II. Sketch the given region R and then find the area. 2. R is the region bounded by the curves y = 0, y = x 2 and x = 3.

Size: px

Start display at page:

Download "II. Sketch the given region R and then find the area. 2. R is the region bounded by the curves y = 0, y = x 2 and x = 3."

Imogene Heath
7 years ago
Views:

1 Math 34 April I. It is estimated that t days from now a farmer s crop will be increasing at the rate of.5t +.4t + bushels per day. By how much will the value of the crop increase during the next 5 days if the market price remains fixed at $5 per bushel? II. Sketch the given region R and then find the area.. R is the region bounded by the curves y =, y = x and x = 3.. R is the region bounded by the curves y =, y = x and x = R is the region bounded by the curves y = x, y = x and x =. 4. R is the region bounded by the curves y = x, y = x and x =. 5. R is the region bounded by the curves y = x + and y = x 6. R is the region bounded by the curves y = e x, y = e x and x = ln(). 7. R is the region bounded by the curves y = /x, y = x and y = x/8. 8. R is the region bounded by the curves y = x 3x and the x-axis. 9. R is the region bounded by the curves y = x 3 and y = x.. R is the region bounded by the curves y = x 3 4x and y = x + 6x. R is the region bounded by the curves y = x 3 4x and y = x 4

2 . R is the region bounded by the curves y = xe x, y = and x = ln(4). 3. R is the triangle with vertices ( 4, ), (, ), and (, 6) 4. R is the trapezoid bounded by the lines y = x +, y = 8 x, x =, and the y-axis.. 5. R is the region bounded by the curves y = x and x = 4 III. Find the average value of the given function f(x) over the specified interval a x b. f(x) = x 3x + 5 over x.. f(x) = e 3x + e x+ over x ln(). 3. f(x) = x + x + x + 6 over x. IV. After t months on the job a postal clerk can sort Q(t) = 7 4e.5t letters per hour. What is the average rate at which the clerk sorts mail during the first 3 months on the job? V. The number of bacteria present in a certain culture after t minutes of an experiment was Q(t) = e.5t. What was the average number of bacteria present during the first 5 minutes of the experiment?

3 I. It is estimated that t days from now a farmer s crop will be increasing at the rate of.5t +.4t + bushels per day. By how much will the value of the crop increase during the next 5 days if the market price remains fixed at $5 per bushel? 5.5t +.4t+dt = (/6)t 3 +(/5)t +t 5 = (/6)5+(/5)5+5 = 55 6 The farmer s crop will increase 55 bushels, therefore the value will increase = 55 = The farmer s crop value will increase by $77.5. II. Sketch the given region R and then find the area.. R is the region bounded by the curves y =, y = x and x = x dx = x3 3 3 = 7 3 = 9. R is the region bounded by the curves y =, y = x and x = 3. 3

4 5 5 3 ( x )dx = x3 3 3 = 7 3 = 9 3. R is the region bounded by the curves y = x, y = x and x =. x ( x)dx = xdx = x = 4. R is the region bounded by the curves y = x, y = x and x =. 4

5 x ( x )dx = x dx = 3 x3 = 3 5. R is the region bounded by the curves y = x and y = x x (x )dx = x dx = x 3 x3 = 3 ( + 3 ) = = R is the region bounded by the curves y = e x, y = e x and x = ln(). 5

6 3 ln() e x e xdx = e x + e x ln() = e ln() + e ln() (e + e ) = / = / 7. R is the region bounded by the curves y = /x, y = x and y = x/ x x/8dx + /x x/8 = ( x x 6 ) + ( x = = 4 = 9 4 x 6 ) ( ) + ( ( )) 8. R is the region bounded by the curves y = x 3x and the x-axis. 6

7 3 3 (x 3x)dx = 3 3x x dx = 3x x3 3 3 = 7 9 ( ) = R is the region bounded by the curves y = x 3 and y = x x3 xdx + x3 dx = ( x4 4 ) + ( x x4 4 ) = ( ) ( ) + ( ) 4 4 = 4 =. R is the region bounded by the curves y = x 3 4x and y = x + 6x 7

8 8 6 4 x3 4x (x + 6x)dx+ 6 x + 6x (x 3 4x )dx = x3 5x 6xdx + 6 x3 + 5x + 6xdx = ( x4 5 x x ) + ( x4 + 5 x x ) 6 = ( ) ( + 5 3) ( + + ) 4 3 = = = R is the region bounded by the curves y = x 3 4x and y = x x3 4x (x 4)dx + x 4 (x 3 4x)dx = x3 x 4x + 4dx + x3 + x + 4x 4dx = ( x4 x3 4 3 x + 4x) ( x4 + x x 4x) = ( + 4) ( ) + ( ) ( + 4) = = R is the region bounded by the curves y = xe x, y = and x = ln(4). 8

9 8 6 4 ln(4) xe x dx = (ln(4)) (/)e u du = (/)e (ln(4)) (/) 3. R is the triangle with vertices ( 4, ), (, ), and (, 6) The line that connects ( 4, ) and (, 6) is y = x x + 4dx 4 = x + 4x 4 = + 8 (8 6) = 8 4. R is the trapezoid bounded by the lines y = x +, y = 8 x, x =, and the y-axis.. 9

10 x (x + )dx = 6 xdx = 6x x = 4 ( ) = 8 5. R is the region bounded by the curves y = x and x = x ( x)dx = 4 xdx = 4 3 x3/ 4 = / = III. Find the average value of the given function f(x) over the specified interval a x b. f(x) = x 3x + 5 over x.

11 f avg = ( ) x 3x + 5dx = ( x3 3 x x = ( () ( ( ()) = = = = f(x) = e 3x + e x+ over x ln(). f avg = ln() e 3x + e x+ dx ln() = ( ln() 3 e3x e x+ ) ln() = ( ln() 3 e3 ln() e ln()+ ) ( ( ln() 3 e3() e )) 3. f(x) = = eln(3 ) e e ln( ) + e 3 ln() ln() 3 ln() ln() = 8 3 ln() e ln() 3 ln() + e ln() = 7 3 ln() + e ln() x + x + x + 6 over x. u = x + x + 6 du = x + (/)du = x + ( ) = du 5 u = ln u 4 5 = 4 = ln() 4 f avg = x+ dx x +x+6 (ln() ln(5)) IV. After t months on the job a postal clerk can sort Q(t) = 7 4e.5t letters per hour. What is the average rate at which the clerk sorts mail during the first 3 months on the job?

12 Q avg = e.5t dt = 7t 4 3(.5) e.5t 3 = 35(3) 8 3 e.5 (35() 8 = e.5 = e.5 = e The average rate the clerk sorts is 57.7 letter per hour. V. The number of bacteria present in a certain culture after t minutes of an experiment was Q(t) = e.5t. What was the average number of bacteria present during the first 5 minutes of the experiment? Q avg = 5 5 e.5t dt = ( 5.5 e.5t ) 5 = 5 4e.5 5 4e = 5 4(e.5 ) = 7. The average number of bacteria during the first 5 minutes of the experiment is 7..

Estimating the Average Value of a Function

Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and