49 | P a g e
12. (a) – Refer to Lecturer – (b) f 1 ( ) x 3(e x 1) , x
0
5
13. (a) x , x 1
3
x
x
(b) (i) (h h )( ) (ii) h 1 ( ) x
x
x 1
14. (a) – Refer to Lecturer –
(b) −1 ( ) = ( ) = [3, ∞); −1 ( ) = ( ) = [2, ∞)
(c) y
-1
f (x) y = x
f(x)
3
2
x
2 3
x 4 1
15. (a) ( )q x (b) x
2 2
(c)
y
2
y =1
x
( ) is one-to-one function because the horizontal line cuts at one point only.
( ) = (−∞, ∞) ; ( ) = (1, ∞)
50 | P a g e
16. domain 1, range 3,
3
0
1
17. (a) – Refer to Lecturer –
(b)
1
0 1
D R 0,
g
f
R D
,
g
f
1 1
2x 1 , x 7x 10 , x
18. (a) f x 2 h x 2
2x 1 , x 1 x 14 , x 1
2 2
51 | P a g e
(b)
y
h(x
)
14
27/
2 x
0
1/2
(c) k x x 2 4 ; x 1
1
19. (a) f g ( ) 5x px (b) p
5
20. (a) - Refer to lecturer for graph - ; R ,1 4 3,
f
(b) ( 3)f 4 ; (0) 3 ; (1)
f
f
4
x
21. (a) (i) f f ( ) ; f 1 ( ) x f ( ) x (ii) a 1, b 1, c
2
x
x
x 1
(b) q 1
22. p 4
4
23. (a) x (b) b ln5
3
2
2
24. (a) ( )f x x 2 1 ; h
(b) - Refer to lecturer for graph –
f is a one to one function because a horizontal lone cut the graph at 1 point only.
R 1,
f
2
x
(c) f 1 ( ) x 1 ; f 1 (3) 4
2x 5
x
25. (a) g 1 ( ) (b) a 2
a
52 | P a g e
26. (a) Refer to Lecturer - (b) D [ 4, ) , R [ 2, )
f
f
1
27. (a) (i) f g ( ) 4bx (ii) b (b) 7
x
4
1
x
28. f 1 ( ) log x - Graph refers to Lecturer –
2 2
8x 2 5 p
2
29. (a)(i) ( )g x (ii) ( )g x x (b) 3
6
30. (a) - Graph refers to Lecturer -
D f 1, D
,
g
R R g 0,
,
f
(b) x 3 ln2
2 x 2
g x
31. x 1
32. p 1,q 2
33. (a)
= { : ∈ } −1 = { : > 0}
= { : > 0} −1 = { : ∈ }
(b) 403.43
34. (a) g f x x 1
1
(b) (i) f 1 x x 1 (ii) g f x x 2 1
(c) D 1 R 1 1,
(g f ) (g f )
35. (a) (i) -refer to lecturer-
e 2
x
(ii) f 1 x
3
(iii) D R f 1 2 , R D f 1
f
f
3
53 | P a g e
mn 52
(b) x 7 4 n
54 | P a g e
CHAPTER 6 : POLYNOMIALS
1
2
3
1. Find the values of a and b if ( )f x ax bx 12 is divisible by x and x 2
.
By using these values of a and b, solve the equation ( )f x 0.
3
2
2. Find the value of r for which x 2 is a factor of 3p x x 4x rx 8.
Hence, factorize ( )p x completely.
2
3. Given that ( )Q x px p 2 x 3p 2 has a remainder of 5 when divided by x 2 .
(a) Find the values of p .
(b) By using the positive value of p from part (a), find all the zeroes of ( )Q x .
4. Given a polynomial ( ) 2P x x 7x 10x 24.
2
3
(a) Show that x 2 is a factor of ( )P x .
(b) Factorize ( )P x completely.
210
(c) Express as a partial fractions.
P ( )
x
2
3
5. The polynomial ( )P x is defined by ( ) 2P x x ax 4x b
.
(a) If x 2 is a factor of ( )P x , and 4 is the remainder when ( )P x is divided by x 1 ,
find the values of a and b.
(b) Find all the factors of ( )P x .
x 1
(c) Express as partial fractions.
P x
2
2
3
6. Determine the integers p and q such that x x 4x x p x q . Hence, express
4
5x 2
in the form of partial fractions.
x x 4x 4
3
2
2
3
.
7. Given x 1 is a factor of the polynomial 2P x x ax 2x b When ( )P x is divided
by x 2 , the remainder is – 6. Find the values of a and b. Hence, factorise ( )P x completely.
55 | P a g e
2
3
8. The polynomial 2P x x 3ax ax b has x 1 as a factor and leaves a remainder of
–54 when divided by x 2 . Find the values of a and b.
3
9. Express as partial fractions.
x 1 x 2 1
2
3
P
P
10. The function P is defined as x px 2x qx 2 where ,p q . x is divisible by
x 2 and 8 is the remainder when x is divided by x 1 .
P
P
(a) Find the value of p and q. Hence, factorize x completely.
2x 2 6x 1
(b) Express in partial fraction.
P x
4
3
P
7
P
11. Given the polynomial x x ax bx 4x . When x is divided byx 1 x 1
, the remainder is 2x 5. Find the values of aand b.
3
2
P
12. The polynomial x x x 10x a is divisible byx 1 .
(a) By using the Factor Theorem, determine the value of a.
P
(b) Factorize x completely.
4x 1
(c) Express in partial fractions.
