Math Assignment Class VIII | Square & Square Root

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Question 1
A particle moves along the curve x2 = 2y. At what point,
ordinate increases at the same rate as abscissa increases?
Answer (1, 1/2)
Solution Hint:
Differentiating the given equation
w. r. t. t
Now putting dy/dt = dx/dt and
find the value of x.
Putting the value of x in the
given equation and find the value of y
Question 2
The side of an equilateral triangle is increasing at the
rate of 2 cm/s. At what rate is its area increasing when the side of triangle
is 20 cm?
Answer: 20√3 cm2/s
Question 3
Volume of sphere is increasing at the rate of 3cm3/s.
Find the rate of increase of surface area, when radius is 2 cm
Answer: 3cm2/s
Question 4
For the curve y = 5x – 2x3. If x increases at the
rate of 2 unit/sec., find the rate of change of the slope of the curve when x =
3.
Answer: -72 unit /sec
Question 5
A man 2m high walks at a uniform speed of 6km/h away from
the lamp post 6m high. Find the rate at which the length of his shadow increases.
Answer : 3km/h
Solution Hint
AB = 6m, CD = 2m dy / dt = 6 km/h,
Let AC = y, CE = x, AE = x + y,
To find : dx/dt = ?
Without using derivative show that the function f(x) = 4x3 – 18x2
+ 27x – 7 is always increasing in R
Strictly Decreasing in (-2, 6)
Question 8
Find the intervals in which the function f(x) = x4 / 4 – x3 – 5x2 + 24x + 12 is
i) Strictly increasing ii) Strictly decreasing
Answer:
Strictly Decreasing in (-∞, 3)⋃(2, 4)
Question 9
Find the interval in which the function:
f(x) = sin4x + cos4x, 0 ≤ x < π/2
is strictly increasing or strictly decreasing.
Answer:
Strictly increasing in (π/4, π/2)
Strictly decreasing in (0, π/4)
Question 10
Find the intervals in which f(x) = sin3x - cos3x, 0<x<π, is strictly increasing or strictly decreasing.
Answer:
Strictly increasing in (0, π/4) ⋃ (7π/12, 11π/12)
Strictly decreasing in (π/4, 7π/12) ⋃ (11π/12, π)
Question 11
Strictly increasing in (-∞, -2)⋃
Strictly Decreasing in (-2, 3)
Question 13
Find the interval in which the following functions are strictly increasing and strictly decreasing : f(x) = 2x3 – 12x2 + 18x + 5
Answer:
Strictly increasing in (-∞, 1)⋃
Strictly Decreasing in (1, 3)
Strictly increasing in
Strictly Decreasing in
Question 15
Find the intervals in which the function given by f(x) = sin3x, x∈ [0, π/2] is increasing and decreasing.
Answer
Increasing on [0, π/6]
Decreasing on [π/6, π/2]
Question 16
Question 17
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