Application of Derivatives

34 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Application of Derivatives' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Application of Derivatives'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Application of Derivatives'? medium MCQ Problem Solving & Optimization
• 4 Find the minimum and maximum values of the function f(x)=x^2-4x+5 on the interval [1,4]. easy 2 marks Find extrema of a quadratic on a closed interval
• 5 A company has the cost function C(x)=1000+50x+2x^2 rupees, where x is the number of units produced in thousands. Determine the production level x that minimises the average cost and compute the minimum average cost. hard 4 marks Optimising average cost
• 6 Water is being poured into a conical tank of height 12 m and radius 4 m. The water level rises at a constant rate of 0.5 m/min. Find the rate of increase of the volume of water when the water level is 6 m. hard 4 marks Related rates for a conical tank
• 7 A rectangle is to be inscribed in a right‑angled triangle whose base is 12,cm and height is 8,cm. The rectangle’s base lies on the triangle’s base and its top side is parallel to that base. Find the dimensions of the rectangle that give the maximum area. medium 3 marks Optimization of area in a triangle
• 8 Consider the function f(x)=x^3-3x^2+4. On which intervals is f increasing? easy MCQ 1 mark Increasing / decreasing intervals
• 9 A 5 m ladder is leaning against a vertical wall. The foot of the ladder is sliding away from the wall at a speed of 0.6 m/s. Find the rate at which the top of the ladder is descending when the foot is 3 m from the wall. medium 2 marks Related rates – sliding ladder
• 10 A cylindrical can is to hold exactly 500 cm^3 of liquid. The material for the top and bottom costs Rs 0.02 per cm^2, while the material for the curved side costs Rs 0.01 per cm^2. Find the radius and height of the can that minimise the total cost. Give your answer correct to two decimal places. hard 4 marks Optimization – cost minimisation
• 11 Find the absolute maximum and minimum values of the function f(x)=x^3-6x^2+9x+1 on the closed interval [0,4]. medium 2 marks Absolute extrema on a closed interval
• 12 A cylindrical can is to hold exactly 2000 cm^3 of liquid. The material for the top and bottom costs Rs 0.03 per cm^2, while the material for the curved side costs Rs 0.015 per cm^2. Find the radius and height of the can that minimise the total cost. Give your answer correct to two decimal places. medium 4 marks Optimization – cost minimisation of a cylinder
• 13 A 6 m ladder leans against a vertical wall. The foot of the ladder is sliding away from the wall at a speed of 0.8 m/s. Find the rate at which the top of the ladder is descending when the foot is 4 m from the wall. medium 3 marks Related rates – ladder problem
• 14 Find the intervals on which the function f(x)=4x^3-9x^2+3x+5 is increasing. easy 2 marks Increasing/decreasing intervals
• 15 Find the dimensions (length and width) of the rectangle with sides parallel to the axes that has maximum area when it is inscribed in the ellipse x^2/25+y^2/9=1. Give the maximum area to two decimal places. hard 5 marks Optimization – maximum area of rectangle in ellipse
• 16 An 8 m ladder is leaning against a vertical wall. The foot of the ladder slides away from the wall at 0.7 m/s. Find the speed at which the top of the ladder descends when the foot is 6 m from the wall. medium 3 marks Related rates – ladder problem
• 17 Find the intervals on which the function f(x)=2x^3-9x^2+12x-3 is increasing. easy 2 marks Increasing/decreasing of a function
• 18 A cylindrical can must hold exactly 1500 cm^3 of oil. The material for the top and bottom costs Rs 0.04 per cm^2 and the material for the curved side costs Rs 0.02 per cm^2. Find the radius r and height h (in cm) of the can that minimise the total cost. Give your answers correct to two decimal places. hard 5 marks Optimization with a volume constraint
• 19 Find the absolute maximum and minimum values of f(x)=2x^4-8x^3+9x^2 on the closed interval [0,3]. hard 3 marks Absolute extrema on closed interval
• 20 A cylindrical can must hold exactly 2000 cm^3 of oil. The material for the top and bottom costs Rs,0.05 per cm^2 and the material for the curved side costs Rs,0.02 per cm^2. Find the radius r and height h (in cm) that minimise the total cost. Give your answers correct to two decimal places. hard 4 marks Optimization of cylindrical cost
• 21 Which of the following intervals is the function f(x)=3x^3-12x^2+9x+5 increasing? (a) (-infty,1) (b) (1,2) (c) (2,3) (d) (3,infty) easy MCQ 1 mark Increasing/decreasing intervals
• 22 A 10 m ladder leans against a vertical wall. The foot of the ladder slides away from the wall at 0.5 m/s. Find the speed at which the top of the ladder descends when the foot is 7 m from the wall. medium 2 marks Related rates
• 23 A 6 m ladder is leaning against a vertical wall. The foot of the ladder slides away from the wall at 0.4 m/s. Find the speed at which the top of the ladder descends when the foot is 5 m from the wall. medium 2 marks Related rates for a ladder
• 24 Which of the following intervals is the function f(x)=2x^3-9x^2+12x-4 increasing? easy MCQ 1 mark Increasing intervals of a polynomial
• 25 A company’s total cost (in rupees) for producing x thousand units is given by C(x)=2000+80x+3x^2. Find the production level x that minimises the average cost per unit and state the minimum average cost (in rupees per unit). hard 3 marks Minimising average cost using derivatives
• 26 An open‑top rectangular box is to be made from a sheet of material having total surface area 1500 cm^2. Find the dimensions (length l, width w, height h) of the box that give the maximum possible volume. Give the dimensions correct to two decimal places. hard 3 marks Maximum volume with a surface‑area constraint
• 27 Find the interval(s) on which the function f(x)=2x^3-9x^2+12x-5 is increasing. easy MCQ 1 mark Increasing intervals of a polynomial
• 28 An open‑top rectangular box is to be made from a sheet of material having total surface area 1200 cm^2. Find the dimensions (length l, width w, height h) that give the maximum possible volume. Give the dimensions correct to two decimal places. medium 3 marks Optimization – maximum volume with given surface area
• 29 A cylindrical can must hold exactly 2500 cm^3 of oil. The material for the top and bottom costs Rs 0.06 per cm^2 and the material for the curved side costs Rs 0.02 per cm^2. Find the radius r and height h (in cm) that minimise the total cost, and state the minimum cost (in rupees). Give r and h correct to two decimal places. hard 4 marks Optimization – cost minimisation with volume constraint
• 30 A 7m ladder leans against a vertical wall. The foot of the ladder slides away from the wall at 0.3m/s. Find the speed at which the top of the ladder descends when the foot is 4m from the wall. easy 2 marks Related rates – ladder problem
• 31 Use linear approximation (the tangent-line estimate) to find √2.01 correct to four decimal places. Write f(x)=√x and take the base point as a=2. easy 3 marks Linear approximation using the first-order Taylor expansion $f(a+h)\approx f(a)+h\,f'(a)$
• 32 Use linear approximation to estimate sqrt[3]8.2 correct to three decimal places. Write f(x)=x^(1/3) and take base point a=8. easy 1 mark Linear approximation (tangent line estimate)
• 33 On which intervals is the function f(x)=x^3-3x^2+4x-5 increasing? easy 1 mark Determine intervals of increase using derivative
• 34 Use linear approximation to estimate sqrt[3]9.3 correct to three decimal places. Write f(x)=x^(1/3) and take the base point a=8. medium 2 marks Linear approximation for cube root