2 (12 marks)

The number of hours of sunlight for the planet Mars per day can be modelled by the following circular function:

where is the number days after 31 December 2012 and .

a. Find the average rate of change in hours of sunlight per day between the 31 Dec 2012 and the 15 Jan 2013. Give your answer correct to 2 decimal places.

2 marks

b. Find the derivative of the function .

1 mark

c. What is the instantaneous rate of change, in hours of sunlight per day, on 15th Jan 2013? Give your answer correct to 2 decimal places.

1 mark

A spaceman liked to walk at sunset in the evenings. He noticed that at some times of the year the number of hours of sunlight per day did not change very much.

d. Find the value(s) of which gives no change in hours of sunlight for .

1 mark

e. Sketch the graph of the derivative: for . Label the -intercepts and the turning points in exact coordinate form.

3 marks

f. For what values of for is negative?

1 mark

The Winter Solstice occurs on the shortest day of the year. [i.e the day that has the least number of hours of sunlight]

g. Find the instantaneous rate of change of one day after the Winter Solstice. Give your answer correct to 2 decimal places.

1 mark

The Spring Equinox occurs exactly halfway between the Winter Solstice and the following longest day of the year. [the longest day has the most number of hours of sunlight]

h. Find the instantaneous rate of change of one day after the Spring Equinox. Give your answer correct to 2 decimal places.

1 mark

i. By comparing your answers to g and h, explain how the number of hours of sunlight changes near the Winter Solstice compared to the Spring Equinox.

1 mark

Question 3 (15 marks)

The cross-sectional area of a piece of cabling is shown in the diagram below.

There are four main cables inside the outer casing. Each of these four cables has one point of contact with two other cables. The cross sectional areas of each of the four cables (shaded part in diagram above) are symmetrical about the vertical and horizontal dotted lines indicated.The diagram below shows a cross-sectional view of the cabling with a set of Cartesian axis positioned so that the origin passes through the centre of the cavity between the four cables. The cross-sectional area of the cavity is symmetrical about the x and y axis.

The equation of the outer edge of the top left quadrant of the cross-section of cable 1 is given by

f : [a,4] R, f (x) = loge (x 1) + 2

The graph of y = f (x) is shown below.

a. Show that a = 1 + e -2.

1 mark

The inverse function f -1 models the outer edge of the bottom right quadrant of the cross-section of cable 2.

b.i. Find f -1(x), the inverse function of f (x).

2 marks

ii. Find the domain of f -1(x).

1 mark

iii. Write down the range of f -1(x).

1 mark

c. Using calculus and algebra, find the equation of the tangent to the function at the point where .

4 marks

d. Hence find the coordinates of the point of contact between the outer edge of cable 1 and cable 2.

1 mark

e. Using calculus, show that .

2 marks

f. Hence find the area of the cavity (shown in the diagram), between the four cables. Express your answer as an exact value.

3 marks

End of SAC1