MMCAS34SB 2ed 07 3pp

Review 1Review 1

77

MMCAS34SB_2ed_07_3pp.fm Page 277 Wednesday, June 17, 2009 3:58 PM

MathsWorld

Mathematical Methods CAS Units 3 & 4

278

Chapters 1–6 revision questions

1

Given that (

x

+

3) and (

x

−

2) are factors of 2

x

3

+

ax

2

+

bx

−

6, find the constants

a

and

b

. Hence solve the equation 2

x

3

+

ax

2

+

bx

−

6

=

0.

2

Three cubic storage containers sit side by side as shown with their visible faces in a vertical plane. The edge lengths of the boxes are

m

,

x

and

n

, where

m

<

x

<

n

. The front top corners

P

,

Q

and

R

are collinear.

a

If

m

=

4 and

x

=

6, find

n

.

b

Express

x

in terms of

m

and

n

.

3

Solve for

x

the following equations, giving answers in exact form.

a

log

10

(1

−

x

)

=

−

1

+

log

10

(2

x

+

1)

b

e

2

x

−

e

x

−

2

=

0

c

3

e

2

x

−

1

=

5

d

2 log

e

x

=

4

−

log

e

9

4

Simplify, giving answers in factorised form:

a

(

x

+

y

)

2

−

(

x

−

y

)

2

b

(

x

+

y

)

2

+

(

x

−

y

)

2

c

(

x

+

y

)

3

−

(

x

−

y)3 d (x + y)3 + (x − y)3

5 Find, giving your answer in exact form:

a {x : 2 + 4 sin x = 0, −π ≤ x ≤ π}

b

6 The solution to the equation 42x − 1 = 36 can be written in the form , where a and b are positive integers. Find a and b.

7 Let f : [−π, π] → R, .

a Sketch the graph of y = f(t).

b Give the period and amplitude.

c State the domain and range of f.

d i Show that is the smallest positive solution to the equation f(t) = 2.5.

ii Hence or otherwise give all the exact values of t for which f(t) = 2.5.

e Find, correct to 2 decimal places, the smallest positive t-axis intercept of the graph of y = f(t).

8 The temperature, y°C, at time t hours from midnight on a day in autumn is modelled by

the function .

a Calculate the temperature at 10:00 am.

b What were the minimum and maximum temperatures that day, and when did they occur?

c When did the temperature first reach 20°C?

d Find an expression for and hence determine the rate at which the temperature was changing at 10:00 am.

P

Q

R

x : 3 sin 2x 3 cos 2x x π2---–

π2---,⎝ ⎠

⎛ ⎞∈,=⎩ ⎭⎨ ⎬⎧ ⎫

alog e blog e 4---------------+

f t( ) 3 cos 2t π2---–⎝ ⎠

⎛ ⎞ 1+=

tπ12------=

y 17 6 sinπt12------ 0 t 24≤ ≤,–=

dydt------

MMCAS34SB_2ed_07_3pp.fm 78 Wednesday, June 17, 2009 3:58 PM

7 Review 1

279

e When was the temperature increasing most rapidly?

f What was the average rate of change in temperature during the period over which the temperature was increasing?

9 The fraction can be expressed in the form .

a Find a and b.

b Sketch the graph of , giving:

i the coordinates of all axes intercepts. ii the equations of the asymptotes.

10 a Consider the statement: ‘2x is an increasing function of x.’ Sketch the graph of y = 2x, and hence explain what is meant by the statement.

b Consider the statement: ‘sin x is a periodic function of x.’ Sketch the graph of y = sin x, and hence explain what is meant by the statement.

c Let f : R → R, f(x) = 2sin x.

i Is f an increasing function? Why or why not? (Hint: consider the values that the exponent, sin x, can take.)

ii Is f a periodic function? Why or why not?

iii Does f have any local maximum values? If so, what are they and for what values of x do they occur. (Do not use calculus.)

iv Does f have any local minimum values? If so, what are they and for what values of x do they occur. (Do not use calculus.)

v Use the information above to help sketch the graph of y = f(x).

11 Let f(x) = x4 − 2x2 − 8.

a Factorise f(x), and hence find {x : f(x) = 0}.

b Find f ′(x), and hence find {x : f ′(x) = 0}.

c Use the information in parts a and b to sketch the graph of y = f(x).

12 Let f(x) = 4 − x2.

a On the same set of axes, sketch the graphs of y = f(x) and y = 4 − .

b For what values of x is 4 − x2 ≥ 4 − ?

c The graph of y = 4 − a and the graph of y = f(x) have identical axes intercepts for a certain value of a. What is this value?

