Second Semester Examination 2018/2019 Academic Session
June 2019
MAA101 - Calculus for Science Students I ( Kalkulus untuk Pelajar Sains I)
Duration : 3 hours (Masa : 3 jam)
Please check that this examination paper consists of TEN (10) pages of printed ma- terial before you begin the examination.
[Sila pastikan bahawa kertas peperiksaan ini mengandungi SEPULUH (10) muka surat yang bercetak sebelum anda memulakan peperiksaan ini.]
Instructions: Answer SIX (6) questions.
[Arahan: Jawab ENAM (6) soalan.]
In the event of any discrepancies, the English version shall be used.
[Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai].
(a) The height in feet (unit symbol: ft) of a baseball hit straight up from the ground with an initial velocity of 64 ft / s is given by h = f (t) = 64t − 16t2, where t is measured in seconds (unit symbol: s) after the hit.
(i) Is f (t) one-to-one on the interval 0 ≤ t ≤ 4? By analyzing the curve of f (t) and considering when the ball travels upward and downward, discuss how f(t) can be made one-to-one function and give its corresponding domains and ranges.
(ii) Find the inverse, f−1, which gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = f−1(h).
(iii) At what time is the ball at a height of 30 ft on the way up?
(b) Find the domain and range and sketch the graph h(x) =√
4 − x2. How is the graph of h(x) =√
4 − x2related to the graph of g(x) =√
4 − x2+ 1? Use the graph of h(x) to sketch the graph of g(x).
[ 30 marks ] Soalan 1
(a) Ketinggian dalam kaki (simbol unit: kaki) suatu bola besbol yang dipukul secara terus ke atas daripada kawasan dataran dengan halaju awal 64 kaki / s diberikan oleh h= f (t) = 64t − 16t2, yang mana t diukur dalam saat (simbol unit: s) selepas suatu pukulan.
(i) Adakah f(t) ini satu-ke-satu pada selang masa 0 ≤ t ≤ 4? Dengan menganal- isis lengkung f(t) dan mempertimbangkan apabila bola bergerak ke atas dan ke bawah, bincangkan bagaimana f(t) boleh dijadikan fungsi satu-ke-satu dan berikan domain dan julat yang sepadan.
(ii) Cari songsangan, f−1, yang dapat memberikan masa t pada ketika bola be- rada di ketinggian h apabila bola ini bergerak ke atas. Nyatakan jawapan anda dalam bentuk t= f−1(h).
(iii) Pada masa berapakah bola ini akan berada di ketinggian 30 kaki dalam per- jalanan ke atas?
(b) Cari domain dan julat serta lakarkan graf h(x) =√
4 − x2. Bagaimanakah h(x) =
√
4 − x2berkait dengan graf g(x) =√
4 − x2+1? Gunakan graf h(x) untuk melakarkan graf g(x).
[ 30 markah ]
...3/-
Question 2
(a) Compute the following limits:
(i) lim
t→1
t4− 1 t3− 1
(ii) lim
h→0
1
(x + h)2− 1 x2 h (iii) lim
x→−4
√
x2+ 9 − 5 x+ 4 (b) Find lim
x→−2
2 − |x|
2 + x , if it exists. If the limit does not exist, explain why.
(c) If a function f represents a system that varies in time, the existence of lim
t→∞f(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value:
(i) The population of a culture of tumor cells is given by f (t) =3500t t+ 1 . (ii) The value of an investment in dollars is given by f (t) = 1000e0.005t. (iii) The population of a colony of squirrels is given by f (t) = 2100
3 − 2e−0.5t .
(d) Let g(x) =
x2+ x, x< 1
a, x= 1
3x + 5, x> 1 .
(i) Determine the value of a for which g is continuous from the left at 1.
(ii) Determine the value of a for which g is continuous from the right at 1.
(iii) Is there a value of a for which g is continuous at 1? Explain.
[ 40 marks ] Soalan 2
(a) Kirakan had-had berikut:
(ii) had
h→0 h
(iii) had
x→−4
√
x2+ 9 − 5 x+ 4
(b) Cari had
x→−2
2 − |x|
2 + x , jika wujud. Sekiranya had ini tidak wujud, terangkan sebabnya.
(c) Jika fungsi f mewakili suatu sistem yang berubah mengikut masa, kewujudan had
t→∞ f(t) memberi makna bahawa sistem telah mencapai keadaan mantap (atau keseimban- gan). Untuk sistem berikut, tentukan sama ada keadaan mantap wujud dan beri nilai keadaan mantap tersebut:
(i) Populasi sel tumor diberikan oleh f(t) =3500t t+ 1 .
(ii) Nilai pelaburan dalam dolar diberikan oleh f(t) = 1000e0.005t. (iii) Populasi suatu koloni tupai diberikan oleh f(t) = 2100
3 − 2e−0.5t .
(d) Biar g(x) =
x2+ x, x< 1
a, x= 1
3x + 5, x> 1 .
(i) Tentukan nilai a yang mana g adalah selanjar dari kiri pada 1.
(ii) Tentukan nilai a yang mana g adalah selanjar dari kanan pada 1.
(iii) Adakah wujud nilai a yang membuatkan g selanjar pada 1? Terangkan.
[ 40 markah ]
...5/-
Question 3
(a) Find the first derivative of the function f (x) =√
9 − x using the definition of deriva- tive. State the domains of f (x) and f0(x).
(b) Find the first derivatives of the following functions:
(i) y= x2− x + 2
√x
(ii) y= x2+ 14
(2x + 1)3(3x − 1)5 (iii) y= ln | sec (5x) + tan (5x)|
(c) The curve y x2+ 4 = 8 is an example of Witch of Agnesi equation. This equation can be used to approximate the spectral energy distribution of x-ray lines in physical sciences.
