Application of Derivatives Class 12 Notes

• f is increasing on an interval where
• $f^{\prime }(x)>0$
• and decreasing where
• $f^{\prime }(x)<0$
• .
• If f'(x) = 0 throughout an interval, f is constant there.
• Stationary points satisfy f'(x) = 0 and separate increasing from decreasing regions.

• Word problems: identify the quantity to optimise, write it as a function of one variable using the constraints, find critical points, and test them.
• Approximation:
• $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\Delta x$
• .
• The largest or smallest value on a closed interval is found by comparing critical points with the endpoints.

Example: Find the intervals on which $f(x)=x^3-3x+1$ is increasing.

Solution: $f^{\prime }(x)=3x^2-3=3(x-1)(x+1)$. This is positive for $x<-1$ and $x>1$, so f is increasing on $(-\infty ,-1)\cup (1,\infty )$ and decreasing on $(-1,1)$.

Example: Show that every edge of the largest rectangle of fixed perimeter 20 cm is 5 cm.

Solution: Let sides be x and 10 − x. Area $A=x(10-x)=10x-x^2$. Then $A^{\prime }=10-2x=0\Rightarrow x=5$ and $A^{\prime \prime }=-2<0$, so x = 5 is a maximum — a square of side 5 cm.

Exercises (NCERT Q1–Q96)

