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Mathematics · Class 12 Science

Ch 6Application of Derivatives — Class 12 Mathematics, concept-first.

In Chapter 5 we developed the tools to differentiate composite functions, inverse trigonometric functions, implicit functions, and exponential and logarithmic functions. This chapter puts those tools to work: we study how the derivative is applied across engineering, the sciences, social science, and many other fields.

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Rate Of Change

The very first application of the derivative is to measure how fast one quantity changes when another changes. If , then the derivative

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

6.1

Introduction

In Chapter 5 we developed the tools to differentiate composite functions, inverse trigonometric functions, implicit functions, and exponential and logarithmic functions.

6.2

Rate of Change of Quantities

24 Q

When one quantity depends on another quantity through , the derivative (or ) gives the instantaneous rate of change of with respect to — how fast is changing at a particular value of . Evaluated at :

+Worked Examplesi6 questions
  1. Example 1Find the rate of change of the area of a circle per second with respect to its radius $r$ when $r = 5$ cm.Free
  2. Example 2The volume of a cube is increasing at a rate of $9$ cubic centimetres per second. How fast is the surface area increasing when the length of…Free
  3. Example 3A stone is dropped into a quiet lake and waves move in circles at a speed of $4$ cm per second. At the instant, when the radius of the circu…Preview
  4. Example 4The length $x$ of a rectangle is decreasing at the rate of $3$ cm/minute and the width $y$ is increasing at the rate of $2$ cm/minute. When…Preview
  5. Example 5The total cost $C(x)$ in Rupees, associated with the production of $x$ units of an item is given by $C(x) = 0.005\,x^3 - 0.02\,x^2 + 30x + 5…Preview
  6. Example 6The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. Find the marginal revenue,…Preview
+Exercise 6.1i18 questions
  1. Q1Find the rate of change of the area of a circle with respect to its radius $r$ when (a) $r = 3$ cm (b) $r = 4$ cmFree
  2. Q2The volume of a cube is increasing at the rate of $8 \text{ cm}^3/\text{s}$. How fast is the surface area increasing when the length of an e…Free
  3. Q3The radius of a circle is increasing uniformly at the rate of $3 \text{ cm/s}$. Find the rate at which the area of the circle is increasing…Free
  4. Q4An edge of a variable cube is increasing at the rate of $3 \text{ cm/s}$. How fast is the volume of the cube increasing when the edge is $10…Preview
  5. Q5A stone is dropped into a quiet lake and waves move in circles at the speed of $5 \text{ cm/s}$. At the instant when the radius of the circu…Preview
  6. Q6The radius of a circle is increasing at the rate of $0.7 \text{ cm/s}$. What is the rate of increase of its circumference?Preview
  7. Q7The length $x$ of a rectangle is decreasing at the rate of $5 \text{ cm/minute}$ and the width $y$ is increasing at the rate of $4 \text{ cm…Preview
  8. Q8A balloon, which always remains spherical on inflation, is being inflated by pumping in $900 \text{ cubic centimetres}$ of gas per second. F…Preview
  9. Q9A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the la…Preview
  10. Q10Find the value of the following: A ladder $5 \text{ m}$ long is leaning against a wall. The bottom of the ladder is pulled along the ground,…Preview
  11. Q11A particle moves along the curve $6y = x^3 + 2$. Find the points on the curve at which the $y$-coordinate is changing $8$ times as fast as t…Preview
  12. Q12The radius of an air bubble is increasing at the rate of $\frac{1}{2} \text{ cm/s}$. At what rate is the volume of the bubble increasing whe…Preview
  13. Q13A balloon, which always remains spherical, has a variable diameter $\frac{3}{2} (2x+1)$. Find the rate of change of its volume with respect…Preview
  14. Q14Sand is pouring from a pipe at the rate of $12 \text{ cm}^3/\text{s}$. The falling sand forms a cone on the ground in such a way that the he…Preview
  15. Q15The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000$…Preview
  16. Q16The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 13x^2 + 26x + 15$. Find the marginal revenu…Preview
  17. Q17Find the value of the following: The rate of change of the area of a circle with respect to its radius $r$ at $r = 6 \text{ cm}$ is (A) $10\…Preview
  18. Q18Find the value of the following: The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x…Preview
6.3

Increasing and Decreasing Functions

26 Q

Consider , a parabola opening upward. To the right of the origin (), as we move left to right the height rises — the function is increasing.