(i) Use implicit differentiation to finddy dx.
(ii) Find equations of tangent lines to this curve when y = 1.
(iii) Find explicit expression for y in terms of x and then calculate dy dx. (iv) Verify that your answer in part (iii) is equivalent to part (i).
[ 30 marks ] Soalan 3
(a) Cari terbitan pertama untuk fungsi f(x) =√
9 − x menggunakan takrif terbitan. Ny- atakan domain-domain untuk f(x) dan f0(x).
(b) Cari terbitan pertama bagi fungsi-fungsi berikut:
(i) y= x2− x + 2
√x
(ii) y= x2+ 14
(2x + 1)3(3x − 1)5 (iii) y= ln | sec (5x) + tan (5x)|
(c) Lengkung y x2+ 4 = 8 adalah merupakan suatu contoh persamaan yang dikenal
(ii) Cari persamaan garis-garis tangen untuk lengkung ini apabila y= 1.
(iii) Cari persamaan eksplisit bagi y dalam sebutan x dan kemudian kirakan dy dx. (iv) Sahkan bahawa jawapan anda di bahagian (iii) bersamaan dengan bahagian
(i).
[ 30 markah ]
...7/-
Question 4
(a) Given f (x) =x2+ 12
2x + 1, f0(x) = 2 −12 + x + x2
(1 + 2x)2 and f00(x) = 98
(1 + 2x)3, find (i) all the x- and y-intercepts and asymptotes,
(ii) the intervals on which f is increasing or decreasing, (iii) the local maximum and minimum points, if any, and (iv) the intervals of concavity and the inflection points, if exist.
Then, sketch the graph of f .
(b) Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 m. What are the dimensions and volume of a square-based box with the greatest volume under this condition?
[ 40 marks ] Soalan 4
(a) Diberi f(x) = x2+ 12
2x + 1, f0(x) =2 −12 + x + x2
(1 + 2x)2 dan f00(x) = 98
(1 + 2x)3, dapatkan (i) kesemua pintasan-x dan -y dan asimptot,
(ii) selang bagi f menokok atau menyusut,
(iii) titik maksimum dan titik minimum tempatan, jika ada, dan (iv) selang kecekungan dan titik lengkuk balas, jika wujud.
Seterusnya, lakarkan graf f .
(b) Andaikan satu polisi syarikat penerbangan menetapkan supaya kesemua bagasi dike- hendaki berbentuk kotak dengan jumlah panjang, lebar, dan tinggi tidak melebihi 108 m. Apakah dimensi dan isipadu kotak yang bertapakkan segi empat sama dengan isipadu yang terbesar di bawah syarat ini?
[ 40 markah ]
Evaluate the following integrals:
(a) R(sec w)9(sec w tan w) dw (b) R x2ln2x dx
(c) R
1 3
0
dx (9x2+ 1)32
, ifR dx (x2+ a2)32
= x
a2√
x2+ a2, a ∈ R
(d) R 2x + 4 x3− 2x2dx
[ 30 marks ]
Soalan 5
Nilaikan kamiran berikut:
(a) R(sec w)9(sec w tan w) dw (b) R x2ln2x dx
(c) R
1 3
0
dx (9x2+ 1)32
, jikaR dx (x2+ a2)32
= x
a2√
x2+ a2, a ∈ R
(d) R 2x + 4 x3− 2x2dx
[ 30 markah ]
...9/-
Question 6
Sketch the region bounded by y = 6x and y = x2+ 5. Then, set up, but do not evaluate the integral for
(a) the area of the region,
(b) the volume of the solid generated when the region is revolved about the x-axis, and (c) the volume of the solid generated when the region is revolved about the line y = −1.
[ 20 marks ] Soalan 6
Lakarkan rantau yang dibatasi oleh y= 6x dan y = x2+ 5. Seterusnya, nyatakan kamiran, tanpa menilaikannya untuk
(a) luas rantau tersebut,
(b) isipadu pepejal yang dihasilkan apabila rantau tersebut diputarkan sekitar paksi-x, dan
(c) isipadu pepejal yang dihasilkan apabila rantau tersebut diputarkan sekitar garis y= −1.
[ 20 markah ]
1. tan x = sin x cos x 2. sec x = 1
cos x 3. csc x = 1
sin x 4. cot x = 1
tan x 5. sin2x+ cos2x= 1 6. 1 + tan2x= sec2x 7. 1 + cot2x= csc2x 8. d
dx(xn) = nxn−1 9. d
dx(ex) = ex 10. d
dx(ln x) =1 x 11. d
dx(ln f (x)) = f0(x) f(x) 12. d
dx(sin x) = cos x 13. d
dx(cos x) = − sin x 14. d
dx(tan x) = sec2x
15. d
dx(sec x) = sec x tan x 16. d
dx tan−1x = 1 1 + x2 17. d
dx( f g) = f g0+ g f0 18. d
dx
f g
= g f0− f g0 g2 19. d
dx( f (g(x))) = f0(g(x))g0(x) 20. Rxndx= xn+1
n+ 1+C 21. R 1
xdx= ln |x| +C 22. Rexdx= ex+C
23. Rln x dx = x ln x − x +C 24. Rsin x dx = cos x +C 25. Rcos x dx = sin x +C 26. Rtan x dx = ln | sec x| +C
27. Rsec x dx = ln | sec x + tan x| +C 28. Rsec2x dx= tan x +C
29. Rsec x tan x dx = sec x +C
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