CALCULUS II

TUTORIAL 4 ALL COHORTS 23 SEPTEMBER 2022

1. Redline Roasting has found that the cost, in dollars per pound, of the coffee it roasts is

𝐶 ′ (𝑥) = −0.012𝑥 + 6.50, 𝑓𝑜𝑟 𝑥 ≤ 300,

where 𝑥 is the number of pounds of coffee roasted. Find the total cost of roasting 200 lb. of coffee.

2. Sylvie’s Old-World Cheeses has found that the cost, in dollars per kilogram, of the cheese it
produces is

𝐶 ′ (𝑥) = −0.003𝑥 + 4.25, 𝑓𝑜𝑟 𝑥 ≤ 500,

where 𝑥 is the number kilograms of cheese produced. Find the total cost of producing 400kg of
cheese.

3. Photos from Nature has found that the cost per card of producing note cards is given by

𝐶 ′ (𝑥) = −0.04𝑥 + 85, 𝑓𝑜𝑟 𝑥 ≤ 1000

where 𝐶 ′ (𝑥) is the cost, in cents, per card. Find the total cost of producing 650 cards.

4. Cleo’s Custom Fabrics has found that the cost per yard of producing yards of a particular fabric
is given by

𝐶 ′ (𝑥) = −0.007𝑥 + 12, 𝑓𝑜𝑟 𝑥 ≤ 350

where 𝐶 ′ (𝑥) is the cost in dollars. Find the total cost of producing 200 yd of this material.

5. A concert promoter sells tickets and has a marginal-profit function given by

where 𝑃′ (𝑥) is in dollars per ticket. This means that the rate of change of total profit with respect
to the number of tickets sold, 𝑥, is 𝑃′ (𝑥). Find the total profit from the sale of the first 300 tickets.

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6. Poyse Inc. has a marginal-profit function given by

𝑃′ (𝑥) = −2𝑥 + 80

where 𝑃′ (𝑥) is in dollars per unit. This means that the rate of change of total profit with respect to
the number of units produced, 𝑥, is 𝑃′ (𝑥). Find the total profit from the production and sale of the
first 40 units.

7. Raggs, Ltd., determines that its marginal cost, in dollars per dress, is given by
−2
𝐶 ′ (𝑥) = 𝑥 + 50, 𝑓𝑜𝑟 𝑥 ≤ 450. Find the total cost of producing the first 200 dresses.
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8. The marginal revenue or a certain product is given by 𝑅 ′ (𝑥) = 300 − 2𝑥. Find the total revenue
function, 𝑅′(𝑥), assuming that 𝑅(0) = 0.

9. The marginal cost for a certain product is given by 𝐶 ′ (𝑥) = 2.6 − 0.02𝑥. find the total-cost
function 𝐶(𝑥), and the average cost, 𝐴(𝑥), assuming that fixed costs are $120; that is 𝐶(0) = 0 =
$120.

𝑑𝑃 −200
10. A firm’s marginal profit, 𝑃, as a function of its total cost, 𝐶, is given by 𝑑𝐶 = 3 .
(𝐶+3) ⁄2

a. Find the profit 𝑃(𝐶), if 𝑃 = $10, when 𝐶 = $61.

b. At what cost will the firm break even (𝑃 = 0)?

11. A company determines that its marginal-cost function is given by

𝐶 ′ (𝑥) = 4𝑥 √𝑥 + 3. Find the total cost given that 𝐶(13) = $1126.40.

12. A profit-making company wants to launch a new product. It observes that the fixed cost of the
new product is $7500, and the variable cost is $500. The revenue received on sale of 𝑥 units is
2500𝑥 − 100𝑥 2 . Find: (i) profit function (ii) break-even point.

13. A company paid $16100 towards rent of the building and interest on loan. The cost of
producing one unit of a product is $20. If each unit is sold for $27, find the break-even point.

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14. A company has fixed cost of $26000 and the cost of producing one unit is $30. If each unit
sells for $43, find the breakeven point.

15. A company sells its product for $10 per unit. Fixed costs for the company are $35000 and the
variable costs are estimated to run 30 % of total revenue. Determine: (i) the total revenue function
(ii) total cost function and (iii) quantity the company must sell to cover the fixed cost.

16. The fixed cost of a new product is $30000 and the variable cost per unit is $800. If the demand
function is 𝑝(𝑥) = 4500 − 100𝑥 find the break-even values.

17. If the total cost function 𝐶 of a product is given by

𝑥+7
𝐶 = 3𝑥 (𝑥+5). Prove that the marginal cost falls continuously as the output increases.

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18. The average cost function (𝐴𝐶) for a product is given by 𝐴𝐶 = 𝑥 + 5 + , where 𝑥 is the
𝑥

output. Find the output for which AC is increasing and the output for which AC is decreasing with
increasing output. Also find the total cost C and the marginal cost MC.

19. A firm knows that the demand function for one of its products in linear. It also knows that it
can sell 1000 units when the price is $4 per unit, and it can sell 1500 units when the price is $2 per
unit. Find: (i) the demand function (ii) the total revenue function (iii) the average revenue function
and (iv) the marginal revenue function.

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20. The average cost function AC for a product is given by 𝐴𝐶 = 𝑥 + 5 + . Find the total cost
𝑥

and marginal cost functions. Also find MC when 𝑥 = 10.

21. The demand function for a product is given as 𝑝 = 30 + 2𝑥 − 5𝑥 2 , where 𝑥 is the number

of units demanded and p is the price per unit. Find (i) Total revenue (ii) Marginal revenue

(iii) MR when 𝑥 = 3.

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22. For the demand function 𝑝 = 3+𝑥 , show that the marginal revenue function is an increasing

function.

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23. During a surge in the demand for electricity, the rate, 𝑟, at which energy is used can be
approximated by 𝑣 = 𝑡𝑒 −𝑎𝑡 , where 𝑡 is the time in hours and 𝑎 is a positive constant.

a) find the total energy, 𝐸, use in the first T hours. Give your answer as a function of 𝑎.

b. What happens to 𝐸 as 𝑇 → ∞?

24. The concentration, C, in 𝑛𝑔/𝑚𝑙, of a drug in the blood as a function of the time, t in hours
since the drug was administered is given by 15𝑡𝑒 −0.2𝑡 . The area under the concentration curve is
a measure of the overall effect of the drug on the body, called bioavailability. Find the
bioavailability of the drug between 𝑡 = 0 and 𝑡 = 3.

25. An evergreen nursery usually sells a type of shrub after 5 years of growth and shaping. The
𝑑ℎ 17.6𝑡
growth rate during those 5 years is approximated by = √17.6𝑡 2 , where 𝑡 is time in years and ℎ
𝑑𝑡 +1

is height in inches. The seedlings are 6 inches tall when planted (𝑡 = 0).

(a) Find the height function.

(b) How tall are the shrubs when they are sold?

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