1. Consider the differential equation

 

                 

 

Use EulerÕs method with  to approximate y(2).

 

 

2. Consider the definite integral

 

                 

 

a. Use the midpoint rule with n=2 subintervals to approximate this integral with a Riemann sum.

 

b. Define a Òleft point ruleÓ for Riemann sums to be an approximation to the area under the curve using areas of rectangles whose heights are taken to be the value of the integrand at the left endpoint of each subinterval. Use the left point rule with n=2 rectangles to approximate this integral as a Riemann sum.

 

c. Sketch a graph of  for  showing the rectangles used for the midpoint rule and the rectangles used for the left point rule.

 

 

3. Describe the relationship between (A) solving the differential equation

 

                 

 

using EulerÕs method with a given value of h to approximate  and (B) approximating the integral

 

                 

 

using a left point rule with n subintervals. How should h and n be related to give the same approximation?