• 1) Find the rate of change of the area of a circle with respect to its radius when (a) and (b) .
• 2) The volume of a cube is increasing at the rate of . How fast is the surface area increasing when the edge length is ?
• 3) The radius of a circle is increasing uniformly at the rate of . Find the rate of increase of its area when the radius is .
• 4) An edge of a variable cube is increasing at the rate of . How fast is the volume of the cube increasing when the edge is ?
• 5) A stone is dropped into a quiet lake and ripples move in circles at the speed of . At the instant when the radius of the ripple is , how fast is the enclosed area increasing?
• 6) The radius of a circle is increasing at the rate of . What is the rate of increase of its circumference when the radius is ?
• 7) The length of a rectangle is decreasing at the rate of and the width is increasing at the rate of . When and , find the rates of change of (a) the perimeter and (b) the area of the rectangle.
• 8) A balloon, which always remains spherical on inflation, is being inflated by pumping in . Find the rate of increase of its radius when the radius is .
• 9) A ladder long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of . How fast is its height on the wall decreasing when the foot of the ladder is away from the wall?
• 10) A particle moves along the curve . Find the point on the curve at which the -coordinate is changing
• 11) A particle moves along the curve . Find the point on the curve at which the -coordinate is changing 8 times as fast as the -coordinate.
• 12) The area of a circle, whose radius is , is increasing at the rate of . Find the rate at which its radius is increasing.
• 13) In a right-angled triangle, the two sides that contain the right angle are and . Find the rate at which the other side changes when the 18 cm side is shortened at and the 24 cm side is lengthened at .
• 14) A man tall walks at a uniform speed of away from a lamp post high. Find the rate at which the length of his shadow increases.
• 15) A man tall walks at a uniform speed of towards a street light high. Find the rate at which the length of his shadow decreases.
• 16) A particle moves along the curve . At what point on the curve are the - and -coordinates changing at the same rate?
• 17) Two sides of a triangle have lengths and respectively and the angle between them is increasing at the rate of . Find the rate of change of the area of the triangle when the angle between the two sides is .
• 18) Show that the function given by is strictly increasing on .
• 19) Show that the function given by is strictly increasing on .
• 20) Prove that the function given by is strictly increasing on .
• 21) Prove that the function given by is (a) strictly decreasing on and (b) strictly increasing on .
• 22) Find the intervals in which the function given by is (a) strictly increasing and (b) strictly decreasing.
• 23) Find the intervals in which the function given by is (a) strictly increasing and (b) strictly decreasing.
• 24) Find the intervals in which the function given by is strictly increasing or strictly decreasing.
• 25) Find the intervals in which the function given by is (a) strictly increasing and (b) strictly decreasing.
• 26) Find the intervals in which the function given by is (a) strictly increasing and (b) strictly decreasing.
• 27) Prove that the logarithmic function is strictly increasing on .
• 28) Prove that the function is neither strictly increasing nor strictly decreasing on .
• 29) For which values of is the function given by strictly increasing on ?
• 30) Which of the following functions are strictly decreasing on ? (i) (ii) (iii) (iv) .
• 31) Prove that the function given by is strictly increasing on .
• 32) Prove that the function given by is strictly decreasing on .
• 33) Show that , , is an increasing function of throughout its domain.
• 34) Find the intervals in which the function given by , , is strictly increasing and strictly decreasing.
• 35) Find the intervals in which the function given by is increasing.
• 36) Find the slope of the tangent to the curve at the point whose -coordinate is 2.
• 37) Find the slope of the normal to the curve at the point whose -coordinate is 2.
• 38) Find the slope of the tangent to the curve at the point whose -coordinate is 3.
• 39) Find the slope of the tangent to the curve at the point whose -coordinate is 2.
• 40) Find the slope of the tangent to the curve at the point whose -coordinate is 1.
• 41) Find equation of the tangent to the curve at the point where .
• 42) Find the slope of the normal to the curve at the point .
• 43) Find the slope of the normal to the curve at the point .
• 44) Find the slope of the tangent to the curve , , at the point .
• 45) Find the slope of the tangent to the curve , , at the point .
• 46) Find the slope of the normal to the curve at .
• 47) Find the slope of the normal to the curve at .
• 48) Find points on the curve where the tangent is parallel to the chord joining and .
• 49) Find the equation of the tangent line to the curve which is (a) parallel to the line and (b) perpendicular to the line .
• 50) Find the equation of the tangent to the curve at the point .
• 51) Find the equation of the normal at the point to the curve .
• 52) Show that the tangents to the curve at the points and are parallel.
• 53) Find the equation of the tangent to the curve at the points where the tangent is parallel to the -axis.
• 54) Find the equation of the tangent to the curve at the point .
• 55) Find the equation of the tangent and normal to the curve at the point .
• 56) Find the equation of the tangent to the curve which is parallel to the line .
• 57) Find the equation of the normal to the curve at the point .
• 58) Find the equation of the normal to the curve at the point .
• 59) Find the equation of the tangent to the curve at the point .
• 60) Find the points on the curve at which the tangent makes an angle of with the positive -axis.
• 61) Find the equation of the tangent to the curve at the origin.
• 62) Find the equation of the tangent to the curve at the point where it cuts the -axis.
• 63) Find the equation of the normal to the curve at the origin.
• 64) Prove that the tangent to the curve at the origin cuts the curve again at a point where the tangents are perpendicular.
• 65) Using differentials, find the approximate value of .
• 66) Using differentials, find the approximate value of .
• 67) Using differentials, find the approximate value of .
• 68) Using differentials, find the approximate value of .
• 69) Using differentials, find the approximate value of .
• 70) Using differentials, estimate .
• 71) Using differentials, find the approximate value of .
• 72) Using differentials, find the approximate value of .
• 73) Using differentials, find the approximate value of .
• 74) The approximate change in the volume of a cube of side metres caused by increasing the side by 3%.
• 75) Find the maximum and minimum values, if any, of the function given by .
• 76) Find the maximum and minimum values, if any, of the function given by .
• 77) Find the local maxima and local minima, if any, of the function .
• 78) Find the local maxima and local minima, if any, of the function .
• 79) Find the absolute maximum and minimum values of on .
• 80) Find the absolute maximum and minimum values of on .
• 81) Find the absolute maximum and minimum values of on .
• 82) Find the local maxima and minima of .
• 83) Find the maximum and minimum values of .
• 84) Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
• 85) Find two positive numbers whose product is 36 and whose sum is minimum.
• 86) The sum of the perimeter of a circle and square is . Prove that the sum of their areas is least when the side of the square is double the radius of the circle.
• 87) A rectangular sheet of tin is to be made into a box by cutting equal squares from each corner and folding up the flaps. Find the side of the square so that the box has maximum volume.
• 88) A window is in the form of a rectangle surmounted by an equilateral triangle. The perimeter of the window is . Find the dimensions that will produce the maximum area.
• 89) Show that the right-circular cone of least curved surface for a given volume is one whose height is times the radius of the base.
• 90) Find the dimensions of the rectangle of maximum area that can be inscribed in a circle of radius .
• 91) Find the maximum profit that a company can make, if the profit function is .
• 92) Find the maximum and minimum values of on .
• 93) An open tank with a square base is to be built to hold of water. Find the dimensions that give the least surface area (hence least material).
• 94) Find the shortest distance of the point from the parabola .
• 95) Find the maximum value of the function on .
• 96) Find the maximum volume of a closed cylinder whose total surface area is .

