NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 2

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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.

Question 2-Conical Funnel of Semi-Vertical Angle

Conical Funnel of Semi-Vertical Angle

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

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