+Worked Examplesi7 questions
  1. Example 7Show that the function given by $f(x) = 7x - 3$ is increasing on $\mathbb{R}$.Free
  2. Example 8Show that the function $f$ given by $f(x) = x^3 - 3x^2 + 4x,\ x \in \mathbb{R}$ is increasing on $\mathbb{R}$.Free
  3. Example 9Prove that the function given by $f(x) = \cos x$ is (a) decreasing in $(0, \pi)$ (b) increasing in $(\pi, 2\pi)$, and (c) neither increasing…Free
  4. Example 10Find the intervals in which the function $f$ given by $f(x) = x^2 - 4x + 6$ is (a) increasing (b) decreasing.Preview
  5. Example 11Find the intervals in which the function $f$ given by $f(x) = 4x^3 - 6x^2 - 72x + 30$ is (a) increasing (b) decreasing.Preview
  6. Example 12Find intervals in which the function given by $f(x) = \sin 3x,\ x \in \left[0, \dfrac{\pi}{2}\right]$ is (a) increasing (b) decreasing.Preview
  7. Example 13Find the intervals in which the function $f$ given by $f(x) = \sin x + \cos x,\ 0 \le x \le 2\pi$ is increasing or decreasing.Preview
+Exercise 6.2i19 questions
  1. Q1Show that the function given by $f(x) = 3x + 17$ is increasing on $\mathbf{R}$.Free
  2. Q2Show that the function given by $f(x) = e^{2x}$ is increasing on $\mathbf{R}$.Free
  3. Q3Show that the function given by $f(x) = \sin x$ is (a) increasing in $(0, \frac{\pi}{2})$ (b) decreasing in $(\frac{\pi}{2}, \pi)$ (c) neith…Free
  4. Q4Find the intervals in which the function $f$ given by $f(x) = 2x^2 - 3x$ is (a) increasing (b) decreasingPreview
  5. Q5Find the intervals in which the function $f$ given by $f(x) = 2x^3 - 3x^2 - 36x + 7$ is (a) increasing (b) decreasingPreview
  6. Q6Find the intervals in which the following functions are strictly increasing or decreasing: (a) $x^2 + 2x - 5$ (b) $10 - 6x - 2x^2$ (c) $-2x^…Preview
  7. Q7Show that $y = \log(1+x) - \frac{2x}{2+x}$, $x > -1$, is an increasing function of $x$ throughout its domain.Preview
  8. Q8Find the values of $x$ for which $y = [x(x-2)]^2$ is an increasing function.Preview
  9. Q9Prove that $y = \frac{4\sin\theta}{(2+\cos\theta)} - \theta$ is an increasing function of $\theta$ in $[0, \frac{\pi}{2}]$.Preview
  10. Q10Prove that the logarithmic function is increasing on $(0, \infty)$.Preview
  11. Q11Prove that the function $f$ given by $f(x) = x^2 - x + 1$ is neither strictly increasing nor decreasing on $(-1, 1)$.Preview
  12. Q12Which of the following functions are decreasing on $\left(0, \frac{\pi}{2}\right)$? (A) $\cos x$ (B) $\cos 2x$ (C) $\cos 3x$ (D) $\tan x$Preview
  13. Q13On which of the following intervals is the function $f$ given by $f(x) = x^{100} + \sin x - 1$ decreasing ? (A) $(0, 1)$ (B) $\left(\frac{\p…Preview
  14. Q14Find the value of the following: For what values of $a$ the function $f$ given by $f(x) = x^2 + ax + 1$ is increasing on $[1, 2]$?Preview
  15. Q15Let I be any interval disjoint from $[-1, 1]$. Prove that the function $f$ given by $f(x) = x + \frac{1}{x}$ is increasing on I.Preview
  16. Q16Prove that the function $f$ given by $f(x) = \log \sin x$ is increasing on $\left(0, \frac{\pi}{2}\right)$ and decreasing on $\left(\frac{\p…Preview
  17. Q17Prove that the function $f$ given by $f(x) = \log |\cos x|$ is decreasing on $\left(0, \frac{\pi}{2}\right)$ and increasing on $\left(\frac{…Preview
  18. Q18Prove that the function given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $\mathbf{R}$.Preview
  19. Q19The interval in which $y = x^2 e^{-x}$ is increasing is (A) $(-\infty, \infty)$ (B) $(-2, 0)$ (C) $(2, \infty)$ (D) $(0, 2)$Preview
6.4