P x
5x 2 17x 17
13. Express as a sum of partial fractions.
x 2 x 1 2
2
3
14. Given x 3 is one factor of 9 12P x x 11x 2x . Factorise completely
13x 18
P x , and express as a sum of partial fractions.
P x
56 | P a g e
2
3
15. A polynomial has the form 2P x x 3x px q , with x real and p, q constants.
When x is divided by x the remainder is 2 4x .Find the values of p and q, and
P
1
P
factorize x completely if 2 is one of the roots.
2
3
16. Polynomial 2P x x ax x b has x 1 as a factor and leaves a remainder 12 when
divided by x 3 . Determine the values of a and b.
2
3
P
17. Two factors of the polynomial x x ax bx 6 are x 1 and x 2 . Determine
the values of a and b, and find the third factor of the polynomial.
2x 2 5x 13
Hence, express as a sum of partial fractions.
P x
3
2
P
6
18. (a) Show that x 3 is a factor of polynomial x x 2x 5x .
P
Hence, factorize x completely.
2
f
1
(b) If x ax bx c leaves remainder 1, 25 and 1 on division by x , x 1
and x 2 respectively, find the values of a, b and c. Hence, show that x has
f
two equal real roots.
2x 1
19. Express in partial fractions.
(x 2)(x 2x 4)
2
20. (a) Find a cubic polynomial ( ) (Q x x a )(x b )(x c satisfying the following
)
conditions:
3
The coefficients of x is 1, Q( 1) 0,Q(2) and Q(3) 8.
0
3
2
(b) A polynomial ( )P x ax 4x bx 18 has a factor x 2 and a remainder 16
when divided byx 1 . Find the values of a and b. Hence, factorise P ( )
x
completely.
57 | P a g e
4x 3
21. Express in partial fractions.
2
x 2 x 2x 2
2
3
22. Given a polynomial 2P x x ax bx 30 has factors x 2 and x 5 .
(a) Find the value of the constants a and b.
P
(b) Factorize x completely.
5x 2 14x 13 A B C
23. By writing in the form of the partial fractions , find A,
2
x 3x 3 x 3 x 1 x 1
3
x
B and C .
2
3
24. When 3P x x px qx 3 is divided by x 2 x 2 , the remainder is 8x .
1
By using the Remainder Theorem, find the values of p and q. Hence, obtain the quotient.
2
3
6
x
25. Given 2f x x 5x .
f
(a) Show thatx 2 is a factor of x .
2
f x
(b) Find a, b and c such that x 2 ax bx c .
x 4
(c) Express as partial fractions.
f x
3
2
2
26. Given that ax b is a remainder when polynomial 2x 5x 28x 15 is divided by x 1
. By using the remainder theorem, find the value of a and b .
4
27. Express in the form of partial fractions.
x 1 x 2 9
2
3
28. The polynomial ( ) 5S x x px qx 1 gives a remainder of 26 on division by (x 1) and
remainder of –8 on division by (x . Find the values of the constants p and q.
1)
29. Given ( ) 10P x x 6x 11 and ( ) x 5x 8x
3
2
2
x
Q
4
x
(a) Show that (x 1) is a factor of ( ) .
Q
(b) Factorise ( )Q x completely.
P ( )
x
(c) Express as partial fractions.
Q ( )
x
58 | P a g e
3
2
1)
30. Given (x and (x 2) are factors of the polynomial ( ) 2S x x px qx 2 . Find the
values of constants p and q.
4
2
3
31. Given ( )f x x 5x 8x .
(a) Show that (x is a factor of ( )f x .
1)
)
(b) Find the value of a such that ( ) (f x x 1)(x a .
2
2x 1
2
(c) Hence, express as partial fractions.
f ( )
x
4x 2 9x 17
32. Express ( )S x in partial fractions
(x 2)(x 2 1)
2
3
33. Given a polynomial ( )P x mx nx 1with (2x 1) is one of the factors and ( )P x has a
remainder of 4 when divided by (x-1). Determine the values of m and n. Hence, factorize
completely.
34. A polynomial ( ) 2P x x mx nx 8 gives a remainder of 6x 4 when divided by
3
2
x 1 x 2 . Find the values of m and n.
2
2
3
x
35. Given ( )f x x 3x 1 and ( )q x x 3x 5.
1)
(a) Show that (x is a factor of ( )f x .
(b) Find the values of a, b and c such that ( ) (f x x 1)(ax bx ) c by using long
2
division.
q ( )
x
(c) Hence, express as partial fractions.
f ( )
x
2
3
36. The polynomial x mx nx 2x gives a remainder 14 when divided by x and
3
P
1
a remainder 7 when divided by x 2 . Determine the values of the constants m and n.
2
3
P 7x x 2 5 Q x x 2x 2
x
37. Given and .
(a) Show that x 2 is a factor of x .
Q
59 | P a g e
Q
(b) Factorize x completely.
P x
(c) Express as partial fractions.
Q x
38. (a) When a polynomial x is divided by x 2 andx 3 the remainders
P
P
R
are 5 and l0 respectively. Find the remainder x , when x is divided
by x 2 x 3 where x ax b , a and b are constants.
R
44 10x
(b) State the factors of 2x 2 5x 3. Hence, express as partial
(x 3)(2x 5x 3)
2
fractions.
ax b
P x
39. Given a polynomial of degree 3, x 2 x 3 Q x where a and b are
constant.