13 Find the derivatives of:

a b c d

14 The parabola with equation y = x2 is subject to the following transformations:

(1) a translation so that the turning point moves to (1, −4); then

(2) a dilation by a factor of from the y-axis.

a Sketch the curve after (1) but before (2) has been applied.

b Sketch the curve after the two transformations have been applied.

2x 1–x 2+

--------------- ax 2+------------ b+

y2x 1–x 2+

---------------=

x

x

x

2x 1–1

2x 1–--------------------- 2x 1–( ) 2x 1–× sin 2x 1–

12---

7


MathsWorld Mathematical Methods CAS Units 3 & 4

280

c Find the equation of the curve in part b, and express it in:

i turning point form: y = a(x − h)2 + k. ii intercept form: y = a(x − b)(x − c).

Hence verify your answer to part b.

d Would the results in parts b and c be the same if the order of the transformations was reversed? Why or why not?

15 Differentiate:

a b c

16 The function f has rule f(x) = a(x − h)2 + k. The graph of y = f ′(x) is shown at right.

a State the value of h.

b Find the value of a.

c Sketch the graph of y = f(x) if f(0) = −2.

17 A curve has equation . Find the equation of the tangent to the curve at the point where x = 2.

18 A function f is given by .

a Sketch the graph of f.

b State with reasons which of the following statements is true.

i f is continuous on R. ii f is differentiable on R. iii f is an even function.

c Sketch the graph of the gradient function f ′.

19 Find the exact value of if f(x) is given by:

a f(x) = sin 2x b f(x) = cos2 2x + sin2 2x

c f(x) = cos2 2x − sin2 2x d loge (sin 2x)

20 A cubic polynomial with rule f(x) = ax3 + bx2 + cx + d is defined by the conditions:

i f(−1) = 3 ii f(1) = 4 iii iv

a Write down four linear equations in a, b, c and d given by these four conditions.

b Express the system of simultaneous linear equations in part a in matrix form.

c Find the exact values of a, b, c and d.

21 Farmer Freda wants to construct three identical rectangular pens side by side. She has 120 metres of fencing material. Let each pen have dimensions x metres by y metres.

a Express y in terms of x.

b If A m2 is the total area enclosed by the pens, find an expression for A.

3x1x---+ 3x

1x---+⎝ ⎠

⎛ ⎞2

e3x 1e2x-------+

10–2 –1

–2

–4

2

4

2

y

x

y 2x sinπx4

------ 3–=

f x( )2 x 2+ x 1≥4x2 1– x 1<⎩

⎨⎧

=

f ′ π6---⎝ ⎠

⎛ ⎞

f ′ 1–( ) 136

------= f ′ 1( ) 16---=

y m

x m


7 Review 1

281

c Use calculus to find the value of x for which the enclosed area is a maximum.

d State the maximum area and the dimensions of each pen.

22 Consider the curve with equation y = 4 − 3x2.

a Sketch that part of the curve that lies in the first quadrant (i.e. x ≥ 0 and y ≥ 0).

A rectangle OPQR is drawn with P on the x-axis, Q on the curve and R on the y-axis.

b i If P is the point (1, 0), find the coordinates of Q and R.

ii Calculate the area of this rectangle.

c i If P is the point (x, 0), show the rectangle on the sketch in part a.

ii Express the area of this rectangle in terms of x.

iii Use calculus to determine the value of x corresponding to the rectangle of maximum area, and state the area in this case.

23 The volume V of a sphere of radius r is given by .

a Find .

A spherical balloon is inflated by a gas canister so that its volume increases at the constant rate of 4π cm3 per second.

b Find the rate of change of the radius in cm per second when the radius is 5 cm.

c Describe how the radius changes over time.

24 Find an antiderivative of each of the following:

a b c

25 Use index laws to simplify the expression .

Hence find the exact value of dx.

26 Evaluate the following definite integrals, giving exact answers in each case.

a (x + 1)(2x − 1) dx b

c d

27 Consider the curve with equation y = 2xe−0.5x.

a Show that the curve passes through the origin O.

b Find .

c Find the equation of the normal to the curve at O.

d Find the exact coordinates of M, the maximum point on the curve.

28 Let f(x) = ax2 + bx + c, where a, b and c are constants.

a If f(1) = −1 and f ′(1) = 4, express b and c in terms of a.

b If f(x) dx = −1, find f(x).

c Find the minimum value of f(x).