Maxima and Minima

The derivative locates the highest and lowest points on a graph — the turning points where a function changes from increasing to decreasing or vice versa.

+Worked Examplesi12 questions
  1. Example 14Find the maximum and the minimum values, if any, of the function $f$ given by $f(x) = x^2,\ x \in \mathbb{R}$. ![Graph of y = x², an upward…Free
  2. Example 15Find the maximum and minimum values of $f$, if any, of the function given by $f(x) = |x|,\ x \in \mathbb{R}$. ![Graph of y = |x|, a V-shaped…Free
  3. Example 16Find the maximum and the minimum values, if any, of the function given by $f(x) = x,\ x \in (0, 1)$. ![f(x) = x in (0,1)](/figures/applicati…Free
  4. Example 17Find all points of local maxima and local minima of the function $f$ given by $f(x) = x^3 - 3x + 3$.Preview
  5. Example 18Find all the points of local maxima and local minima of the function $f$ given by $f(x) = 2x^3 - 6x^2 + 6x + 5$.Preview
  6. Example 19Find local minimum value of the function $f$ given by $f(x) = 3 + |x|,\ x \in \mathbb{R}$. ![Graph of f(x) = 3 + |x| (Example 19): upward V]…Preview
  7. Example 20Find local maximum and local minimum values of the function $f$ given by $f(x) = 3x^4 + 4x^3 - 12x^2 + 12$.Preview
  8. Example 22Find two positive numbers whose sum is $15$ and the sum of whose squares is minimum.Preview
  9. Example 23Find the shortest distance of the point $(0, c)$ from the parabola $y = x^2$, where $\dfrac{1}{2} \le c \le 5$.Preview
  10. Example 24Let AP and BQ be two vertical poles at points A and B, respectively. If $AP = 16$ m, $BQ = 22$ m and $AB = 20$ m, then find the distance of…Preview
  11. Example 25If length of three sides of a trapezium other than base are equal to $10$ cm, then find the area of the trapezium when it is maximum. ![Isos…Preview
  12. Example 26Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that…Preview
6.4.1

Maximum and Minimum Values of a Function in a Closed Interval

32 Q

Consider on the open interval . It is continuous there yet has neither a maximum nor a minimum value (and no local extremum either): the values approach as and as , but the endpoints are excluded, so…