(a) ( ) has the remainder −18 and 17 when divided by ( + 2) and ( − 3)
respectively. Find the values of a and b, and state the remainder when ( ) is divided
by ( + 2)( − 3)
(b) Let ( )Q x mx n. Use the values of a and b in part 39(a) to determine ( ) if the
coefficient of x in ( ) is 2 and 4P 42.
3
x 12
(c) Express in partial fraction.
x 2 x 2 3
60 | P a g e
ANSWERS
1. a 3,b 9; x 1,2,2
2
2. r 0, ( )p x x 2 3x 2x 4
1 5 21
3. (a) p or p 1 (b)
3 2
4. (a) - Refer to Lecturer -
(b) x 2 2x 3 x 4
210 5 24 7
(c)
x 2 2x 3 x 4 x 2 2x 3 x 4
0
5. (a) a 2 , b
(b) Factors : 2x , x 2 and x 1
x 1 1 1 1
(c)
P x 4x 12 x 2 3 x 1
5x 2 4x 4 1
6. p 4 , q 1;
2
x x 4x 4 x 4 x 1
2
3
7. a 6 , b ; P 2x x 1 x 1 x 3
6
4
8. a 3 , b
3 3 3 3
9.
2
x 1 x 1 4 x 1 2 x 1 2 4 x 1
61 | P a g e
7
3
P x
10. (a) p , q ; x 2 3x 1 x
1
2x 2 6x 1 1 1 1
(b)
x 2 3x 1 x 1 x 2 2 3x 1 2 x 1
11. a 2 , b 3
12. (a) a 8
x
(b) P x 1 x 2 x 4
4x 1 1 3 1
(c)
P x x 1 2 x 2 2 x 4
5x 2 17x 17 3 2 5
13.
(x 2)(x 1) 2 x 2 x 1 x 1 2
13x 18 1 3 2
14. P ( ) (x 3) (1 2 ) ,
x
2
x
x
P ( ) x 3 (x 3) 2 1 2x
2
15. p 3 , q ; ( ) (P x x 2)(2x 1)(x
1)
6
16. a 5 , b
2x 2 5x 13 1 1 2
3
17. a 2 , b 5; Third Factor = x ;
P ( ) x 1 x 2 x 3
x
4
9
18. (a) ( ) (P x x 3)(x 2)(x (b) a , b 12, c
1)
2x 1 1 (x 4)
19.
(x 2)(x 2x 4) 4(x 2) 4(x 2x 4)
2
2
2
20. (a) ( ) (Q x x 1)(x 2)(x 5) (b) a 1 , b 3 ; ( ) (P x x 2)(x 3)
4x 3 1 x 4
21.
x 2 x 2x 2 2 x 2 2 x 2x 2
2
2
62 | P a g e
P x
22. (a) a 3 , b 29 (b) x 2 x 5 2x 3
4
23. A 2, B 1, C
24. p 4 , q 3, The quotient is 3x 1
25. (a) - Refer to Lecturer –
(b) a , b 1, c 3
2
x 4 2 2 1
(c)
x
f ( ) 3 x 2 2x 3 3 x 1
26. a 26 , b 10
4 1 1 1
27.
x 1 x 2 9 3 x 3 6 x 3 2 x 1
28. p 8 , q 22
29. (a) - Refer to Lecturer -
2
(b) ( ) (Q x x 1)(x 2)
x
P ( ) 15 5 39
(c)
Q ( ) x 1 x 2 x 2 2
x
30. p 1 , q 5
31. (a) - Refer to Lecturer -
2
(b) a
2x 1 1 1 7
2
(c)
(x 1)(x 2) 2 x 1 x 2 x 2 2
3 x 7
32. S ( ) x
x 2 x 2 1
2
33. m 2 , n 3 ; ( ) (2P x x 1)(x 1)
63 | P a g e
8
34. m 8, n
35. (a) – Refer to Lecturer –
(b) a 1, b , c 1
2
q ( ) 3 5x 7
x
(c)
f ( ) 2(x 1) 2 x 2 2x 1
x
m 7 ,n 20
36. 3 3
Q x
37. (a) - (b) x 2 x 1 x 1
P x 11 2 6
(c )
Q x x 2 x 1 x 1
38. (a) The remainder is + 7
44 10x 4 2 2
(b )
(1 2 )(x 3) 2 1 2x x 3 (x 3) 2
x
39. (a) a 7,b 4;R 7x x 4
(b) Q 2x x 5
2 2x 3
(c)
x 2 x 2 3
64 | P a g e
CHAPTER 7: LIMIT
1. Evaluate the following limits if they exist:
x 1 x 3
3
(a) lim (b) lim
x 1 x 1 x x 9
2. Evaluate the following limits if they exist:
2
e 1 x 2
3x
(a) lim (b) lim
x
x 0 e 1 x 3x 6
3. Find the following limits.
x 4 1 1
2
(a) lim (b) lim
x 2 x 2 x 0 x 1 x x
4. Evaluate each of the following limits, if it exists.
x 4 3x x
4
(a) lim (b) lim
2
x 4 x 2 x x 6
e e 2x
x
5. Evaluate lim .
x
x 0 e e x
6. A function f is defined by
34 , x 4
x
0 , 2
f ( ) 17 , x 4
x
4 2
x 3x 4 , x 4, x 3, x 2, x 4
x 2 x 6
Evaluate lim ( )f x .
x 2
7. Evaluate each of the following limits, if it exists.
65 | P a g e
2 x 3x 6
(a) lim (b) lim
x 2 x 2 6x 8 x x 2
2
x 2, x 2
3
17 2
x
8. Given x x 8 A, 2 2
f
x
Bx 5, 2 5
10, x 5
(a) Evaluate
x
(i) lim f ( )
x 3
x
(ii) lim f ( )
x 10
(b) If lim ( )f x and lim ( )f x exists, find the value of A and B .
x 2 x 5
f 5x
9. (a) If lim 1 , find lim f x .
x 4 x 2 x 4
x 4x 2x 8
2
3
(b) Evaluate lim .
x 4 x 4
x 3
x 9 , x 1
2
, 1 x
10. Function f is defined as x 10 px 3
f
2
x , 3 x 5
x
, x 5.