V43---πr3=

dVdr-------

x 1x---–⎝ ⎠

⎛ ⎞ 2

3 x1

x--------+ 1

3 2x–( )3----------------------

e2x 3ex 4+–ex

-------------------------------

−1

0⎛⎜⎝

⌠⎮⌡

e2x 3ex– 4+ex

-------------------------------⎞⎟⎠

⌠⌡1

2 ⌠⎮⌡0

π/3

cos 2x 3 sin3x2

------+⎝ ⎠⎛ ⎞ xd

⌠⎮⌡0

12

2x 1+--------------- xd

⌠⎮⌡0

π/121

cos2 3x------------------- dx

dydx------

⌠⌡0

1

7



282

29 A wooden box, open at the top, is to be made with a square base of side x cm and height of y cm. Its volume will be 4000 cm3.

a i Express y in terms of x.

ii Hence show that the area A cm2 of wood required to make the box is given by

.

b i Find .

ii Hence find the value of x for which the area of wood required is a minimum.

iii What are the dimensions of the box, and how much wood is required to make it?

30 This CAS screenshot shows the graph of a function f on [0, π]. Its rule is f(x) = a sin bx + c, where a, b and c are integers with b > 0.

a State the values of a, b and c.

b Find the exact values of x for which f(x) = 0.5.

c Give the coordinates of the x-axis intercepts correct to three decimal places.

d Find f ′(x) and hence find the exact value of .

e Find the coordinates of the point at which the slope of the graph of y = f(x) is steepest and negative.

Examination-style technology-free questionsFor VCE questions, marks allocated are shown. For additional questions, marks have been included to indicate their relative worth.

1 Use the remainder theorem to determine if 2x4 − 3x3 + 7x + 11 is exactly divisible by x + 1.

[VCE MM 3/4 Exam 1 2003 Part II Q1] [2 marks]

2 Let P(x) = x3 − 3x2 + 2x + d, where d is a constant. P(x) is divisible by x − 3.

a Find d.

b Hence show that the equation P(x) = 0 has exactly one solution.[2 + 1 = 3 marks]

3 Let f : R → R, f(x) = 2x2 + 12x + 10.

a Write f(x) in the form a(x + b)2 + c.

b Hence write down the coordinates of the turning point of the graph of f.

[VCE MM 3/4 Exam 1 2001 Part II Q2] [1 + 1 = 2 marks]

4 If a > 0, x > 0, and 2 loga x + loga 9 = 2 + loga 2, express x in terms of a.[2 marks]

5 Find the exact solutions of the equation .

[VCE MM & MM (CAS) 3/4 Exam 1 2003 Part II Q3 (CAS Q2)] [2 marks]

6 Show that the line with equation y = −x + 2 cuts the curve with equation in exactly one point, and state the coordinates of that point.

[3 marks]

A x2 16 000 x

----------------- +=dAdx-------

f ′ π6---⎝ ⎠

⎛ ⎞

sin 2πx 3– cos 2πx 0 x 1≤ ≤,=

yx

2x 1–---------------=


7 Review 1

283

7

The diagram shows one cycle of the graph of a circular function.

a

State the period of the function.

b

State the amplitude of the function.

c The graph shown can be obtained from the graph of

y

=

cos

x

by a series of transformations. Give a possible sequence.

[1

+

1

+

3

=

5 marks]

8

On the one set of axes, sketch the graphs of

y

=

e

x

,

y

=

e

2

x

and

y

=

e

x

+

0.5

, giving the exact coordinates of any intersection points.

[4 marks]

9 a

The graph of

g

is obtained from the graph of the function

f

with rule

f

(

x

)

=

x

2

by a translation by

+

3 units parallel to the

x

-axis. Write down the rule for

g

.

b

The graph of

h

is obtained from the graph of

g

by a translation by

−

1 unit parallel to the

y


h

.

c

The graph of

k

is obtained from the graph of

h

by a dilation by a scale factor of 0.5 from the

y


k

.

[VCE MM 3/4 Exam 1 2004 Part II Q2] [1

+

1

+

1

=

3 marks]

10

State a possible sequence of transformations so that the graph of

y

=

tan

x

is transformed

into the graph of .

[3 marks]

11

Find the smallest exact value of

x

,

x

>

0, which satisfies the equation

.


12

The graph of the function with rule is transformed as follows:

.

a dilation by a factor of from the

y

-axis

. a reflection in the y-axis

. a translation of +3 units parallel to the x-axis

. a translation of +1 unit parallel to the y-axis.

a Write down the equation of the rule of the transformed function.

b Hence state the domain and range of the transformed function.