+Worked Examplesi3 questions
  1. Example 27Find the absolute maximum and minimum values of a function $f$ given by $f(x) = 2x^3 - 15x^2 + 36x + 1$ on the interval $[1, 5]$.Free
  2. Example 28Find absolute maximum and minimum values of a function $f$ given by $f(x) = 12x^{4/3} - 6x^{1/3},\ x \in [-1, 1]$.Preview
  3. Example 29An Apache helicopter of enemy is flying along the curve given by $y = x^2 + 7$. A soldier, placed at $(3, 7)$, wants to shoot down the helic…Preview
+Exercise 6.3i29 questions
  1. Q1Find the maximum and minimum values, if any, of the following functions given by (i) $f(x) = (2x-1)^2 + 3$ (ii) $f(x) = 9x^2 + 12x + 2$ (iii…Free
  2. Q2Find the maximum and minimum values, if any, of the following functions given by (i) $f(x) = |x+2|-1$ (ii) $g(x) = -|x+1|+3$ (iii) $h(x) = \…Free
  3. Q3Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the…Free
  4. Q4Prove that the following functions do not have maxima or minima: (i) $f(x) = e^x$ (ii) $g(x) = \log x$ (iii) $h(x) = x^3+x^2+x+1$Preview
  5. Q5Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) $f(x) = x^3, x \in [-2…Preview
  6. Q6Find the maximum profit that a company can make, if the profit function is given by $p(x)=41-72x-18x^2$Preview
  7. Q7Find both the maximum value and the minimum value of $3x^4-8x^3+12x^2-48x+25$ on the interval $[0, 3]$.Preview
  8. Q8Find the value of the following: At what points in the interval $[0, 2\pi]$, does the function $\sin 2x$ attain its maximum value?Preview
  9. Q9What is the maximum value of the function $\sin x+\cos x$?Preview
  10. Q10Find the maximum value of $2x^3-24x+107$ in the interval $[1, 3]$. Find the maximum value of the same function in $[-3, -1]$.Preview
  11. Q11It is given that at $x = 1$, the function $x^4 - 62x^2 + ax + 9$ attains its maximum value, on the interval $[0, 2]$. Find the value of $a$.Preview
  12. Q12Find the maximum and minimum values of $x + \sin 2x$ on $[0, 2\pi]$.Preview
  13. Q13Find two numbers whose sum is $24$ and whose product is as large as possible.Preview
  14. Q14Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum.Preview
  15. Q15Find two positive numbers $x$ and $y$ such that their sum is 35 and the product $x^2y^5$ is a maximum.Preview
  16. Q16Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.Preview
  17. Q17A square piece of tin of side $18$ cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to…Preview
  18. Q18A rectangular sheet of tin $45$ cm by $24$ cm is to be made into a box without top, by cutting off square from each corner and folding up th…Preview
  19. Q19Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.Preview
  20. Q20Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.Preview
  21. Q21Of all the closed cylindrical cans (right circular), of a given volume of $100$ cubic centimetres, find the dimensions of the can which has…Preview
  22. Q22A wire of length $28$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should…Preview
  23. Q23Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\frac{8}{27}$ of the volume of the sphere.Preview
  24. Q24Show that the right circular cone of least curved surface and given volume has an altitude equal to $\sqrt{2}$ time the radius of the base.Preview
  25. Q25Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is $\tan^{-1}\sqrt{2}$.Preview
  26. Q26Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $\sin^{-1}\left(\frac{1}{3}\right)$.Preview
  27. Q27The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is (A) $(2\sqrt{2}, 4)$ (B) $(2\sqrt{2}, 0)$ (C) $(0, 0)$ (D) $(2,…Preview
  28. Q28Find the value of the following: For all real values of $x$, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is (A) $0$ (B) $1$ (C) $3$ (D) $…Preview
  29. Q29Find the value of the following: The maximum value of $[x(x-1)+1]^{\frac{1}{3}}$, $0 \le x \le 1$ is (A) $(\frac{1}{3})^{\frac{1}{3}}$ (B) $…Preview

