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Mathematics · Class 12 Science

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

Whenever a quantity is expressed as a function of another quantity , i.e. , the derivative (evaluated at a particular ) measures the instantaneous rate of change of with respect to at that point -- how fast is changing per unit change in , at the instant .

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Key concepts

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Rate of Change of Quantities

Many quantities in geometry and physics are functions of time (or of some other varying quantity), and the derivative measures how fast one quantity changes as another changes.

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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.

1

Rate of Change of Quantities

Whenever a quantity is expressed as a function of another quantity , i.e. , the derivative (evaluated at a particular ) measures the instantaneous rate of change of with respect to at that point -- ho…

2

Increasing and Decreasing Functions

Increasing and decreasing on an interval. Let be a real function defined on an interval . Then is said to be:

3

Tangents and Normals

The tangent as the limiting secant. For a curve , the tangent at a point on the curve is the straight line through whose slope equals -- the limiting position of a secant line through and a second nea…

4

Maxima and Minima -- First Derivative Test

Local maximum and local minimum, defined. A function has a local maximum at if for every in some open interval around -- the graph has a "peak" there, at least compared to its immediate neighbourhood…

5

Maxima and Minima -- Second Derivative Test

A faster alternative at a single point. The first derivative test needs the sign of on both sides of a critical point .

6

Simple Optimization Problems

From "find the extrema of " to "find the best real-world outcome." Many practical situations -- maximizing the area enclosed by a fixed length of fence, minimizing the material used to build a contain…

Summary

Rate of change. If and both vary with time , (related rates, via the chain rule); a positive rate means increasing, a negative rate means decreasing.

Sample & Board Papers

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

+Show 26 questions26 questions
  1. Q1f(x) = 5 - |x - 1|. Find the maximum value of f(x), also find the value of x for which f(x) is maximum.Preview
  2. Q2If x > 0, then show that log(1 + x) > x / (1 + x).Preview
  3. Q3ax^2 + by^2 = 1 and Ax^2 + By^2 = 1 meet each other orthogonally (a not equal to A, b not equal to B, aB - bA not equal to 0). Show that 1/a…Preview
  4. Q4Find maximum and minimum values of (x^2 - x + 1) / (x^2 + x + 1) using Calculus.Preview
  5. Q5The rate of increase of a side of a square is 1 cm/sec. The rate of increase of area of the square, when length of a side of the square is 2…Preview
  6. Q6x > 0, y > 0 and xy = 1, find the least value of x + y.Preview
  7. Q7Find the equation of tangent of x² + y² = 32 at (4, 4).Preview
  8. Q8Using calculus, show that the straight line lx + my + n = 0 touches the circle x² + y² = a² if a²(l² + m²) = n².Preview
  9. Q9x is real. Using differential calculus find the maximum and minimum values of (x² - x + 1) / (x² + x + 1).Preview
  10. Q10The value of the function f(x) = 4x - x² - 3 will be maximum when: (a) x = 3 (b) x = 2 (c) x = -2 (d) x = -3Preview
  11. Q11Slope of tangent of xy = c² at (ct, c/t) is (a) -1/t² (b) 1/t² (c) -1/t (d) -1/t³Preview
  12. Q12Find the equation of tangent of y = 3x⁴ - 4x at x = 1.Preview
  13. Q13Find the equation of tangent and normal of x^(2/3) + y^(2/3) = 2 at (1, 1).Preview
  14. Q14If x + y = 60 find the maximum value of xy³.Preview
  15. Q15Maximum value of 5 - (x-1)² is (a) 5 (b) 4 (c) 6 (d) 3Preview
  16. Q16Using Calculus, find the maximum value of 3 sin x + 4. (0 < x < π)Preview
  17. Q17Using Calculus, find the conditions that y = mx + c be a tangent of x² + y² = a². Hence show that two tangents can be drawn from an outside…Preview
  18. Q18Using Calculus, find perpendicular distance from (0, 0) on 3x + 4y + 5 = 0.Preview
  19. Q19Find the values of 'k' for which the curves x=y² and xy=k cut at right angle.Preview
  20. Q20Show that the maximum value of the function x³+1/x³ is less than its minimum value.Preview
  21. Q21If $y = \dfrac{\log_e x}{x}$ then the maximum value of $y$ is (a) $e$ (b) $e^2$ (c) $\dfrac{1}{e}$ (d) $\dfrac{1}{e^2}$Preview
  22. Q22Statement-I: $f(x) = 3 + |x - 3|$ has a local minimum value 3. Statement-II: $f(x) = \sin x$ has infinite number of maximum and minimum valu…Preview
  23. Q23The volume of a spherical balloon increases at the rate of $10\ \text{cm}^3/\text{sec}$. The rate of change of its surface area when its rad…Preview
  24. Q24Which one of the following is correct for all values of $x$ if $x \in (0, 1)$? (a) $e^x < 1 + x$ (b) $\log_e (1 + x) < x$ (c) $\sin x > x$ (…Preview
  25. Q25Let $f(x) = \dfrac{x}{1 + |x|}$. Then $f(x)$ is monotonically increasing in the interval (where $\mathbb{R}$ is the set of all real numbers)…Preview
  26. Q26If the straight line $y = x$ and the curve $xy = k^2$ cut at a right angle, then ($k$ is a real constant) (a) $k = 0$ (b) $k = \pm 1$ (c) $-…Preview