[VCE MM & MM (CAS) 3/4 Exam 1 2002 Part II Q5] [1 + 2 = 3 marks]

13 Sketch the graph of for 0 ≤ x ≤ 2, showing the coordinates of all

stationary points and axes intercepts.[5 marks]

y

0 x

1.0

1 2 3 4 5 6 7 8

0.5

–0.5

–1.0

–1.5

y 2 tan 3x π2---–⎝ ⎠

⎛ ⎞=

sin 5xπ3---+⎝ ⎠

⎛ ⎞ 22

--------–=

y1x---=

12---

y 2 2 cos π x13---+⎝ ⎠

⎛ ⎞–=

7



284

14 a The graph of the function f with rule f(x) = 2 loge (x + 3) + 1 intersects the axes at the points (a, 0) and (0, b). Find the exact values of a and b.

b Hence sketch the graph of the function with rule f(x) = 2 loge (x + 3) + 1 on the axes shown. Label any asymptote with its equation.

c On the axes shown, sketch the graph of f ′, thederivative of f. Label any point(s) of intersectionwith the axes with its coordinates. Label anyasymptote with its equation.

[VCE MM 3/4 Exam 1 2003 Part II Q4] [2 + 2 + 2 = 6 marks]

15 Let f : D → R, f(x) = 2 loge , where D is the maximal domain of f.

a State D.

b Find the exact coordinates of the points where the graph of y = f(x) intersects the x- and y-axes.

c Sketch the graph of y = f(x). Indicate any asymptote with its equation.

[VCE MM (CAS) 3/4 Exam 1 2003 Part II Q3] [1 + 3 + 2 = 6 marks]

16 If y = (2 + x2)4/3, evaluate at x = 5.[2 marks]

17 Find the exact value of .[3 marks]

18 Let f : , where D is the largest subset of R for which f is defined.

a Express f(x) in the form .

b State D.

c On the axes shown, sketch the graph of the function with equation y = f(x). Clearly mark the coordinates of the points of intersection with the axes. Clearly label any asymptotes with their equations.


1–2–3–4 –1–1

–2

1

2

2 3 4

y

x

3

–3

0

1–2–3–4 –1–1

–2

1

2

2 3 4

y

x

3

–3

0

x 3+

dydx------

⌠⎮⌡0

π/2

3 cosx2--- 4 sin 2x–⎝ ⎠

⎛ ⎞ xd

D R→ f x( ), xx 1+------------=

1–2–3 –1–1

–2

1

2

2 3

y

x

3

–3

0

f x( ) Ax 1+------------ B+=


7 Review 1

285

19 Let f : R → R, f(x) = 3e−x − 2.

a Sketch the graph of y = f(x).

b State the range of f.

c The x-axis intercept can be expressed in the form loge a − loge b, where a and b are positive integers with no common factors. Find a and b.

[2 + 1 + 2 = 5 marks]

20 The diagram shows the graphs of two circular functions, f and g.

State the type of transformation, together with any relevant scale factors, distances or directions, required to transform the graph whose equation is y = f(x) into the graph whose equation is y = g(x).


21 The curve with equation , has horizontal and vertical asymptotes

with equations x = −1 and y = −3. The curve passes through (1, −1).

a Find the values of A, B and C.

b Sketch the curve.

[3 + 2 = 5 marks]

22 a Find the coordinates of the point P on the curve with equation y = x2 − 2x − 1 at which the tangent is parallel to the line y = 3x − 5.

b Find the equation of the normal to the curve at the point P.


23 Let V : [0, ∞) → R be given by V(t) = −0.5(t + 5)2 + 2000.

a Find the value of t for which the instantaneous rate of change of V with respect to t is −10.

b Find the average rate of change of V with respect to t over [0, 2].


24 The graph of is reflected in the x-axis, then dilated by a factor of 2 from the x-axis, and finally translated so that its vertex is at (−1, 3). For the graph that results from this sequence of transformations:

a sketch the graph.

b state the equation of the graph.

[2 + 2 = 4 marks]

π π2 2

y

0x

2

2ππ 3

1

–2

–1

y=f(x)

y=g(x)

yA

x B+( )2-------------------- C x B–≠,+=

y x=

7



286

25 Let f : R → R be a continuous function with the following properties:f(0) = 0 f ′(0) = 0 f ′(x) < 0 for (−∞, 0) ∪ (0, 3)f(4) = 0 f ′(3) = 0 f ′(x) > 0 for (3, ∞) a On the set of axes provided, sketch a

possible graph of f.

b If f(x) = ax3(x − b) and the point (2, −4) also lies on the graph of f, find the values of a and b.

c Use calculus to find the equation of the tangent to the graph of f at the point where x = 4.


26 Marco uses a computer graphing package to draw the curve with equation y = 3 tan 4x in the window with dimensions [0, π] by [−10, 10]. He then draws the graph of on the same set of axes.

a How many intersections of the two graphs would Marco see in this window?

b Find exactly the smallest value of x at which the graphs intersect in this window.

