The given slope field is [math]\frac{dy}{dx}=y[/math][br]y(0)=1
Go to Tools --> More --> Others. Then use the pen to sketch the graph of y at the particular solution (0,1)[br]Make your graph color blue.
Fine the equation of the tangent line at y(0)=1
Graph the equation (Algebra --> Write your equation). If your line is not tangent to your sketch then go back and edit your sketch.[br][br]Use linearization to approximate y(2) . Do you think your answer is accurate ? Why not ?
Go to your line and make the interval [0,0.5] [br]Steps:[br]1) Delete your first graph of the tangent line[br]2) Paste this command in "Algebra": if(0<x<0.5, mx+b)[br]3) Put your equation instead of mx+b [br]4) Increase the thickness to make your graph more visible
Find the equation of the line at x=0.5 using your previous equation and the differential equation above? Then graph it over the interval [0.5,1] using the same steps in Task 4.[br][br]What do you notice about the two lines you graphed?
Find the equation of the tangent line at x=1 ? Then graph it over the interval [1,1.5][br][br]Again, what do you notice?
Find the equation of the tangent line at x=1.5 ? Then graph it over the interval [1.5,2][br][br]Again, what do you notice?
Now find y(2) using the equation you found in Task 7,[br][br]Do you think this value for y(2) is a better approximation than your original approximation in Task 3? Why or Why not?
We just used the Euler Method. After completing this activity, what do you think it does?