Section 4.1 1. Find the indefinite integral.. 2. Find the indefinite integral.
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Transcript of Section 4.1 1. Find the indefinite integral.. 2. Find the indefinite integral.
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section 4.1
1. Find the indefinite integral. 4x dx
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2. Find the indefinite integral. x dx
3 21 2 3 2x 2
x dx x dx C x C3 2 3
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3. Find the indefinite integral.1dxx
1 21 2 1 21 x
dx x dx C 2x C1 2x
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4. Find the indefinite integral. (8x 5)dx
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5. Find the indefinite integral.3
16 x dx
x
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6. Find the indefinite integral.3 2
3 2
110 x dx
x
3 2 2 3 2 3
3 2
110 x dx 10 x dx x dx
x
5 3 1 35 3 1 3x x
10 C 6x 3x C5 3 1 3
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7. Find the indefinite integral.2(x 1) dx
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8. Find the indefinite integral. (1 10x) x dx
1 2 3 2(1 10x) x dx (x 10x )dx
3 2 5 23 2 5 2x x 2
10 C x 4x C3 2 5 2 3
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9. Find the indefinite integral. (x 2)(x 4)dx
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10. Find the indefinite integral.2x 1
dxx 1
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11. a. Verify that 3
2 xx dx C.
3
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 .
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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.
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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.
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14.GENERAL : Learning – A person can memorize words at a rate of 3/t words per minute.a. Find a formula for the total number of words that can be memorized
in t minutes.b. 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 wordsmemorized is N(25) = 6(25) 1/2 = 30 words
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15.ENVIRONMENTAL SCIENCE: Pollution – A chemical plant is adding pollution to a lake at the rate of 40t 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 tonsThus, 512 tons of pollution will enter thelake in the first four years of the plant’soperation.
c. No, since 512 > 400.