[1 + 2 = 3 marks]

27 a Find the exact value of k for which 3x = ekx for all x ∈ R.

b Hence use calculus to find the derivative of y = 3x with respect to x.


28 Given that and , find f(x).

[3 marks]

29 The graph of is shown in the CAS

screenshot. Find in exact form the coordinates of the maximum point on the graph.

[3 marks]30 The graph of the function f, with domain (−2, 4),

is shown at right.

1–2–3–4 –1–5 2 3 4 5

y

x0

y 3=

f ′ x( ) ex 2+( )2= f 0( ) 32---=

ylog e x

x--------------- x 0>,=

–1

–2

1

2

3

–3

1–2–3 –1 2 3 4

y

x0


7 Review 1

287

a On the set of axes below, sketch the graph of the derivative function f ′.

b State the domain of f ′.[VCE MM 3/4 Exam 1 2002 Part II Q7] [2 + 1 = 3 marks]

31 The graph of the function f: R\{−1} → R has a vertical asymptote with equation x = −1 and a horizontal asymptote with equation y = 2. It also has the following properties.

f(0) = 0f ′(x) > 0 for all x < −1f ′(x) > 0 for all x > −1

Sketch a possible graph of y = f(x).


32 The graph of a function f with equation y = f(x) is shown. Its axes intercepts have coordinates (−1.5, 0) and (0, 1). Sketch the curves with equations below.

a y = f(−x)

b y = −f(x)

c y = f(2x)

d y = 2f(x)

[1 + 1 + 1 + 1 = 4 marks]33 Let

a Find f ′(x).b Use the linear approximation formula for f(x + h) to approximate .

[1 + 2 = 3 marks]

34 A cubic polynomial function with rule f(x) = ax3 + bx2 + cx + d, is uniquely defined by the following conditions.

f(−1) = 1 f ′(1) = −2 f ′(0) = 0

a Each of the above conditions can be written as a linear equation in terms of a, b, c and d. Write down the corresponding linear equation for each condition.

b Express this as a system of simultaneous linear equations in matrix form.

c Solve the system of simultaneous linear equations exactly, and state the rule of the cubic polynomial function.

[VCE MM (CAS) 3/4 Exam 1 2002 Part II Q4] [2 + 2 + 2 = 6 marks]

–1

–2

1

2

3

–3

1–2–3 –1 2 3 4

y

x0

1–2 –1

–2

2

2

y

x

4

–4

0

f x( ) x=

26

f 1( ) 53---=

7



288

35 Let f : R → R, f(x) = ax2 + bx + c, where a, b and c are real numbers and a ≠ 0. Suppose that f(1) = 6 and f ′(1) = 4.

a Find the values of a and b in terms of c.

b Suppose f(x) dx = 6. Find the value of c.

[VCE MM (CAS) 3/4 Exam 1 2003 Part II Q4] [2 + 2 = 4 marks]

36 Part of the graph of y = f(x), where f: R → R is given below. For a fixed non-zero real constant p, the three points A, B and C, with coordinates (−p, f(−p)), (0, f(0)) and (p, f(p))respectively, lie on a straight line.

a Show that .

b Let p = 1 and let f(x) = ax3 + bx2 + cx + d, where a, b, c and d are real constants, and a ≠ 0.

i Show that b = 0.

ii Hence show that for all x ∈ R.

[VCE MM (CAS) 3/4 Exam 1 2005 Part II Q4] [1 + 5 = 6 marks]

Examination-style extended response questionsFor VCE questions, marks allocated are shown. For additional questions, marks have been included to indicate their relative worth.

1 Sama runs a small pottery business. Due to capacity constraints, fixed costs and costsof material, when Sama makes a batch of x bowls, the cost per bowl in dollars is x2 − 12x + 48. Sama charges $75 for one bowl.

a What is the cost of making a batch containing a single bowl?[1 mark]

⌠⌡0

1

yy = f(x)

x

A

B

C0

f p–( ) f p( )+2

------------------------------ f 0( )=

f x–( ) f x( )+2

----------------------------- f 0( )=


7 Review 1

289

b Sama finds that by making b bowls in a batch, the cost per bowl is a minimum.i What is the value of b?ii What profit per batch will she make if she makes b bowls in a batch?

[2 + 1 = 3 marks]

c Sama wonders whether she could make a larger profit using a different batch size.i If $C is the total cost of making x bowls in a batch, express C in terms of x.ii If $P is the profit per batch, express P in terms of x.

iii Find and hence find the value of x that maximises the profit per batch. State the

profit made for a batch of this size.