Miscellaneous Examples

+Miscellaneous Examplesi8 questions
  1. Example 30A car starts from a point P at time $t = 0$ seconds and stops at point Q. The distance $x$, in metres, covered by it, in $t$ seconds is give…Free
  2. Example 31A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is $\tan^…Free
  3. Example 32A man of height $2$ metres walks at a uniform speed of $5$ km/h away from a lamp post which is $6$ metres high. Find the rate at which the l…Free
  4. Example 33Find intervals in which the function given by $f(x) = \dfrac{3}{10}x^4 - \dfrac{4}{5}x^3 - 3x^2 + \dfrac{36}{5}x + 11$ is (a) increasing (b)…Preview
  5. Example 34Show that the function $f$ given by $f(x) = \tan^{-1}(\sin x + \cos x),\ x > 0$ is always an increasing function in $\left(0, \dfrac{\pi}{4}…Preview
  6. Example 35A circular disc of radius $3$ cm is being heated. Due to expansion, its radius increases at the rate of $0.05$ cm/s. Find the rate at which…Preview
  7. Example 36An open topped box is to be constructed by removing equal squares from each corner of a $3$ metre by $8$ metre rectangular sheet of aluminiu…Preview
  8. Example 37Manufacturer can sell $x$ items at a price of rupees $\left(5 - \dfrac{x}{100}\right)$ each. The cost price of $x$ items is Rs $\left(\dfrac…Preview

Miscellaneous Exercise on Chapter 6

+Miscellaneous Exercisei16 questions
  1. Q1Show that the function given by $f(x) = \frac{\log x}{x}$ has maximum at $x = e$.Free
  2. Q2The two equal sides of an isosceles triangle with fixed base $b$ are decreasing at the rate of 3 cm per second. How fast is the area decreas…Free
  3. Q3Find the intervals in which the function $f$ given by $f(x) = \frac{4 \sin x - 2x - x \cos x}{2 + \cos x}$ is (i) increasing (ii) decreasing…Free
  4. Q4Find the intervals in which the function $f$ given by $f(x) = x^3 + \frac{1}{x^3}, x \neq 0$ is (i) increasing (ii) decreasing.Preview
  5. Q5Find the maximum area of an isosceles triangle inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with its vertex at one end o…Preview
  6. Q6A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2$ m and volume is $8$ m$^3$.…Preview
  7. Q7The sum of the perimeter of a circle and square is $k$, where $k$ is some constant. Prove that the sum of their areas is least when the side…Preview
  8. Q8A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10$ m. Find the dimension…Preview
  9. Q9A point on the hypotenuse of a triangle is at distance $a$ and $b$ from the sides of the triangle. Show that the minimum length of the hypot…Preview
  10. Q10Find the points at which the function $f$ given by $f(x) = (x - 2)^4 (x + 1)^3$ has (i) local maxima (ii) local minima (iii) point of inflex…Preview
  11. Q11Find the absolute maximum and minimum values of the function $f$ given by $f(x) = \cos^2 x + \sin x, x \in [0, \pi]$Preview
  12. Q12Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4r}{3}$.Preview
  13. Q13Let $f$ be a function defined on $[a, b]$ such that $f'(x) > 0$, for all $x \in (a, b)$. Then prove that $f$ is an increasing function on $(…Preview
  14. Q14Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\frac{2R}{\sqrt{3}}$. Also find the…Preview
  15. Q15Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi vertical angle $\…Preview
  16. Q16A cylindrical tank of radius $10$ m is being filled with wheat at the rate of $314$ cubic metre per hour. Then the depth of the wheat is inc…Preview

Summary

- Rate of Change: The derivative measures the instantaneous rate of change of with respect to . For a quantity , its velocity is and acceleration is .

