Euler's Method
Suppose that we wish to estimate the function Y at some value of t and the
only information concerning Y is the rate of change of Y with respect to t and
an initial value of Y at some value of t. Suppose it is known that Y'(t) = 2t
+ 1 and Y(0) = 1. Pictured below is the graph the function Y.
If you move the cursor over the graph, you will see the Y and the graph of
the line tangent Y at t = 0. The line is graphed over the interval t = 0 to
t = 0.5
Student Questions...
3. What is Y'(0)? That is, what is the slope of the line tangent to the function
Y?
4. Using Y'(0), what is 
y1?
5. Using y1,
what is y1?
6. What is the first estimate of Y at t = .5?
7. Is the value of y1 an over estimate of Y or an underestimate
of Y at t = .5?
8. What is the error? Your answer should be in terms of Y and y1.