[1 + 1 + 4 = 6 marks]

[Total: 10 marks]

2 The temperature, T degrees Celsius, in a greenhouse at t hours after midnight for a typical

November day is modelled by the formula for 0 ≤ t ≤ 24. Use thismodel to answer the following questions.

a State the maximum and the minimum temperatures in the greenhouse.[2 marks]

b At what time does the maximum temperature occur?[1 mark]

c At what times will the temperature equal 23°C?[2 marks]

d Consider the hottest four-hour interval in the greenhouse for a typical November day. What is the minimum temperature in this four-hour interval? Give your answer correct to 2 decimal places.

[2 marks]

e In the greenhouse, there is an automatic switch for a root watering system. Whenever the rate of change of temperature, with respect to time, is greater than or equal to +0.2°C per hour, the system switches on. It switches off again when the rate of change of temperature, with respect to time, is less than +0.2°C per hour.i Use calculus to find an expression for the rate of change of temperature with respect

to time.ii Hence find the range of values of t (correct to 2 decimal places) for which the

watering system will be on.[1 + 3 = 4 marks]

[VCE MM 3/4 Exam 2 2001 Q1] [Total: 11 marks]

3 Let f(x) = −x3 + 3x2 + 9x − 11.

a Show that the graph of y = f(x) passes through the point (1, 0).[1 mark]

b Find, in exact form, the remaining intercepts of the graph of y = f(x).[2 marks]

dPdx------

T 25 4 cosπ t 3–( )

12-------------------–=

7



290

c Find:

i f ′(x) ii {x : f ′(x) = 0}

[1 + 2 = 3 marks]

d Find and classify the turning points of the graph of y = f(x).[2 marks]

e Use the information above to sketch the graph of y = f(x), labelling all key features.

[3 marks]

f There is a point on the graph of y = f(x) at which the gradient is a maximum. Find the equation of the tangent at this point, and show it on your sketch in part e.

[3 marks]

[Total: 14 marks]

4 Consider the function f: [0, 2π] → R, f(x) = 2 sin (x). The graph of f is shown below, with tangents drawn at points A and B.

a i Find f ′(x).

ii Find the maximum and minimum values of |f ′(x)|.

[1 + 2 = 3 marks]

b i The gradient of the curve with equation y = f (x), when x = , is 1.

Find the other value of x for which the gradient of the curve, with equation y = f (x), is 1.(The exact value must be given.)

ii Find the equation of the tangent to the curve at x = .(Exact values must be given.)

iii Find the axes intercepts of the tangent found in b ii.(Exact values must be given.)

[1 + 2 + 3 = 6 marks]

–2

2

0 2

y

A

B

x

π3---

π3---


7 Review 1

291

c The two tangents to the curve at points A and B have gradient 1. A translation of m units in the positive direction of the x-axis takes the tangent at A to the tangent at B. Find the exact value of m.

[2 marks]

d Let h: R → R, h(x) = 2 |sin (x)|.Find the general solution, for x, of the equation h(x) = 1.

[2 marks]

[VCE MM (CAS) Exam 2 2006 Section 2, Qn 1] [Total: 13 marks]

5 The population of a waterborne parasite in a small sample of water from a water supply is measured each month from the beginning of the year. The number N is modelled by the rule N(t) = Aet + B, where t is the number of months since the first measurement. At the beginning of January the number was 200, and at the beginning of April it was 250.

a i Use the information above to find two equations in A and B.

ii Hence show that, correct to 2 decimal places, A = 2.62 and B = 197.38.

[2 + 3 = 5 marks]

b What does the model predict for the number of parasites present at the start of June?

[1 mark]

c Sketch the graph of N against t, showing all key features.[2 marks]

The monthly area under the graph of part c is used to determine at what stage it will be necessary to treat the water to eliminate the parasite. As the treatment is expensive, it will only commence if the increase in the area from one month to the next is greater than 50%.

d Approximate the area during the month of February, using a single left rectangle, giving the answer correct to 1 decimal place.

[2 marks]

e Use a single left rectangle to approximate the area during the month of March, giving the answer correct to 1 decimal place.

[2 marks]

f What is the percentage increase between the February area and the March area, as approximated in parts d and e, correct to the nearest 1 per cent?

[1 mark]

g Predict at the beginning of which month the treatment is likely to commence using

i the approximate method above.

ii a definite integral and formal antidifferentiation.

[1 + 2 = 3 marks]

[Total: 16 marks]

7



292

6 Since the early 1970s, global population growth rates have been declining. The World Bank uses the following model for the population, P(t), in thousand millions of a particular group of people (for example, the low income group) at any time after the beginning of 1990.