2
2x 1
(a) Evaluate:
(i) lim f x (ii) lim f x (iii) lim f x .
x 5 x 3 x
(b) Find the value of p if lim f x exists.
x 3
11. Evaluate each of the following limits, if exists.
66 | P a g e
x 1 1 x 2 4x 5
(a) lim . (b) lim .
x 2 x 2 x 1 x 1
2x 4, x k
12. (a) Given x
f
k
kx 5 , x k .
Find the value of k if lim f x exist.
0
x k
f
(b) The function x is defined as:
x , x 1
f x x 1, 1 x 3
x
8 2 , 3 x 6
4 , x 6.
Find
(i) lim f x (ii) lim f x (iii) lim f x
x 1 x x 3
13. Evaluate each of the following limits, if exists.
16x 4 x 7
2
(a) lim (b) lim
2
x 4x 3 x 7 3x 4 5
a x , x b
1
14. Given x x 6 , x b
b
f
3
x 5a , x b
Find the values of constants a and b if lim f x and lim f x exist.
x b x b
x 25
2
15. Evaluate lim .
x 5 2x 1 3
67 | P a g e
2x 3 x 2 1
16. (a) Evaluate lim .
3
x 2x 1
(b) Determine the values of A and B if the limit of ( )f x exist at the points x 1 and
x 4.
4 , x 1
4
f x Ax B , 1 x
2
Ax Bx 1 , x 4
2 x
17. (a) Evaluate lim .
x 4 4 x
2
(b) If lim ( )f x and lim ( ) 64g x , find lim f ( ) x g ( ) x ( )f x .
2
x 1 x 1 x 1
18. Function f is defined as
x 2 p , x 1
q , 1 x 2
2
( )f x x 4 , 2 x 6
x 2
2x 2 x 2
, x 6
x 2 3x 1
(a) Find lim f ( ).
x
x
(b) Find the value of p if lim ( ) 5f x .
x 2
(c) Find the value of q if lim ( )f x exists
x 2
19. Evaluate each of the following limits, if exists.
2
2x 6 3x 11x 1
3
x
(a) lim (b) lim
x 3 x 2 2x 3 x 4x 2x 3
ax 7, x k
3,
k x 1
2
20. Given x x 1, 1 x k .
f
3x 1, k 5
x
16, x 5
Find the values of k if lim f x exists.
x k
68 | P a g e
21. (a) Find lim x x 2 5x .
x
x 4, x 1
22. Given x , A 1 x B , where A and B are constants.
f
x 1, x
B
f
(i) Find the values of A and B if the limit of x exist at x 1 and x B .
(ii) Hence, find lim f x .
x 6
23. A function is defined as
3 x
2x 2 7x 3 , x 3
, m x 3
x
f x n , 3 x
5
7
x 7
, 5 x 7
x 7
p 2, x 7
Where m, n and pare constants.
(a) Find lim f x
x 3
(b) Determine the values of and such that lim f x f 3 lim f x
x 3 x 3
(c) Find the values of such that lim f x lim f x
x 7 x 7
ANSWERS
1. (a) 3 (b) 0
1
2. (a) 3 (b)
3
69 | P a g e
1
3. (a) –4 (b)
2
4. (a) 4 (b) 3
3
5.
2
6. 4
1
7. (a) 2 (b) 3
641 ,B
A
8. (a) (i) 29 (ii) 10 (b) 1
64
9. (a) lim ( ) 7 (b) 18
f
x
x 4
1 1
10. (a) (i) 25 (ii) (iii) 0 (b) p
6 3
x 1 1 1 x 2 4x 5
11. (a) lim (b) lim 6
x 2 x 2 2 x 1 x 1
12. (a) k 1
(b) (i) Does not exist.
(ii) lim f x 4
x
(iii) lim f 2x
x 3
10
13. (a) 4 (b)
3
14. a 2, b
3
15. 30
3
16. (a) 1 (b) A , B 7
1
17. (a) (b) 20
4
4
18. (a) 2 (b) p 1 (c) q
1 3
19. (a) (b)
2 2
70 | P a g e
20. k 0,k 3
5
21.
