CQMS702 - Assignment 6 Questions with Verified Solutions

CQMS702 - Assignment 6 Questions with Verified Solutions


1. (10 marks) Find the second derivative of C
y = 2x 2 +1 3 3/ 2
.



Solution: Using the Chain rule, we get

y' =C 23 3 C 2x 2 +13
1/ 2 1/ 2
(4x) =6x(2x2 +1)

Then by the product rule, we get
1/ 2
y''=(6x)'(2x2 +1) + (6x) C (2x2 +1)- 1/2 (4x)3
1
2

=6(2x2 +1)
1/ 2
+12x C 2x2 +13 - 1/ 2

, dy
2. (10 marks) If y = f (x) is defined by the following equation, find
dx
=x + 2 y 2



Solution: Taking derivative with respect to x on both sides, we get
1
(2x + 2 y dy ) =1 + 4 y dy
2 x2 + y2 dx dx


⟶ 1
(x + y dy ) =1 + 4 y dy
x2 + y2 dx dx



⟶ x + ydxdy = x2 + y2 (1+ 4 y dydx )

⟶ ( y - 4 y x2 + y2 ) dy
dx
= - x

⟶dxdy =( - x) /( y - 4 y x2 + y2 )




3. (10 marks) Determine the interval where f (x) =x3 - 5x 2 - 8x +1 is increasing and
the interval where it is decreasing.

Solution: Taking the 1st derivative, we get
f '(x) =3x2 - 10x - 8