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Indefinite Integrals, Functions, and Pollution Analysis in Environmental Science

This section covers various calculus operations including finding indefinite integrals for multiple functions, verifying derivatives at a specific point, and analyzing business costs and revenues through marginal functions. Additionally, it explores the rate of memorization in a learning context and pollution impacts from a chemical plant, calculating total pollution over time and its ecological implications. Key applications include evaluating the effects of marginal cost and revenue functions and determining environmental health in relation to pollution thresholds.

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Indefinite Integrals, Functions, and Pollution Analysis in Environmental Science

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  1. section 4.1 1. Find the indefinite integral.

  2. 2. Find the indefinite integral.

  3. 3. Find the indefinite integral.

  4. 4. Find the indefinite integral.

  5. 5. Find the indefinite integral.

  6. 6. Find the indefinite integral.

  7. 7. Find the indefinite integral.

  8. 8. Find the indefinite integral.

  9. 9. Find the indefinite integral.

  10. 10. Find the indefinite integral.

  11. 11. a. Verify that b. Graph the five functions 1/3 x 3 – 2, 1/3 x 3 – 1, 1/3 x 3 , 1/3 x 3 + 1, 1/3 x 3 + 2, on the window [-3, 3] and [-5, 5]. c. Find the derivatives at x = 2 of the five functions in part b. c. For x = 2, the slope of each curve is 4. Thus, the derivative of each curve is x2 .

  12. BUSINESS: Cost – A company’s marginal cost function is • MC = 20 x 3/2 – 15 x 2/3 + 1, where x is the number of units, and fixed cost are $4000. Find the cost function.

  13. 13. BUSINESS: Revenue – A company’s marginal revenue function is MR = 12 x 1/3 + 3 x ½ , where x is the number of units. Find the revenue function.

  14. GENERAL : Learning – A person can memorize words at a rate of 3/t words per minute. • Find a formula for the total number of words that can be memorized in t minutes. • Use that formula to find the total number of words that can be memorized in 25 minutes. b. After 25 minutes, the total number of words memorized is N(25) = 6(25) 1/2 = 30 words

  15. ENVIRONMENTAL SCIENCE: Pollution – A chemical plant is adding pollution to a lake at the rate of 40t 3 tons per year, where t is the number of years that the plant has been in operation. • a. Find a formula for the total amount of pollution that will enter the lake in the first t years of the plant’s operation. • b. Use the formula that you just found to find out how much pollution will enter the lake in the first 4 years of the plant’s operation. • c. If all life in the lake will cease when 400 tons of pollution have entered the lake, will the lake live beyond 4 years? b. P(4) = 16(4) 5/ 2 = 512 tons Thus, 512 tons of pollution will enter the lake in the first four years of the plant’s operation. c. No, since 512 > 400.

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