2
22. (i) A 5,B 4 (ii) lim f 7x
x 6
1 1 8
23. (a) (b) m ,n (c) p 2 7 2
5 5 35
71 | P a g e
CHAPTER 8: DIFFERENTATION
dy
1. Find when x for each of the following:
0
dx
2 e 2x (2x 1)
3
(i) y ln x x 1 (ii) y .
x 1
dy
2. Given ln y e xy , find
dx
2
d x e x
x
3. If y x e , show that 0
dy 2 1 e x 3
dy
4. Find the value of at x 1 if
dx
1 2 2 1 3x
(a) y 2x 3x (b) y
x x 2 x 3
2
2
4
5. Given that x 2xy 2y . Solve for x and y if dy 0 .
dx
2
2
3
0
6. Let y x y 2x for x .
(a) If y =1, find the value of x.
dy dy
(b) Find . Hence, evaluate when y = 1.
dx dx
B
7. Given y Ax 2 , where A and B are constants and x .
0
x
dy d y d y dy
2
2
0
(a) Find and . Hence, show that x 3 x 2 3B .
dx dx 2 dx 2 dx
dy
(b) Find the values of A and B if y = 3 and 3 when x = 1.
dx
72 | P a g e
x
e e x dy 2
8. If y , show that y 2 1.
2 dx
2
B 2 d y dy
9. Given y Ax 2ln x, show that x x y 2ln x .
x dx 2 dx
10. Differentiate the following with respect to x .
3x 1 3
(a) y (b) y 2x 3 ln 4x 5 .
x 2
dy dy
2
2
11. (a) Find if x 3y 2xy 19 0. Hence, evaluate at 2,3 .
dx dx
3
(b) Given y Ax B e x , find the constants A and B if y and dy 5 when
dx
x 0 .
2
12. Find '( )f x for ( ) 3f x x 2x 1 by using the first principle.
dy
13. (a) Find for e 4x (x y 2 1) 1.
dx
ln(x 1) dy 1 1 ln(x 1)
(b) Given y , show that
x dx x x 1 2x
14. Differentiate the following with respect to x :
2 2
x
(a) f ( ) 3x x
3
3x 3
(b) ( )f x 2x 1 x 2 1
x
2 x
(c) ( ) ln(f x x e 2 5x 1 )
73 | P a g e
dy
y
15. Find for 2x y y 2 2 2e by using implicit differentiation.
dx
16. (a) Find the slope of a curve y 5x 2 3 at the point x by using the first principle
2
of differentiation.
2
2
x d y dy
(b) If y Ax 2 Bx , show that x y 0 .
2 dx 2 dx
17. Differentiate the following with respect to . Give your answer in the simplest form.
xe 3x
3
x
(a) f ( ) (b) ( )f x (3x 2)(x 4)
x 3
18. Find '(2)f for ( ) 2f x x 2 x by using the first principle.
19. Differentiate the following with respect to x. Give your answer in the simplest form.
4
(a) f ( ) x 9 x 2
x
2x 4 3
x
(b) f ( )
x 3
3
x
(c) f ( ) ln 2x 3 e 3x
dy
3
2
4y
2
20. (a) Find for y 2x y xe by using implicit differentiation.
dx
dy
0
Hence, evaluate at y .
dx
2
(b) Prove that if y Ax B e 2x then d y 4 dy 4y .
0
dx 2 dx
dy
2
21. Find for y 3x y 7xy 1 0 by using implicit differentiation. Hence, determine all
2
2
dx
dy
values of when y 1.
dx
22. Differentiate the following with respect to x. Give your answer in the simplest form.
(a) y x 2 3 e (b) y ln x 1 3
5x
5x 2 2
2
f
' f
x
23. Find x for x 2x 1 by using the first principle. Hence, find 2f ' .
74 | P a g e
' f
24. Find x for 3f x x 2 2 by using the first principle. Hence, or otherwise, find the
value of x at x 1.
' f
25. Given y Ae x ln x , where A is a constant.
2
d y
2
0
(a) Show that x 2 yx Ae x 2x 1 .
dx 2
dy
(b) Find the value of A if 1 when x 1. Give your answer in exact value.
dx
26. Use suitable rules of differentiation to find the derivative of the following functions. Give
your answer in the simplest form.
(a) x x e
3 7x
f
(b) x h x ln x e ln x
dy 2
27. Apply implicit differentiation to find for the equation x y xy 2. Hence, solve for
dx
dy
x when 0. Give your answer in exact value.
dx
2
2
3
28. Determine the values of A, B and C for y Ax B x 1 Cx , if dy 2and d y 1
dx dx 2
at the point (2, 1).
dy dy
2
2
29. If 4y 3x 5xy , find in terms of and . Hence evaluate when = 0.
8
dx dx
x 3 dy Ax 2 x B
30. (a) Given y and . Find the values of and .
x 1 2 dx x 1 3
1 dy
(b) Given y 4x 3x 3 2 , find in the simplest form.
2
dx
2
2
31. (a) Given y 4 5x e 3x , find d y in the simplest form. Hence evaluate d y when
dx 2 dx 2
= 0.
dy
(b) Find in terms of x and y, given that y ln x where > 0 and > 0.
ln 2y
dx
dy
Give your answer in the simplest form. Hence evaluate when = 1.
dx
75 | P a g e
ANSWERS
3
1. (i) 1 (ii)
2
2 xy
dy y e
2.
dx 1 xye xy
3. – Refer to Lecturer –
9
4. (a) x 20 (b) x
2
5. x 2, y or x 2, y 2
dy 2x 2 dy
6. (a) x 2 (b) ; 2
2
dx 3y 2y dx
2
dy B d y B
7. (a) 2Ax ; 2A ; – Refer to Lecturer –
dx x 2 dx 2 x 3
(b) A , B 1
2
8. – Refer to Lecturer –
9. – Refer to Lecturer –
dy 7 dy 4 2x 3 3 2
10. (a) (b) 6 2x 3 ln 4x 5
dx x 2 2 dx 4x 5
dy x y 1
11. (a) , (b) A 2, B 3
dx x 3y 7
12. f '( ) 6x 2
x
2
dy 4 2xye 4x dy 2 xy 2x y 2
13. (a) or (b) – Refer to Lecturer –
2 4x
dx x e dx x 2
dy 2 2 1
14. (a) 1
dx x 4 x 3 2 x
dy 1
(b) 4x 1
dx x 2
dy 2
(c) 2x
5
dx x
dy 2xy
15.
