NEET (2024 Updated)-National Eligibility cum Entrance Test (Medical) Chemistry Coaching Programs
πΉ Video Course 2024 (8 Lectures [4 Hrs : 33 Mins]): Offline Support
Click Here to View & Get Complete Material
Rs. 100.00
1 Month Validity (Multiple Devices)
β³ π― Online Tests (5 Tests [50 Questions Each]): NTA Pattern, Analytics & Explanations
Click Here to View & Get Complete Material
Rs. 500.00
3 Year Validity (Multiple Devices)
π Study Material (159 Notes): 2024-2025 Syllabus
Click Here to View & Get Complete Material
Rs. 350.00
3 Year Validity (Multiple Devices)
π― 3111 MCQs (& PYQs) with Full Explanations (2024-2025 Exam)
Click Here to View & Get Complete Material
Rs. 650.00
3 Year Validity (Multiple Devices)
NCERT Class 12- Mathematics: Chapter β 6 Application of Derivatives Part 7
Question 26:
Let have second derivative at such that and , then is a point of ________.
Answer:
Local minima.
Question 27:
Minimum value of if in is ________.
Answer:
Question 28:
The maximum value of is ________.
Answer:
Question 29:
The rate of change of volume of a sphere with respect to its surface area, when the radius is cm, is________
Answer:
6.3 EXERCISE
Short Answer (S. A)
Question 1:
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.
Answer:
We have, rate of decrease of the volume of spherical ball of salt at any instant is surface. Let the radius of the spherical ball of the salt be .
And surface area
[where, k is the proportionality constant]
Hence, the radius of ball is decreasing at a constant rate.
Question 2:
If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.
Answer:
Given: circle where its area is increasing at a uniform rate
To prove perimeter varies inversely as the radius
Explanation: Let the radius of the circle be .
Let A be the area of the circle,
Then
As per the given criteria the area is increasing at a uniform rate, then
Now substituting the value from equation (i) in above equation, we get
Now differentiating with respect to t we get
Now let P be the perimeter of the circle, then
Now differentiating perimeter with respect to t, we get
Applying the derivatives, we get
Now substituting value from equation (ii) in the above equation we get
Cancelling the like terms, we get
Converting this to proportionality, we get
Hence the perimeter of the circle with given condition varies inversely as the radius.
Hence Proved