P(t) = P0eG(t), where P0 thousand million is the population of the given group at the beginning of 1990, t is the number of years after the beginning of 1990, and G is a function of t, where G ′(t) = a + bt, and a and b are real constants. This model can be applied to various population groups by using appropriate values of P0, a and b.

a i This model can be used when G′(t) ≥ 0. For a particular group, a > 0 and b < 0. Find, in terms of a and b, the number of years for which the model can be used for this group.

ii Show that G(0) = 0.

iii Show that .[2 + 2 + 2 = 6 marks]

For the low income group, as defined by the World Bank, the population at the beginning of 1990 was three thousand million, with a = 0.02 and b = −0.0002.

b i Show that P(t) = 3e(0.02t − 0.0001t2).

ii What is the greatest number of years for which this model for the low income group can be used?

iii Using this model, find the rate of change of the population of the low income group with respect to time.

iv At the beginning of the year 2010, what will be the rate of change of the population (in thousand millions per year) of the low income group? Give your answer correct to 3 decimal places.

[1 + 1 + 1 + 1 = 4 marks]

c For another group, P = 3e0.01t, t ≥ 0, where P thousand million is the population t years after the beginning of 1990.

i Find a rule for t in terms of P.

ii Find the year during which the population of this group will be double what it was at the beginning of the year 1990.

[2 + 1 = 3 marks]

[VCE MM & MM (CAS) 3/4 Exam 2 2004 Q3 [Total: 13 marks]

7 An open fish tank is to be made with glass sides, a glass base and a metal frame bounding the five faces. The rectangular metal frame will be constructed first and the glass panes added later. A length of 18 metres of metal is used for the frame. The specifications call for the height of the finished tank to be half its length.

Let the tank have length x metres and width y metres.

a Show that 3x + 2y = 9.[1 mark]

G t( ) at12---bt2+=


7 Review 1

293

b Find an expression for the total surface area of glass required, A m2, in terms of x, and hence determine the largest area of glass that could possibly be required to construct the tank. State the dimensions of the tank in this case.

[5 marks]

In practice, it is desired to make the fish tank with the largest possible volume consistent with the stated specifications.

c Find an expression for the total volume, V m3, of the tank in terms of x.

[1 mark]

d i Find .

ii Hence, determine the tank with maximum possible volume and state the dimensions of this tank.

[2 + 3 = 5 marks]

e What percentage saving is there in glass used compared to the maximum that might have been required?

[2 marks]

[Total: 14 marks]

8 Victoria Jones wants to construct a time capsule in which to bury some of her treasures. The time capsule will be a right circular cylinder of height h cm, and radius r cm, with hemispherical caps of radius r cm on each end, as shown in the diagram. Let the total volume of the capsule be V cm3.

a Express V in terms of r and h.

[1 mark]

b The total volume of the capsule will be 8000 cm3.

i Show that .

ii The values which r may take lie in an interval. Find the endpoints of this interval, correct to 2 decimal places.

[2 + 2 = 4 marks]

c The material for the cylindrical part of the capsule costs 2 cents per cm2 of surface. The material for the hemispherical caps costs 3 cents per cm2 of surface.

Find an expression for C cents, the total cost of the materials for the capsule, in terms of r.

(The surface area of a sphere of radius r is 4πr2.)

[2 marks]

dVdx-------

r cm

h cm

h 8000πr2

------------ 4r3-----–=

7



294

d Sketch the graph of C over an appropriate domain on the axes shown. Label any horizontal or vertical asymptote with its equation. You are not required to show the coordinates of any turning point.

[3 marks]

e Use calculus to find the value of r, correct to 2 decimal places, for which C is a minimum. [You do not need to justify that the value you find is a minimum.]

[3 marks]


9 In an outback region of Australia, the population of a certain predator varies according to the population of its prey. If P(t) is the population of the predator and Q(t) is the population

of its prey at time t months, the respective populations are modelled by

and , where a, b, c and d are positive constants.

a Over a 12 month period, the predator population fluctuates between 700 and 1300.

i Find the values of a and b.

ii Sketch the graph of P versus t for the 12 month period given by 0 ≤ t ≤ 12.

[1 + 2 = 3 marks]

b Over a 12 month period, the population of the prey fluctuates between 15 000 and 25 000.

i Find the values of c and d.

ii Sketch the graph of Q versus t for the 12 month period given by 0 ≤ t ≤ 12.

[1 + 2 = 3 marks]

c There is a period of time during the year when the prey population is increasing while the predator population is decreasing. For what values of t is this true?

[1 mark]

d Describe how the two populations interact over the 12 month period.