dx x 2 y e y
76 | P a g e
16. (a) 20 (b) – Refer to Lecturer –
1
3x
2
dy 3e x 3x 1 dy 3 2x 3 x 4 2
17. (a) (b)
dx x 3 2 dx 3x 2
18. f '(2)
7
(x 2) (9x 3 70)
19. (a) '( )f x
2 x 9
8(x 2) (2x 2 11)
(b) '( )f x
(x 3) 2
3(2x 5)
(c) '( )f x
2x 3
dy 4xy e 4 y dy 1
20. (a) ;
dx 3y 2 2x 2 4xe 4 y dx 16
(b) - Refer to Lecturer -
dy 7y 6xy 2 5
21. , ,15
dx 6x y 2 2y 7x 2
2
e 5x 5x 2 x 15 3 5x 10x 2
22. (a) (b)
x 3 x 1 5x 2 2
2
7
23. ' f x 4x 1, f ' 2
24. ' f 6x x, 6
25. (a) - (b) A e
2 7x
26. (a) x e (7x 3) (b) ln x
77 | P a g e
18
27. x
5
3
28. A , B 5, C 1
4
dy 6x 5y 2 dy
29. ; 5
dx 4 10xy dx
3
dy 2 15x 4
30. (a) A 1,B 3 (b)
dx 1
3
3x 2 2
2
d y dy y 2 1
31. (a) 3e 3x 22 15 ;66x (b) ;
dx 2 dx x 1 y ln x 4
78 | P a g e
CHAPTER 9: APPLICATION OF DIFFERENTIATION
1. A furniture company produces units of table daily. The demand function and average
cost function (in RM) are given 400 2p q q and q 2q 400 respectively.
4
C
q
Find the
(a) revenue function, ( )
(b) cost function, ( ), and
(c) maximum profit of the company
2
3
f
2. Given that x x 9x 15x 11. Find the maximum and minimum points,
3. Given that the demand function is p 300 4x x and the cost function is
2
C x x 150x 5000 where is the number of products. Determine
(a) the revenue function, ( ) and the profit function x ,
(b) the maximum profit,
(c) the selling price to get the maximum profit,
2
5
4. Given y x 7x . Find the equation of normal to the curve at the point (1,3).
5. A company produces and sells pots each year with cost function,
2
x
p
C 4000 4x x 0.005x and demand function, 160 0.2x , where x is the
number of pots, ( ) and ( ) are in RM. Find:
(a) the revenue function and the number of pots that should be sold to maximize the
revenue,
(b) the profit function and the maximum profit,
(c) the selling price to ensure maximum profit.
2
3
6. Find the equation of tangents to the curve x 3xy y at point = 2.
8
79 | P a g e
7. The average cost function and the demand function in RM for a furniture company is given
40
by x 20 and 100 2p x x . The company produces x units of table daily.
C
x
(a) Find the cost function.
(b) Find the revenue function
(c) The level of production to achieve the maximum profit
(d) Find the maximum profit
2
2
8. Find the equation of the normal to the curve 2x 2y 4xy 3 0 at the point (1, −4).
9. A firm has determined that the price function from the sale of q units of product is
p 30 0.5q . Find
(a) the marginal revenue when the price is RM 20,
(b) the quantity that should be sold to maximize its revenue,
(c) the selling price to ensure maximum revenue.
10. Find the equation of a tangent line at a curve 4xy 2 3x 5y when = 1.
11. Given the profit function and the average cost function (in RM), of a mechanical pencil are
35
2
x 0.001x 50x 35 and 50C x respectively. Determine
x
(a) the cost function,
(b) the demand function,
(c) the maximum profit
12. An oil well has a maximum daily production of 100 barrels. The owner estimate that the
daily profit x (in RM) from a production of x barrels is
100x x 5 ,0 100.
x
x
2
(a) Determine whether profit or loss will be incurred from a production 0f 100 barrels
80 | P a g e
(b) Calculate the daily production that maximizes the profit and what is the maximum
profit.
13. The equation of the tangent to the curve x y ay 2 2 b at the point (1,2) is 4x 3y 7
where and are constants. Find the values of and .
C
14. The demand function p(x) and the average cost function x for Syarikat ABC are given
as 240 20p x x and 5C x x 40 . Find
(a) the cost function, the revenue function, and the profit function
(b) determine the quantity and the price needed to maximize the total revenue
(c) The profit and total revenue that will give maximum profit.
dy
2
2
15. Given that the curve 2x 4xy 4y 8,find the expression for in term of and .
dx
Hence, find the points in the curve where the tangent is parallel to -axis.
16. The demand function for a product is x 3000 25p where is the number of unit
demanded and is the price in RM.
(a) Determine the price that should be charged to maximize the total revenue.
(b) Find the maximum revenue.
2
17. Find the equation of the normal line to the curve y x xy 3 at the coordinate (1,1).
3
18. Given a curve y 12x 2 2x .
(a) Determine the two stationary points of the above curve. Hence, determine whether
the two stationary points are maximum or minimum points.
(b) The gradient at a point of the above curve is 24. Find the equation of the tangent
line at the point .
2
4
19. Find the equation of the normal line to the curve 2y xy 2x at the coordinate (2,1).
81 | P a g e
2
4
20. Consider the curve y x 2x 2.
(a) Determine the stationary points of the above curve and hence determine whether
the stationary points are maximum or minimum points.
