Application of Calculus

If $R(x) = 8x - x^2$, find the value of $AR$ at $x=4$.

If $C(x) = x^2 + 4$, what is the Marginal Cost at $x=3$?

The Marginal Cost function is the _____ of the Total Cost function.

For a demand function $p = 10 - x$, at what $x$ is the Total Revenue maximized?

If the price is given by $p = 100 - 3x$, find the marginal revenue function.

What is the elasticity of demand for the demand function $x = 10/p$?

If the marginal cost is $MC = 20 + 2x$ and fixed costs are zero, what is the average cost function?

For a product, the demand is $x = 100 - p$. At what price $p$ is the price elasticity of demand equal to 2?

If the demand function is $p = 10e^{-x/2}$, find the Consumer's Surplus when $x = 2$.

The marginal cost is $MC = 2 + 5e^{x}$. Find the total cost function if the fixed cost is $C(0) = 10$.

A company's total revenue function is represented by the curve $R(x) = 20x - 0.5x^2$, where $x$ represents the number of units sold. The graph below shows the revenue function on a coordinate plane. Find the marginal revenue when 10 units are sold.

The total cost function for producing $x$ units of a commodity is given by $C(x) = 2x^2 + 15x + 50$. The graph below shows the Marginal Cost curve $MC(x)$. Find the Marginal Cost when $x = 10$ units.

If the marginal cost is $MC = 4x + 1$, find the change in total cost when production increases from $x=2$ to $x=4$.

In a competitive market, price $p$ is constant at 15. If $C(x) = x^2 + 5x + 10$, find the output $x$ that maximizes profit.

Given the demand function $p = 120 - 3x$ and the cost function $C(x) = x^2 + 40x + 50$, the diagram displays the Profit function $P(x)$. Determine the price $p$ at which the profit is maximized.

The average cost function for a manufacturer is given by $AC = \frac{100}{x} + 20 + x$. The diagram shows the total cost curve. Find the level of output $x$ at which the Marginal Cost (MC) is equal to the Average Cost (AC).

A company has a cost function $C(x) = 500 + 30x$ and a revenue function $R(x) = 100x - x^2$. Calculate the output level $x$ where the profit is increasing at a rate of 10 units per production of one extra unit.

A tech company manufactures a specialized microchip. The daily total cost $C(x)$ (in thousands of Rupees) is given by $C(x) = 50 + 20x$, where $x$ is the number of chips produced. Market research shows the demand function is $p(x) = 100 - x$, where $p$ is the price per chip in thousands of Rupees. The following graph shows the Marginal Revenue (MR) and Marginal Cost (MC) curves for this production process. At what level of production $x$ is the daily profit maximized, and what is the maximum daily profit (in thousands of Rupees)?

A firm's marginal revenue is $MR = 20 - 2x$ and marginal cost is $MC = 4$. What is the total change in profit if production increases from $x=2$ to $x=6$? Use the area between curves in the diagram.

A company has a total cost function $C(x) = 50 + 10x - 0.5x^2 + \frac{1}{3}x^3$. Determine the range of $x$ for which the marginal cost is decreasing. Look at the $MC$ curve vertex.

If $p = 20 - \frac{x}{4}$, find the marginal revenue when $x = 20$.

The total cost function is $C(x) = 0.2x^2 + 5x + 100$. Find the marginal cost at $x = 20$.

If $R(x) = 150x - 2x^2$, find the marginal revenue when $x = 25$.

Find the marginal cost at $x = 1000$ for the linear cost function $C(x) = 3x + 100$.

If $C(x) = 4x + \frac{x^2}{10}$, find the marginal cost at $x = 10$.

For the demand function $p = 10 - 2x$, find the price elasticity of demand when $x = 2$.

If the total cost function is $C(x) = 5x + 100$, what is the marginal cost?

The Marginal Revenue function is $MR = a - 2bx$. Find the value of the price elasticity of demand when $x = a/4b$.

If $C(x) = \frac{x^2}{4} + 3x + 400$, at what output is $MC = AC$?

The revenue function is $R(x) = 200x - 0.1x^2$. Find the percentage change in $MR$ if $x$ increases from 100 to 110.

Identify the break-even value of $x$ when $R(x) = 200x$ and $C(x) = 100x + 1000$.

If the profit function is given by $P(x) = 100x - 5000$, find the output $x$ for the break-even point.

Find the break-even level $x$ if $AC = 50 + \frac{500}{x}$ and $AR = 100$.

If $R(x) = 40x$ and $C(x) = 30x + 200$, find the break-even quantity.

Find the break-even points if $p = 200$ and $C(x) = 2x^2 + 40x + 3000$.

A company has a cost function $C(x) = 40x + 800$ and a revenue function $R(x) = 130x - x^2$. Calculate the break-even points.

Determine the break-even quantity for a firm with profit function $P(x) = -x^2 + 65x - 1000$.

The total revenue function is $R(x) = 800x - 4x^2$ and the total cost function is $C(x) = x^2 + 100x + 15000$. Find the break-even levels.

A manufacturer has $MR = 500 - 2x$ and $MC = 4x + 80$. If fixed costs are ₹5,000, find the break-even points.

If $p = 600 - 3x$ and $TC = x^2 + 60x + 12000$, determine the break-even points.