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Business Mathematics and Statistics · Class 11 Commerce

Ch 6Applications of Differentiation (incl. Business/Economics applications, Maxima/Minima, Partial Derivatives) — Class 11 Business Mathematics and Statistics, concept-first.

The previous chapter built up the machinery of differentiation — limits, derivatives, and the rules for differentiating standard functions. This chapter puts that machinery to work on real business and economic questions: how fast is cost changing as output rises, at what output is profit largest, and how does a firm's…

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Key concepts

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Maxima and Minima of a Function

A stationary point () is classified as a local maximum or minimum using the first derivative test (sign change of ) or the equivalent second derivative test ( for a maximum, for a minimum).

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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.

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Marginal Cost and Marginal Revenue

The previous chapter built up the machinery of differentiation — limits, derivatives, and the rules for differentiating standard functions.

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Elasticity of Demand

A demand function relates the quantity demanded to the price (usually falls as rises). A manager rarely cares only about the direction of that relationship — a much sharper question is how sensitive d…

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Maxima and Minima of a Function — First and Second Derivative Tests

Many business questions boil down to finding the single best value of something — the output that maximises profit, the price that minimises average cost.

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Applications: Maximising Profit and Minimising Average Cost

The maxima/minima tests of the previous section become genuinely useful business tools once applied to profit and cost.

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Partial Derivatives — Functions of Two Variables

Every function differentiated so far has depended on a single variable. Many genuine business quantities depend on two or more variables at once — a firm's output depends on both labour and capital; a…

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Applications of Partial Derivatives — Marginal Productivity

A firm's output is commonly modelled as a function of two inputs — labour and capital — written , called a production function.

Exercises

+Show 5 questions5 questions
  1. Q7A manufacturer's profit function is $P(x) = -2x^2 + 40x - 150$ (in thousand rupees), where $x$ is the number of units produced (in hundreds)…Free
  2. Q8The demand function for a product is $x = 200 - 4p$. At $p=30$, determine whether demand is elastic, inelastic, or unit elastic, and state w…Free
  3. Q9If $z = x^2y + 3xy^2 - 5x + 2y$, find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{\partial y}$, and evaluate both at $(x,y)=(1,…Preview
  4. Q10If $z = x^3 + x^2y^2 + y^3$, find $\dfrac{\partial^2 z}{\partial x^2}$, $\dfrac{\partial^2 z}{\partial y^2}$, and the mixed partial derivati…Preview
  5. Q11A firm's output is given by the production function $Q(L,K) = 4L^2 + 3LK + 2K^2$, where $L$ is labour and $K$ is capital. Find the marginal…Preview

