ECON 220 (8)

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Quant Tools for Bus & Econ – ECON 220-02 (11273)

Homework 8

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Section 11.1 (20 points)

Find the derivative of each function

1. 𝑓(𝑥) = 3𝑥 3 − 2𝑥 2 + 𝑥

2. 𝑓(𝑥) = 𝑥 0.5 + 2𝑥 −0.5

3. 𝑔(𝑥) = 1

𝑥2 +

1

𝑥3

4. ℎ(𝑥) = − 1

2𝑥0.2

5. ℎ(𝑥) = 2

𝑥 −

2

𝑥3 +

1

𝑥4

2

6. 𝑡(𝑥) = 3|𝑥| − √𝑥

7. 𝑠(𝑥) = 𝑥 + 7

√𝑥

Find the indicated derivative

8. 𝑦 = 𝑥1.2

3 −

𝑥0.9

2 ;

𝑑𝑦

𝑑𝑥

9. 𝑠 = 2

𝑡 1.1 + 𝑡 −1.2;

𝑑𝑠

𝑑𝑡

10. 𝐴 = 4𝜋𝑟2; 𝑑𝐴

𝑑𝑟

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Section 11.2 (14 points) The cost of producing x Teddy Bears per day is given by

𝐶(𝑥) = 350 + 0.10𝑥 + 0.002𝑥2 𝑑𝑜𝑙𝑙𝑎𝑟𝑠

a. Find the marginal cost function and use it to estimate how fast the cost is going up at a

production level of 100 Teddy Bears. Compare this with the exact cost of producing 101st

Teddy bear (3 points)

b. Find the average cost function �̂�, and evaluate �̂�(100). (2 points)

c. Interpret your result (1 point)

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11. The Economic Student Association is planning its annual eating competition fundraising

where it will charge students for $1.10 per serving of pasta. ESA estimates that the total

cost of producing x servings of pasta a the event will be given by

𝐶(𝑥) = 350 + 0.10𝑥 + 0.002𝑥2 𝑑𝑜𝑙𝑙𝑎𝑟𝑠

a. Calculate the marginal revenue and marginal profit functions (2 points)

b. Compute the (i) revenue (ii) profit (iii) marginal revenue and (iv) marginal profit if you

produced and sold 200 servings of pasta. Interpret your results (5 points)

c. For which value of x is the marginal profit zero? (1 point)

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Section 11.3 (11 points)

Calculate 𝑑𝑦

𝑑𝑥 and simplify your answer (2 points each)

12. 𝑦 = 3𝑥2(2𝑥 + 1)

13. 𝑦 = 3𝑥

−3𝑥+2

14. The monthly sales of Sunny electronics new sound system are given by 𝑞(𝑡) = 2,000𝑡 −

100𝑡2 units per month, t months after its introduction.

The price Sunny Charges is 𝑝(𝑡) = 1,000𝑡 − 𝑡2 dollars per sound system, t months after introduction.

a. Find the rate of change of monthly sales 5 months after the introduction of the sound

system (2 point)

b. the rate of change of the price 5 months after the introduction of the sound system

(2 point)

c. Find the rate of change of monthly revenue 5 months after the introduction of the

sound system. (3 points)