section 4.1

1. Find the indefinite integral. 4x dx

2. Find the indefinite integral. x dx

3 21 2 3 2x 2

x dx x dx C x C3 2 3

3. Find the indefinite integral.1dxx

1 21 2 1 21 x

dx x dx C 2x C1 2x

4. Find the indefinite integral. (8x 5)dx

5. Find the indefinite integral.3

16 x dx

x

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

7. Find the indefinite integral.2(x 1) dx

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

9. Find the indefinite integral. (x 2)(x 4)dx

10. Find the indefinite integral.2x 1

dxx 1

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 .

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. 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.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

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.