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Monday 3/29
Today we talked about slope fields which are graphical representations of a differential equation. The first questions asked of us are shown in the graphic below.

y = .5x^2 + 1
 * 1) 4 If the initial condition is y = 1 when x = 0 then to value of c would be zero, The solution to the differential equation would then be

Last week we solved differential equations algebraically by separating variables and integrating both sides. Unfortunately, it is not always possible to solve algebraically. In this case, a graphical approach is an option.

Consider the differential equation This implies that the slope of tangent lines to the solution curve at any point (x,y) is equal to the opposite of x/y

At (1,1) the slope of the tangent line will be -1; at (2,2) the slope of the tangent line will be -1; in fact, the slope of the tangent line will be -1 at all points where the x and y coordinates are the opposite of each other.

If the x and y coordinates are opposites, e.g. (1, -1), (-2,2) ..., the slope of the tangent lines will be 1.

All points on the x axis have a y coordinate equal to zero. Using the formula for dy/dx, these tangent lines will have undefined slopes implying that the tangent lines will be vertical. [Note: the origin is not included here because the slope of the tangent line will be of indeterminate form.]

All points on the y axis have x coordinate equal to zero. Using the formula for dy/dx, these tangent lines will have zero slope implying that the tangent lines will be horizontal. [Note: the origin is not included here because the slope of the tangent line will be of indeterminate form.].

Very small tangent lines, with appropriate slopes, can be sketched on a grid as shown below.

This is called a SLOPE FIELD.

The slope field suggests that the general solution of the differential equation may be one of a family of concentric circles. Which circle is the particular solution required will depend on the initial condition given.

In the graphic below, the curve sketched in red is for the initial condition when x = 1, y =1. The curve in black is for the initial condition y = 3 when x = 0.



The problem we did together in class was to sketch the slope field for the differential equation dy/dx = x and this is shown below.

Note:

The slopes of the tangent lines for all points with x coordinate equal to one are all 1.

The slopes of the tangent lines for all points with y coordinate 1 increase as x increases.

The slope field is symmetric to the y axis.

After this introduction to the idea of a slope field, we turned our attention to the exploration connected to our textbook. The questions and answers will appear shortly.

Tuesday 3/30
Exploration 7.4a) and b) can be seen below.

Wednesday 3/31
Today and Thursday we worked on some AP Problems which combined graphical interpretations of differential equations (slope fields) and the algebraic solutions of 1st order, variable separable differential equations.

Friday 4/1
Today we worked some multiple choice problems from the D&S books. SInce these books canot be taken out of Ms. Gentry's room, we are not going to post the problems nor the answers. Tutorial is the option if you want help with any of these!