Slope Fields Sect. 6-1

Calc AB Name ______Sect.______Slope Fields sect. 6-1 x2 1. Find the slopes given by the differential equation y = at each of the y - 2 following points: a. (0,0) b. (1,1) c. (- 2,4) d. (4,- 2) e. (- 3, - 3) f. (5,12)

2. For the differential equation in Problem 1, why are there no slopes when y = 2 ?

x 3. The slope field for y = is shown at right.  y y a. Plot the following points on the slope field: i. (1, 2) ii. (3, 1) iii. (0, 3)  iv. (0,- 2) v. (- 2, - 1) x b. Plot a solution curve through each of the points from Part a. Remember that the curves have to    

be functions.  c. What would a solution curve containing (2, 2) look like? x  d. Solve the differential equation y = . y dy 4. For the differential equation = y y dx a. Use dot paper to draw the slope field ...... m= for the differential equation. . . . 2. . . . m= b. Graph the particular solutions passing ...... m= through the points (0,1) and (0,- 1) . x ...... m= -2 2 c. Solve the differential equation, and find ...... m= the particular solutions that contain the . . . -.2 . . . m= points (0, 1) and (0,- 1) ...... m= dy x 5. For the differential equation = y dx 2 ...... a. Use dot graph paper with [-2, 2] as your interval . . . .2 . . . for both x and y to draw the slope field for the ...... differential equation. x ...... b. Graph the particular solution passing through . .-2 . . . . 2 . (- 1,1) the point . . . . -.2 . . . c. Solve the differential equation, and find the particular ...... solution that contains the point (- 1,1) . 1 6. Repeat the three parts of Problem 5 for the differential equation y = . For this 2y problem, draw your graph and write your solution as a function of x.

7. Repeat the three parts of Problem 5 for the differential equation y y2 =1.

dy 8. Repeat the three parts of Problem 5 for the differential equation =2x ( y - 1) , dx but use the origin (instead of (- 1,1) ) for Parts b. and c.

 y 9. The slope field for a certain differential equation is

shown at right. Which of the following could be a  specific solution to that differential equation. x

1     a. x3 + y = 2 b. x2+ y 2 = 4 3  4 y c. x2- y 2 = 4 d. y = e. = 4 x x 

10. Which of the differential equations below  y matches the slope field shown at right?  dy dy a. =x - y b. =y - x x dx dx     dy x dy y dy c. = d. = e. = xy dx y dx x dx 

 11. The slope field for a certain differential equation is shown at the right. Which of the following could be a specific solution to the differential equation? y a. y= ex b. y= e-x c. y= - e x  d. y= - ln x e. y= ln x 

 