EXERCISE: APPLICATIONS OF DERIVATIVE
1.
Find the gradient of 4y2 = 3 ( x + 2xy ) at y = 1.
2.
Determine the equations of tangent and normal of 6xy + 2y 2 = 2e x + 2x 2 at (0,1).
3.
The surface area of a cube is increasing at a rate of 12 cm2/s where the edge of the cube is r. Find the rate of increase in volume of the cube when the edge is of length 10 cm.
4.
The circumference of a circular patch of oil on the surface of a pond is assumed to be increasing at the constant rate of 1.5 m/s. When the radius is 3.5 m, at what rate is the area of the oil changing?
5.
List down the intervals where the graph of f ( x ) = x 4 − 8x2 + 4 is concave up or concave down.
6.
Given: − x3 + 1 g(x ) = 2 3 18 x − x − 96 x + 79
a)
;
x 1
;
x 1
For x 1, solve the following: i)
If given, g ' (x ) = 36 x − 3 x 2 − 96 g ' (4 ) = g ' (8 ) = 0 g ' (x ) 0 for 4 x 8 and g ' (x ) 0 for 1 x 4 and 8 x
Locate the extremum points. ii)
Differentiate g' ( x ) .
iii)
State the intervals where g(x ) is concave up and concave down if g(x ) changes concavity at x = 6 .
b)
Sketch the graph of g(x ) .
ALL RIGHTS RESERVED MOOC MAT099
1
EXERCISE: APPLICATIONS OF DERIVATIVE
ANSWERS:
1.
27 16
2.
tan gent :
normal :
y = −x + 1
y = x +1
3.
30cm3 / s
4.
5.25 m2/s
5.
Concave up intervals: (-∞, -
3
2
Concave down interval : (6.
2
3
,
) and (
2 3
2 3
, ∞)
)
a i) ( 4, −81) minimum point
(8, −49) maximum point ii) g" ( x ) = 36 − 6x iii) concave up : ( −,6 ) concave down : ( 6, )
b)
y 1 1
4
8
x
-49
-81
ALL RIGHTS RESERVED MOOC MAT099
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