(b) A point (2, ) is on the curve. Determine the value of , and hence find the equation
of the tangent line at the point.
21. The demand function for a product is 200 2p x x and the average cost function is
200
C 0.4x x 8 where is the number of units produced. Find
x
(a) the cost, the revenue and the profit function
(b) the level of output and the price at which the profit is maximized
2
22. The revenue function for the sales of x unit of product is x ax bx ,where and
R
are constants. If the optimal sale that maximizes revenue is 10 units, and the maximum
revenue is RM 500, find the value of a and . Hence, find the price when the revenue is
maximum.
2
3
23. The equation of a curve is y 10x ax 12x b where and are constants. The curve
,
has a minimum value of 6 at = 1.
(a) Find the values of and
(b) The gradient at the point ( , ) on the curve is 12, where > 0. Find the equation
of the normal line to the curve at M.
1
24. ( , ) is a point on a curve y 2x 3 5x 2 with > 0. Given the tangent line to the
4
curve at the point is parallel to the line = , find the values of and . Hence find the
equation of the normal line at the point .
2
x
25. At a particular point of the curve y 2x , q the equation of the tangent is
= 3 − 5. Find the value of the constant .
82 | P a g e
C
26. The demand function p and the average cost function x for a product are given as
3000
p 0.1x 40 and 0.5C x x 20 , where x is the number of units of the product.
x
(a) Find the marginal revenue function and the profit function
(b) Find the average revenue and the profit or loss when the cost is minimized.
27. An electronic company produces calculators for the local market. The company finds that
the cost function is 50C x x 20000 and the demand function is 100 0.02 ,
x
p
x
where represents the number of calculators and ( ) represents the price per calculator
in RM.
(a) Find the revenue function and the profit function
(b) Calculate the level of production that will maximize the profit. Hence, find the
maximum profit.
(c) Determine the price per unit when the profit is maximum.
2
3
x
28. Find the equation of tangent to the curve y x 2x 1 at = 1.
1 1
3
2
f
29. Given x x x 2x
3 2
(a) Find all the stationary points
(b) Determine the local maximum and minimum points
30. A manager of a washing machine manufacturer finds that the total cost function C(q) (in
RM) and the total revenue function R(q) (in RM), of producing and selling q units of
2
q
washing machine per week is given as C 300q q 500 and 400q q
R
respectively. Find the
(a) marginal cost, marginal revenue and demand function
(b) profit function and number of units per week that should be produced in order to
maximize the profit.
(c) revenue, profit and selling price per unit at maximum profit.
83 | P a g e
2
3
4
31. Find the equation of the normal line for the curve y x 3x at = 1.
x 3 x 2 4
32. Determine the maximum point and the minimum point of the curve y 2x
3 2 3
by using the second derivative test.
33. The total revenue function and total cost function of a company that produced one model
2
of handphone are 2000R x x x and 4000 200x respectively where x is the
C
x
number of units of handphone produced. Find
(a) the marginal cost function and the profit function
(b) the number of handphone to be produced to maximize the profit
(c) the price of handphone per unit when the profit was maximized
x 4
2
f
34. Given x
8
x 3
2 x 2 16
' f
(a) Show that x 11 by using quotient rule. Hence find all the stationary
3x 3
points.
(b) Using the second derivatives test, determine the relative minimum and maximum
for the curve ( ) at all stationary points.
35. The management of a company estimates that the cost (in RM) to produce x units of a
2
certain products is given by 0.015C x x 10x 300. The revenue generated after
2
selling x units of this products is 60R x x 0.01 . Find
x
(a) the maximum profit
(b) the selling price per unit in order to maximize the profit
(c) the level of production so that the average cost is minimum.
3
2
36. Find the equation of normal to the curve y 2x 4x at point (2, −4).
x
6
84 | P a g e
6
2
3
37. Determine the maximum and minimum points of the curve y 4x 15x 18x by
using the first derivative test.
38. The total revenue function and cost function of a company that produced car components
are 0.2R x x 2 80x and 0.1x 40x 6000 respectively where x is the number
2
C
x
of unit components produced. Find
(a) the total profit function
(b) the demand function
(c) the profit obtained when 1000 unit components are sold
(d) the number of units component to be produced to minimizes the cost. Hence, find
the minimum cost and price per unit components.
39. The equation of a curve is given by y 3 2x y 3 3. Find the equation of the tangent to the
curve at = 1.
40. An electronic company is producing microchips for local market. The company that
C
average cost function x and the demand function ( ) where is the quantity of
20x 10000
microchips are given by x and 100 0.01 .p x x Find
C
x
(a) The total cost function and fixed cost
(b) The total revenue function
(c) The total profit function
(d) The price per microchip when the profit was maximized. Hence, calculate the
maximum profit.
2
41. The curve x ax bx c has a maximum point at (2,10). Find the values of , and
f
if f passes through point (0,2).
2
3
42. (a) Find the local maximum and minimum point of the curve y 6x 9x 2.
85 | P a g e
1
(b) Find the equation tangent to the curve x 2 3y 2 6y at point 1, .
2
43. A company which produces cakes and find that the total cost function and the demand
function are 1000 10C x x and 100 x respectively where x is the number of
D
x
cakes produced.
(a) Construct the total revenue function and find the number of cakes that should be
produced in order to maximize the revenue.
(b) Determine the profit and the number of cakes that should be produced in order to
maximize the profit. Hence, find the maximum profit.