Appendix 1 — Proofs in Mathematics

A proof transforms a guess or a pattern into an unshakable truth. It is a logical chain of reasoning that starts from accepted statements (axioms, definitions, or previously proven theorems) and, thro…

+Examples A.1i9 questions
  1. Example 1Show that if $x^2 - 5x + 6 = 0$, then $x = 3$ or $x = 2$.Free
  2. Example 2Prove that the function $f : \mathbf{R} \to \mathbf{R}$ defined by $f(x) = 2x + 5$ is one-one.Free
  3. Example 3Show that if $A = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$, then $A^{n} = \begin{pmatrix} \cos n\t…Free
  4. Example 4Show that in any triangle $ABC$, $\ a = b\cos C + c\cos B$. ![Triangle ABC with the perpendicular AD from A to BC](/figures/application-of-d…Preview
  5. Example 5Show that the set of all prime numbers is infinite.Preview
  6. Example 6Prove that the function $f : \mathbf{R} \to \mathbf{R}$ defined by $f(x) = 2x + 5$ is one-one, using the contrapositive.Preview
  7. Example 7Show that "if a matrix $A$ is invertible, then $A$ is non-singular".Preview
  8. Example 8For each $n$, $\ 2^{2^{n}} + 1$ is a prime $(n \in \mathbf{N})$. Examine whether this generalisation is true.Preview
  9. Example 9Every continuous function is differentiable. Examine whether this statement is true.Preview

Introduction

This introductory section sets the stage for the appendix: it recalls the logical building blocks you already know from earlier classes and previews a systematic study of how to prove mathematical sta…

What is a Proof?

A proof is a sequence of statements, each justified by a definition, an axiom, or a previously established proposition, using only allowed logical rules.

Appendix 2 — Mathematical Modelling

Mathematical modelling translates a real-world problem into mathematical language, solves the mathematical problem, and interprets the solution back in the real-world context.

+Examples A.2i7 questions
  1. Example 1Find the height of a given tower using mathematical modelling. ![Vertical tower AB and an observer, with angle of elevation to the top](/fig…Free
  2. Example 2Let a business firm produce three types of products $P_1$, $P_2$ and $P_3$ that use three types of raw materials $R_1$, $R_2$ and $R_3$. Let…Free
  3. Example 3Interpret the model in Example 2, in case $$A = \begin{pmatrix} 10 & 15 & 6 \\ 10 & 20 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 3 & 4 &…Free
  4. Example 4Suppose $P_1$, $P_2$, $P_3$ and $R_1$, $R_2$, $R_3$ are as in Example 2. Let the firm have $330$ units of $R_1$, $455$ units of $R_2$ and $1…Preview
  5. Example 5A manufacturer of medicines is preparing a production plan of medicines $M_1$ and $M_2$. There are sufficient raw materials available to mak…Preview
  6. Example 6Suppose a company plans to produce a new product that incurs some costs (fixed and variable) and let the company plan to sell the product at…Preview
  7. Example 7Let a tank contain $1000$ litres of brine which contains $250$ g of salt per litre. Brine containing $200$ g of salt per litre flows into th…Preview

Introduction

Mathematical modelling is the process of converting a real-life problem into mathematical language. Instead of studying the physical situation directly, we create a model — a simplified representation…

Why Mathematical Modelling?

Mathematical modelling is the process of translating a real-world problem into mathematical language, solving it, and interpreting the solution back in context.

Principles of Mathematical Modelling

Mathematical modelling is a principled process: start with a real-world problem, translate it into mathematics, solve it, then interpret the solution back in the real world.

