# Ch06. Applications of Derivatives

## Achievements

The following badges can be earned while learning.

## Certificate

Certificate on successful completion of this course.

THIS IS A COMBO OF TWO COURSES

COURSE – 1

In this online course, you will learn applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations). For further understanding of concepts and for examination preparation, this course has explanation of all NCERT Exercise questions and important NCERT Examples.

## Course Content

Rate of Change
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Increasing and Decreasing Functions
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Tangents and Normals
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Approximations
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Maxima and Minima
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NCERT Miscellaneous Exercise
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NCERT Examples
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COURSE – 2

This online course is an extended part of Ch06. Applications of Derivatives. The course is based on the assignments by Ashish Kumar (Agam Sir), which have questions from NCERT Exemplar, Board’s Question Bank, R. D. Sharma etc. The PDF of assignments can be downloaded within the course.

## Course Content

The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

• Question 1. An open box with square base is to be made out of a given iron sheet of area 27 sq. meter; show that the maximum volume of the box is 13.5 cubic meters.
• Question 2. Prove that the surface area of solid cuboid of a square base and given volume is minimum, when it is a cube.
• Question 3. A given quantity of metal is to be cast half cylindere., with a rectangular base and semicircular ends. Show that the total surface area is minimum when the ratio of the length of cylinder to the diameter of its semicircular ends is {\displaystyle \pi :(\pi +2)} .
• Question 4. What is the largest possible area for a right triangle whose hypotenuse is 5 cm long?
• Question 5. Show that the line {\displaystyle \frac{x}{a}+\frac{y}{b}=1} touches the curve {\displaystyle y=b{{e}^{\frac{-x}{a}}}} at the point, where the curve intersects the axis of y.
• Question 6. Show that the curves {\displaystyle y={{a}^{x}}} and {\displaystyle y={{b}^{x}},\,\,\,\,\,\,a>b>0} intersect at an angle of {\displaystyle {{\tan }^{-1}}\left( \left| \frac{\log \left| \frac{a}{b} \right|}{1+\log a\log b} \right| \right)} .
• Question 7. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is {\displaystyle 6\sqrt{3}r} .
• Question 8. An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when {\displaystyle \theta =\frac{\pi }{6}} .

### Assignment – 2

Find the intervals in which the following functions increasing or decreasing:

• Question 1. {\displaystyle f(x)={{x}^{4}}-8{{x}^{3}}+22{{x}^{2}}-24x+5}
• Question 2. {\displaystyle f(x)=2{{x}^{2}}-lnx}
• Question 3. {\displaystyle f(x)=\frac{1-x+{{x}^{2}}}{1+x+{{x}^{2}}}}
• Question 4. {\displaystyle f(x)=\frac{x}{{{x}^{2}}+1}}
• Question 5. {\displaystyle f(x)=2log(x-2)-{{x}^{2}}+4x+1}
• Question 6. {\displaystyle f(x)=si{{n}^{4}}x+co{{s}^{4}}x,\,x\in \left[ 0,\,\,\frac{\pi }{2} \right]}
• Question 7. Prove that the function f given by {\displaystyle f(x)=\log |\cos x|} is decreasing on {\displaystyle \left( 0,\,\,\frac{\pi }{2} \right)} and increasing on {\displaystyle \left( \frac{3\pi }{2},\,\,2\pi \right)} .   (NCERT Exercise 6.2 Q17)

### Assignment – 3

• Show that the local maximum value of {\displaystyle x+\frac{1}{x}} is less than local minimum value.
• Find the difference between the greatest and the least values of the function {\displaystyle f(x)=sin2x-x,} on {\displaystyle \left[ \frac{-\pi }{2},\,\,\frac{\pi }{2} \right]} .
• Find the maximum and minimum values of             {\displaystyle f(x)=secx+logco{{s}^{2}}x,\,\,\,0<x<2\pi } .
• Find the area of greatest rectangle that can be inscribed in an ellipse {\displaystyle \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1} .
• An open box with the square base is to be made of a given quantity of cardboard of area {\displaystyle {{c}^{2}}} . Show that the maximum volume of the box is {\displaystyle \frac{{{c}^{3}}}{6\sqrt{3}}} cubic units.
• The sum of the surface areas of a rectangular parallelepiped with sides {\displaystyle x,\,\,2x} & {\displaystyle \frac{x}{3}} and a sphere is given to be constant. Prove that the sum of their volumes is minimum if x is equal to three times the radius of the sphere. Also, find the minimum value of the sum of their volumes.
Syllabus medium English Hinglish (Hindi + English) 12 Online learning Self-learning from videos Mathematics