[2 marks]

A measure of the relative sizes of the two populations is the ratio, R(t), of prey to predator

populations, where .

e i Calculate R at times t = 0, 3, 6, 9 and 12.

ii Sketch the graph of R against t.

[1 + 2 = 3 marks]

10 000C

r0

P t( ) a b sinπt6-----+=

Q t( ) c d cosπt6-----+=

R t( ) Q t( )P t( )-----------=


7 Review 1

295

A small value of R indicates that the prey population is at a low point compared to the predator population; a large value of R indicates that the prey population is abundant in comparison to the predator numbers.

f Find, correct to 2 decimal places, the minimum and maximum values of R.

[2 marks]

g A prey to predator ratio of 20 is considered to be optimal for the interacting populations.

i The equation R(t) = 20 can be expressed in the form tan , where k is a constant. Find k.

ii Without solving the equation, explain why the solutions to the equation occur at times that are 6 months apart.

iii For the 12 month period, find the values of t for which the prey to predator ratio is optimal. Give your answers correct to 2 decimal places.

[2 + 1 + 1 = 4 marks]

[Total: 18 marks]

10 On an adventure park ride, riders are strapped into seats on a platform that starts 15 metres above the ground and goes up and down repeatedly. The distance, x metres, of the platform above the ground, t seconds after the ride starts, can be modelled by the

formula .

a i According to this model, find the maximum height above the ground reached by the platform.

ii According to this model, how many seconds after the ride starts is the platform first closest to the ground, and how high above the ground is it at that time?

[1 + 2 = 3 marks]

Tasmania Jones is redesigning the ride so that the platform moves further up and down each cycle. During the first 60 seconds of the redesigned ride, y metres, the distance of the platform above the ground t seconds after the ride starts, can be modelled by the formula

.

b i According to this model, the platform is exactly 6 metres above the ground for the first time about 58 seconds into the ride. Find this time correct to 2 decimal places of a second.

ii According to this model, how many times is the platform exactly 15 metres above the ground from t = 40 to t = 59?

iii According to this model, find the time from when the ride starts until the platform first reaches 24 metres above the ground. Give your answer correct to the nearest second.

[1 + 1 + 2 = 4 marks]

πt6----- k=

x t( ) 15 6 sinπt3-----+=

y t( ) 15 e0.04t sinπt3----- 0 t 60≤<,+=

7



296

c i Find an expression for .

ii Hence write down an equation, one solution of which is the value of t when the platform is closest to the ground. Find this value of t, correct to 2 decimal places. Also find, according to the model, the distance in metres of the platform above the ground at that time, correct to 2 decimal places.

[2 + 3 = 5 marks]

Safety regulations for the ride require that .

d Find the range of values of t, correct to 3 decimal places, for which during the first 60 seconds of the ride.

[2 marks]

Tasmania can adjust the ride so the model for the distance, in metres, of the platform above

the ground, t seconds after the ride starts, becomes , where a is a positive constant.

e Find, correct to 3 decimal places, the greatest value of a such that is never more than 11 during the first 60 seconds of the ride.

[3 marks]


11 The pollution level, y units, along a straight road between two factories, A and B, which are 10 km apart, is given by

, where 0 ≤ x ≤ 10

where x km is the distance from Factory A, and p and q are positive constants.

a On a particular day the values of p and q are measured to be p = 9 and q = 4.

i Find

ii Find the exact value of x at which the pollution level y is a minimum and the exact value of this minimum.

iii Sketch the graph of , where 0 ≤ x ≤ 10. Clearly label the

endpoints and turning points with their coordinates. Exact values are required.

iv Jack travels from Factory A to Factory B along the road. For what length of his journey (in kilometres correct to three decimal places) is the pollution level less than 5?

[1 + 3 + 2 + 1 = 7 marks]

b On another day only the value of q is known, q = 4. The pollution level, y units, is given by

, where 0 ≤ x ≤ 10 and p is a positive constant.

For what values of p does

i the maximum value of y occur when x = 10?

ii the minimum value of y occur when x = 10?

[3 + 2 = 5 marks]

[VCE MM (CAS) 3/4 Exam 2 2005 Q4] [Total: 12 marks]

dydt------

11–dydt------ 11≤ ≤

11–dydt------ 11≤ ≤

h t( ) 15 ae0.04t sinπt3----- 0 t 60≤<,+=

dhdt-------

yp

x 1+------------ q

11 x–---------------+=

dydx------

y9

x 1+------------ 4

11 x–---------------+=

yp

x 1+------------ 4

11 x–---------------+=


MMCAS34SB 2ed 07 3pp

Documents

Transcript of MMCAS34SB 2ed 07 3pp