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 42 questions42 questions
  1. Q1A 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 t…Free
  2. Q2If the area of a circle increases at a uniform rate, then prove that the perimeter varies inversely as the radius.Free
  3. Q3A kite is moving horizontally at a height of $151.5$ metres. If the speed of the kite is $10$ m/s, how fast is the string being let out when…Free
  4. Q4Two men $A$ and $B$ start with velocities $v$ at the same time from the junction of two roads inclined at $45^\circ$ to each other. If they…Preview
  5. Q5Find an angle $\theta$, $0 < \theta < \dfrac{\pi}{2}$, which increases twice as fast as its sine.Preview
  6. Q6A man, $2$ m tall, walks at the rate of $1\tfrac{2}{3}$ m/s towards a street light which is $5\tfrac{1}{3}$ m above the ground. At what rate…Preview
  7. Q7A swimming pool is to be drained for cleaning. If $L$ represents the number of litres of water in the pool $t$ seconds after the pool has be…Preview
  8. Q8The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.Preview
  9. Q9$x$ and $y$ are the sides of two squares such that $y = x - x^2$. Find the rate of change of the area of the second square with respect to t…Preview
  10. Q10Show that $f(x) = 2x + \cot^{-1}x + \log\left(\sqrt{1+x^2} - x\right)$ is increasing in $\mathbb{R}$.Preview
  11. Q11Show that for $a \geq 1$, $f(x) = \sqrt{3}\,\sin x - \cos x - 2ax + b$ is decreasing in $\mathbb{R}$.Preview
  12. Q12Show that $f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in $\left(0, \dfrac{\pi}{4}\right)$.Preview
  13. Q13At what point is the slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ maximum? Also find the maximum slope.Preview
  14. Q14Prove that $f(x) = \sin x + \sqrt{3}\,\cos x$ has maximum value at $x = \dfrac{\pi}{6}$.Preview
  15. Q15If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum wh…Preview
  16. Q16Find the points of local maxima, local minima and the points of inflection of the function $f(x) = x^5 - 5x^4 + 5x^3 - 1$. Also find the cor…Preview
  17. Q17A telephone company in a town has $500$ subscribers on its list and collects fixed charges of Rs $300$ per subscriber per year. The company…Preview
  18. Q18An open box with a square base is to be made out of a given quantity of cardboard of area $c^2$. Show that the maximum volume of the box is…Preview
  19. Q19Find the dimensions of the rectangle of perimeter $36$ cm which will sweep out a volume as large as possible when revolved about one of its…Preview
  20. Q20If the sum of the surface areas of a cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere w…Preview
  21. Q21$AB$ is a diameter of a circle and $C$ is any point on the circle. Show that the area of $\triangle ABC$ is maximum when it is isosceles.Preview
  22. Q22A metal box with a square base and vertical sides is to contain $1024$ cm$^3$. The material for the top and bottom costs Rs $5$/cm$^2$ and t…Preview
  23. Q23The sum of the surface areas of a rectangular parallelopiped with sides $x$, $2x$ and $\dfrac{x}{3}$ and a sphere is given to be constant. P…Preview
  24. Q24The values of $a$ for which the function $f(x) = \sin x - ax + b$ increases on $\mathbb{R}$ are ______.Preview
  25. Q25The function $f(x) = \dfrac{2x^2 - 1}{x^4}$, $x > 0$, decreases in the interval ______.Preview
  26. Q26The least value of the function $f(x) = ax + \dfrac{b}{x}$ ($a > 0$, $b > 0$, $x > 0$) is ______.Preview
  27. Q27The sides of an equilateral triangle are increasing at the rate of $2$ cm/sec. The rate at which the area increases, when the side is $10$ c…Preview
  28. Q28A ladder, $5$ m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate o…Preview