Answers at a glance

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    Q18 → f'(x) = 3 > 0, so f is strictly increasing on the set of real numbers.
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    Q19 → f'(x) = 2e^(2x) > 0 for all x, so f is strictly increasing on the set of real numbers.
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    Q20 → f'(x) = 3(x - 1)^2 >= 0 with equality only at x = 1; f is strictly increasing on the set of real numbers.
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    Q27 → f'(x) = 1/x > 0 for all x > 0, so f is strictly increasing on the interval (0, infinity).
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    Q28 → f'(x) changes sign at x = 1/2, so f is neither strictly increasing nor strictly decreasing on the set of real numbers.
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    Q31 → f'(x) = cot x > 0 on (0, pi/2), so f is strictly increasing there.
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    Q32 → f'(x) = -tan x < 0 on (0, pi/2), so f is strictly decreasing there.
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    Q33 → y' = x^2 / ((1+x)(2+x)^2) >= 0, zero only at x = 0, so y is an increasing function of x.
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    Q52 → The tangent slopes at x = 0 and x = 1 are 2 and -1, which are not equal.
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    Q64 → The tangent at the origin is y = x, which meets the curve again at a point with slope -1, giving perpendicular tangents.
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MCQ Questions

1 Mark Questions

1. 1
   State the sign of f′(x) for a decreasing function.
2. 2
   Write the slope of the normal to the curve y = x² at x = 1. (Ans: −1/2)
3. 3
   Find the point where the tangent to y = x³ has slope 3.
4. 4
   When is a critical point called a point of inflection?

2 Mark Questions

1. 1
   Find the equation of the tangent to y = x² at (1, 1). (Ans: y = 2x − 1)
2. 2
   Find the intervals in which f(x) = x³ − 3x is increasing. (Ans: increasing on [−1, 1] and [1, ∞) — using f′ = 3x² − 3, increasing for x ≤ −1 and x ≥ 1)
3. 3
   A stone dropped from rest has s = 4.9t² as distance in metres. Find the velocity at t = 2 s. (Ans: 19.6 m/s)

3 Mark Questions

1. 1
   Find the maximum value of f(x) = −x² + 4x + 5 and the point at which it is attained, using the first derivative test. (Ans: max 9 at x = 2)
2. 2
   Find the approximate change in the volume of a cube due to a 2% increase in its side.
3. 3
   Find the interval(s) of increase and decrease for f(x) = x³ − 6x² + 9x + 15. (Ans: increasing on (−∞, 1) and (3, ∞), decreasing on (1, 3))

4 Mark Questions

1. 1
   A window is in the form of a rectangle surmounted by a semicircle and has a perimeter of 10 m. Find the dimensions that admit maximum light. Show the steps of optimisation and justify the maximum using the second derivative.
2. 2
   A tank is to be built with a square base and open top holding a fixed volume V. Find the side of the base and the height that minimise the material needed; give the ratio height : base side. (Ans: h = a/2)