Sample & Board Papers

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

+Show 24 questions24 questions
  1. Q1If $u = e^{x^2}$ then $\dfrac{\partial u}{\partial x}$ is equal to : (a) $0$ (b) $2xe^{x^2}$ (c) $e^{x^2}$ (d) $2e^{x^2}$Preview
  2. Q2If $u(x, y)$ is a continuous function of $x$ and $y$, then $\dfrac{\partial^2 u}{\partial y \, \partial x}$ is equal to : (a) $\dfrac{\parti…Preview
  3. Q3Find the elasticity of supply for the supply function $x = 2p^2 + 5$ when $p = 3$.Preview
  4. Q4If $f = x^3y + y^4z - z^3x^2y$, find $\dfrac{\partial^2 f}{\partial x^2}$ and $\dfrac{\partial^2 f}{\partial y^2}$.Preview
  5. Q5(a) The total cost function of a firm is $C(x) = \dfrac{x^3}{3} - 5x^2 + 28x + 10$, where $x$ is the output. A tax at the rate of ₹ 2 per un…Preview
  6. Q6(a) The demand for a commodity A is $q = 250 - P_1^2 + 3P_2 - P_1 P_2$. Find the partial elasticities $\dfrac{Eq}{EP_1}$ and $\dfrac{Eq}{EP_…Preview
  7. Q7If the demand function is said to be elastic, then : (a) $|\eta_d| < 1$ (b) $|\eta_d| > 1$ (c) $|\eta_d| = 0$ (d) $|\eta_d| = 1$Preview
  8. Q8Relationship among MR, AR and $\eta_d$ is : (a) $MR = AR = \eta_d$ (b) $\eta_d = \dfrac{AR}{AR - MR}$ (c) $AR = \dfrac{MR}{\eta_d}$ (d) $\et…Preview
  9. Q9For the function $y = x^{3} + 19$, find the values of $x$ when its marginal value is equal to $27$.Preview
  10. Q10Find the values of $x$, when the marginal function of $y = x^{3} + 10x^{2} - 48x + 8$ is twice the $x$.Preview
  11. Q11(a) Find the stationary points and stationary values for the function : $f(x) = 2x^{3} + 9x^{2} + 12x + 1$. OR (b) As the number of units pr…Preview
  12. Q12(a) For the cost function $C = 2x\left(\dfrac{x+5}{x+2}\right) + 7$, prove that Marginal Cost (MC) falls continuously as the output $x$ incr…Preview
  13. Q13If $u = e^{x^2}$, then $\dfrac{\partial u}{\partial x}$ = ______ . (a) $2e^{x^2}$ (b) $2x\,e^{x^2}$ (c) $0$ (d) $e^{x^2}$Preview
  14. Q14The maximum value of $f(x) = \sin x$ is : (a) $\dfrac{1}{\sqrt{2}}$ (b) $1$ (c) $\dfrac{-1}{\sqrt{2}}$ (d) $\dfrac{\sqrt{3}}{2}$Preview
  15. Q15The total cost C in Rupees of making $x$ units of a product is $C(x) = 50 + 4x + 3\sqrt{x}$. Find the marginal cost of the product at 9 unit…Preview
  16. Q16The total cost function $y$ for $x$ units is given by $y = 3x\left(\dfrac{x + 7}{x + 5}\right) + 5$. Show that the Marginal Cost [MC] decrea…Preview
  17. Q17For the function $y=x^3+19$, find the values of $x$ when its marginal value is equal to $27$. (a) $\pm 3$ (b) $\pm 1$ (c) $\pm 4$ (d) $\pm 2…Preview
  18. Q18If $q=1000+8p_1-p_2$, then $\dfrac{\partial q}{\partial p_1}$ is : (a) $1000$ (b) $-1$ (c) $1000-P_2$ (d) $8$Preview
  19. Q19A manufacturing company has a contract to supply 4000 units of an item per year at uniform rate. The storage cost per unit per year amounts…Preview
  20. Q20Find the interval in which the function $f(x)=x^2-4x+6$ is strictly increasing and strictly decreasing.Preview
  21. Q21If demand and the cost function of a firm are $p=2-x$ and $c=-2x^2+2x+7$, then its profit function is : (a) $-x^2+7$ (b) $x^2+7$ (c) $-x^2-7…Preview
  22. Q22If $u=e^{x^2}$, then $\dfrac{\partial u}{\partial x}$ is equal to : (a) $2e^{x^2}$ (b) $2xe^{x^2}$ (c) $0$ (d) $e^{x^2}$Preview
  23. Q23For the given demand function $p=40-x$, find the output when $\eta d=1$.Preview
  24. Q24If $u=x^2(y-x)+y^2(x-y)$, then show that $\dfrac{\partial u}{\partial x}+\dfrac{\partial u}{\partial y}=-2(x-y)^2$.Preview

More questions

+Show 6 questions6 questions
  1. Example 1The total cost function of a firm is $C(x) = x^3 - 3x^2 + 5x + 100$, where $x$ is the number of units produced. Find the marginal cost and t…Free
  2. Example 2The demand function for a commodity is $p = 100 - 2x$, where $p$ is the price per unit and $x$ is the quantity demanded. Find the total reve…Free
  3. Example 3If the demand function is $x = 50 - p$, find the price elasticity of demand when $p = 20$, and state whether demand is elastic, inelastic, o…Preview
  4. Example 4Find the points of local maximum and local minimum of $f(x) = x^3 - 6x^2 + 9x + 15$, and state the corresponding maximum and minimum values.Preview
  5. Example 5The total cost function of a firm is $C(x) = 2x^2 + 3x + 50$. Find the output at which average cost is minimum, and verify using the conditi…Preview
  6. Example 6A firm's total revenue and total cost functions are $R(x) = 40x - x^2$ and $C(x) = x^2 + 8x + 20$. Find the output level that maximises prof…Preview