More questions

32 Q
+Show 3 questions3 questions
  1. Q30A stone is dropped into a quiet lake and circular waves move outward at a speed of $4\ \text{cm/s}$. At the instant when the radius of the c…Free
  2. Q31Find the equation of the tangent to the curve $y = x^2 + 4x + 1$ at the point where it crosses the $y$-axis.Preview
  3. Q32Show that for a closed right circular cylinder of given volume $V$, the total surface area is minimum when the height equals the diameter of…Preview
+Show 8 questions8 questions
  1. Example 1The side of a square is increasing at the rate of $4\ \text{cm/s}$. Find the rate of increase of the area of the square when the side is $10…Free
  2. Example 2A spherical balloon is being inflated so that its volume increases at the rate of $900\ \text{cm}^3/\text{s}$. Find the rate at which the ra…Free
  3. Example 3Show that the function $f(x) = x^3 - 3x^2 + 4x$ is increasing on $\mathbf{R}$.Free
  4. Example 4Find the intervals in which the function $f(x) = 2x^3 - 15x^2 + 36x + 1$ is (a) increasing (b) decreasing.Preview
  5. Example 5Find the equations of the tangent and the normal to the curve $y = x^2$ at the point $(1, 1)$.Preview
  6. Example 6Find the points on the curve $y = x^3 - 3x^2 - 9x + 7$ at which the tangent is parallel to the $x$-axis.Preview
  7. Example 7Find the local maximum and local minimum values of $f(x) = x^3 - 3x$ using the first derivative test.Preview
  8. Example 8Find the local maximum and local minimum values of $f(x) = x^3 - 6x^2 + 9x + 15$ using the second derivative test.Preview
+Show 5 questions5 questions
  1. Q9The radius of a circle is increasing at the rate of $3\ \text{cm/s}$. Find the rate of increase of its area when the radius is $10\ \text{cm…Free
  2. Q10A ladder $5\ \text{m}$ ($500\ \text{cm}$) long is leaning against a vertical wall. The bottom of the ladder is being pulled away from the wa…Free
  3. Q11A spherical soap bubble's radius is increasing at the rate of $2\ \text{cm/s}$. Find the rate of increase of its volume when the radius is $…Preview
  4. Q12The length of a rectangle is increasing at the rate of $3\ \text{cm/s}$ and its width is increasing at the rate of $4\ \text{cm/s}$. Find th…Preview
  5. Q13The edge of a cube is increasing at the rate of $5\ \text{cm/s}$. Find the rate at which the volume of the cube is increasing when the edge…Preview
+Show 5 questions5 questions
  1. Q14Show that the function given by $f(x) = 7x - 3$ is increasing on $\mathbf{R}$.Free
  2. Q15Find the intervals in which $f(x) = x^2 - 4x + 6$ is (a) increasing (b) decreasing.Free
  3. Q16Find the intervals in which the function $f(x) = -2x^3 - 9x^2 - 12x + 1$ is increasing or decreasing.Preview
  4. Q17Find the intervals in which $f(x) = \dfrac{x^4}{4} - x^3 - 5x^2 + 24x + 12$ is increasing or decreasing.Preview
  5. Q18Show that the function $f(x) = x - \sin x$ is increasing on $\mathbf{R}$.Preview
+Show 5 questions5 questions
  1. Q19Find the equations of the tangent and the normal to the curve $y = x^3$ at the point $(1, 1)$.Free
  2. Q20Find the point on the curve $y = x^2$ at which the tangent is parallel to the line $y = 4x - 5$, and write the equation of the tangent.Free
  3. Q21Find the equations of the normals to the curve $y = x^3 + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$.Preview
  4. Q22Find the slope of the tangent to the curve $y = 3x^4 - 4x$ at $x = 4$.Preview
  5. Q23Find the point on the curve $y = x^2 - 2x + 7$ at which the tangent is parallel to the $x$-axis, and write the equation of that tangent.Preview
+Show 6 questions6 questions
  1. Q24Find the local maximum and local minimum values of $f(x) = 2x^3 - 15x^2 + 36x + 10$ using the first derivative test.Free
  2. Q25Find the local maximum and local minimum values of $f(x) = x^4 - 8x^2 + 2$ using the second derivative test.Free
  3. Q26Find the local maximum and local minimum values of $f(x) = x^3 - 3x^2 - 9x + 5$ using the second derivative test.Preview
  4. Q27A rectangle has a fixed perimeter of $40\ \text{m}$. Express its area as a function of one side, and find the dimensions of the rectangle fo…Preview
  5. Q28An open box is to be made from a square piece of cardboard of side $18\ \text{cm}$ by cutting equal squares of side $x$ from each corner and…Preview
  6. Q29Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum.Preview