  29. Q29The interval on which the function $f(x) = 2x^3 + 9x^2 + 12x - 1$ is decreasing is: (A) $[-1, \infty)$ (B) $[-2, -1]$ (C) $(-\infty, -2]$ (D…Preview
  30. Q30Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 2x + \cos x$, then $f$: (A) has a minimum at $x = \pi$ (B) has a maximum at $x = 0…Preview
  31. Q31$y = x(x - 3)^2$ decreases for the values of $x$ given by: (A) $1 < x < 3$ (B) $x < 0$ (C) $x > 0$ (D) $0 < x < \dfrac{3}{2}$Preview
  32. Q32The function $f(x) = 4\sin^3 x - 6\sin^2 x + 12\sin x + 100$ is strictly: (A) increasing in $\left(\pi, \dfrac{3\pi}{2}\right)$ (B) decreasi…Preview
  33. Q33Which of the following functions is decreasing on $\left(0, \dfrac{\pi}{2}\right)$? (A) $\sin 2x$ (B) $\tan x$ (C) $\cos x$ (D) $\cos 3x$Preview
  34. Q34The function $f(x) = \tan x - x$: (A) always increases (B) always decreases (C) never increases (D) sometimes increases and sometimes decrea…Preview
  35. Q35If $x$ is real, the minimum value of $x^2 - 8x + 17$ is: (A) $-1$ (B) $0$ (C) $1$ (D) $2$Preview
  36. Q36The smallest value of the polynomial $x^3 - 18x^2 + 96x$ in $[0, 9]$ is: (A) $126$ (B) $0$ (C) $135$ (D) $160$Preview
  37. Q37The function $f(x) = 2x^3 - 3x^2 - 12x + 4$ has: (A) two points of local maximum (B) two points of local minimum (C) one maxima and one mini…Preview
  38. Q38The maximum value of $\sin x \cdot \cos x$ is: (A) $\dfrac{1}{4}$ (B) $\dfrac{1}{2}$ (C) $\sqrt{2}$ (D) $2\sqrt{2}$Preview
  39. Q39At $x = \dfrac{5\pi}{6}$, $f(x) = 2\sin 3x + 3\cos 3x$ is: (A) maximum (B) minimum (C) zero (D) neither maximum nor minimumPreview
  40. Q40Maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is: (A) $0$ (B) $12$ (C) $16$ (D) $32$Preview
  41. Q41$f(x) = x^x$ has a stationary point at: (A) $x = e$ (B) $x = \dfrac{1}{e}$ (C) $x = 1$ (D) $x = \sqrt{e}$Preview
  42. Q42The maximum value of $\left(\dfrac{1}{x}\right)^x$ is: (A) $e$ (B) $e^e$ (C) $e^{1/e}$ (D) $\left(\dfrac{1}{e}\right)^{1/e}$Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 12 questions12 questions
  1. Q1A function 𝑓(𝑥) = 10 − 𝑥 − 2𝑥2 is increasing on the interval (A) (−∞, − 1/4] (B) (−∞, 1/4) (C) [− 1/4, ∞) (D) [− 1/4, 1/4]Preview
  2. Q2Case Study -2 LED bulbs are energy-efficient because they use significantly less electricity than traditional bulbs while producing the same…Preview
  3. Q3The cost (in rupees) of producing x items in factory, each day is given by 𝐶(𝑥) = 0.00013𝑥3 + 0.002 𝑥2 + 5𝑥 + 2200 Find the marginal cost wh…Preview
  4. Q4A kite is flying at a height of 3 metres and 5 metres of string is out. If the kite is moving away horizontally at the rate of 200 cm/s, fin…Preview
  5. Q5Let $f(x)$ be a continuous function in the interval $[a, b]$ and differentiable in the interval $(a, b)$. Then $f(x)$ is strictly increasing…Preview
  6. Q6If $f(x) = a(x - \cos x)$, $x \in \mathbb{R}$ is a strictly decreasing function, then 'a' lies in which of the following intervals? (A) $\{0…Preview
  7. Q7The equation of the normal to the parabola $y^2 = 8x$ at its origin is ________. OR The radius of a circle is increasing uniformly at the ra…Preview
  8. Q8Find the equation of the normal to the curve $x^2 = 4y$ which passes through the point $(-1, 4)$.Preview
  9. Q9Find the points on the curve $9y^2 = x^3$, where the normal to the curve makes equal intercepts with both the axes. Also find the equation o…Preview
  10. Q10The total cost $C(x)$ associated with the production of $x$ units of an item is given by $C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000$. Find the…Preview
  11. Q11An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show tha…Preview
  12. Q12Find the equations of the tangent and the normal, to the curve $16x^2 + 9y^2 = 145$ at the point $(x_1, y_1)$, where $x_1 = 2$ and $y_1 > 0$…Preview