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NCERT Class 12- Mathematics: Chapter β 6 Application of Derivatives Part 2
Question 2:
Water is dripping out from a conical funnel of semi-vertical angle at the uniform rate of the surface area, through a tiny hole at the vertex of the bottom. When the slant height of cone is , find the rate of decrease of the slant height of water.
Answer:
If s represents the surface area, then
Therefore,
When
Question 3:
Find the angle of intersection of the curves and .
Answer:
Solving the given equations, we have and
Therefore,
i.e.. points of intersection are and
Further
And
At , the slope of the tangent to the curve is parallel to axis and the tangent to the curve is parallel to axis.
Angle of intersection and that of is .
Question 4:
Prove that the function is strictly decreasing on
Answer:
When
Therefore,
Thus for
Hence is strictly decreasing on .
Question 5:
Determine for which values of x, the function is increasing and for which values, it is decreasing.
Answer:
Now,
Since and is continuous in and . Therefore is decreasing in and is increasing in .
Note: Here f is strictly decreasing in and is strictly increasing in .
Question 6:
Show that the function has neither maxima nor minima.
Answer:
Hence is a point of inflexion i.e.. neither a point of maxima nor a point of minima.
is the only critical point, and has neither maxima nor minima.
Question 7:
Using differentials, find the approximate value of
Answer:
Let
Using . , taking and , we get