(c) Compute the maximum profit if each cake is taxed at RM5.
44. (a) Find the local maximum or minimum point using the second derivative test for the
curve y 9x 3x 2 1 where > 0.
(b) Find the gradient of the tangent line to the curve x 2 y 2 10 at = 1, where
> 0.
2
45. The revenue function, (in RM) for sales of x units of a product is x ax bx, where
R
and are constants. Given that the sale that maximizes revenue is 300 units and the
maximum revenue is RM 27,000.
(a) Find the values of and
(b) Find the demand function of the product. Hence, find the price of a unit of the
product when the revenue is maximum.
2
(c) Given the cost function 0.05C x x 9x 2500. Find the maximum profit.
ANSWERS
2
1. (a) R 400q q 2q
2
(b) C 2q q 4q 400
(c) Maximum profit is RM 9401
86 | P a g e
2. Minimum point is (1,4) and maximum point is (5,36).
2
2
3. (a) R 300x x 4 ; x 5x 450x 5000
x
(b) Maximum profit is RM5125
(c) RM120
1 28
4. Normal equations: y x
9 9
2
5. (a) R 160x x 0.2x ; 400 units to maximize the total revenue
2
(b) x 0.205x 164x 4000 ; Maximum profit is RM28800
(c) RM 80
x 10
x
6. Equation of tangent: y ; y
2
11
7. (a) C 40 20x x
2
(b) R 100x x 2x
(c) 20 units will maximize he profit
(d) RM 760
x
3
8. Equation of normal: y
9. (a) RM 10
(b) 30 units will maximize the revenue
(c) RM 15
1 8
10. Equation of tangent: y x
35 7
87 | P a g e
11. (a) C 50x x 35
70
(b) p x 0.001x 100
x
(c) Maximum profit is RM 625035
12. (a) The company will loss RM 40000 from a production of 100 barrels
(b) 10 units will maximize the profit. The maximum profit is RM 500
1
13. a ,b 4
2
2
2
2
14. (a) C 5x x 40 ;R 240x x 20 ; 200x x 25x
x
x
(b) The quantity is 6 units and the price is RM 120 when the total revenue is maximized
(c) The profit is RM 400 and the total revenue is RM640.
dy y x
15. ; The points are (2,2) and (−2, −2)
dx 2y x
16. (a) The price is 160
(b) The maximum revenue is RM 90 000
3 1
17. y x
2 2
18. (a) (0,0) is the minimum point and (4,64) is the maximum point
(b) Equation of tangent: y 24x 16
19. y 2x 3
20. (a) stationary points are (0,0), (-1,-3) and (1,-3); Maximum point: (0,-2), Minimum
point: (1,-3) and (-1,-3)
(b) y 24x 42
88 | P a g e
2
2
x
21. (a) C 0.4x x 8x 200,R 200x x 2 , 192x x 2.4x 200
2
(b) The output level is 40 units and the price is RM120 when the profit is maximized
22. = 5, = 100; 50
3
2
23. (a) = −21, = 5; y 10x 21x 12x 5
1 2459
(b) y x
12 300
24. = 2, = −1; = − + 1
25. = −3
2
26. (a) ' R x 0.2x 40, x 0.4x 60x 3000
(b) The average revenue is RM 42 and the loss is RM 1960 when the cost is minimized
2
2
27. (a) R x 0.02x 40; x 0.02x 50x 20000
(b) The production level of 1250 will maximize the profit. The maximum profit is
RM 11 250.
(c) The price per unit is RM 75
28. = 19 − 23
,
29. (a) Stationary points: 1, 7 2, 10
6 3
(b) Minimum point is 1, 7 and maximum point is 2, 10
6 3
30. (a) C ' 300, 'q R 400 2 ,q q D 400q q
2
(b) 100q q q 500; 50 units per week shall be produced
(c) Total revenue is RM 17500, profit is RM2000 and selling price per unit is RM350
89 | P a g e
31. 3 = − + 19
5
32. Minimum point is 1, , maximum point is 2,2
2
33. (a) C ' 200,x 1800x x x 4000
2
(b) 900 units will maximize the profit
(c) The price per unit is RM 1100
34. (a) (4,0.2976), (−4, −0.2976)
(b) (4,0.2976) and (−4, −0.29776) are maximum points
35. (a) The maximum profit is RM 24700
(b) The selling price per unit is RM 50
(c) The level of production is 141 units
1 34
36. Equation of normal: y x
9 9
37. Maximum point is 3,87 and minimum point is 1 5 ,
2 4
2
38. (a) 0.1x x 120x 6000
(b) p 0.2x x 80
(c) the profit obtained is RM 214 000
(d) 200 units of components need to be produced to minimize the cost. The minimum
cost is RM 2000. Price per unit component is RM 120 at minimum cost.
18 25
39. y x
7 7
90 | P a g e
40. (a) C 20x x 10000, fixed cost is RM 10000
2
(b) R 100x x 0.01x
2
(c) x 0.01x 80x 10000
(d) price per unit is RM 60 and the maximum profit is RM 150 000
8
41. a 2,b
1,5
42. (a) 0,2 is a minimum point and is a maximum point
2 1
(b) Equation of tangent: y x
3 6
43. (a) = 50
(b) 45 RM 1025
(c) 42.5 RM 806.25
1
44. (a) , 2 is a relative minimum point of ( )
3
1
(b) m
T
3
x
45. (a) a 0.3,b 180 (b) p ( ) 0.3x 180;RM 90
(c) RM 23,015
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