Quick Review 2012 Pearson Education, Inc.

Quick Review Solutions 2012 Pearson Education, Inc.

First, a little review: Consider: or then: It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given: find We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

If we have some more information we can find C. Given: and when , find the equation for . This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.

We’ve plotted points on graphs before… • • Plot the following points on the empty grid at the top of your worksheet. • • • 1 1 This is an introduction to the first problem on the worksheet. 2 4 –1 1 –2 4

• • Now, instead of plotting points on a graph, try plotting slopes. Plot the slopes of y on top of the points you just plotted. How would you do that? • • • 1 1 2 This is an introduction to the first problem on the worksheet. If students are stuck, among the questions I will ask them are: What kind of graph do you think of when you hear the word slope? At the point (x, y) what is the general direction of the graph? About how big should the line segment or arrow be to indicate the direction of the graph at that point? 2 4 4 –1 1 –2 –2 4 –4

Suppose all that you knew was y Suppose all that you knew was y. You don’t know what y is but you do know it’s derivative. Use what we just did to plot the slope of y in the following problem that is on your worksheets…

Plug 0 into y for x 1 Plug 1 into y for x 2 3 1 2 1 1 2 2 4 -1 -2 -2 Draw a segment with slope of 2. 1 Plug 1 into y for x 2 3 1 2 Draw a segment with slope of 0. 1 1 2 This is the first problem on the worksheet. Draw a segment with slope of 4. 2 4 -1 -2 -2 -4

If you know an initial condition, such as (1,2), you can sketch the curve. By following the slopes, you get a rough picture of what the curve looks like. In this case, it is a parabola. How could we have seen this coming? A parabola is exactly what you would get when you integrate

What would the graph of these slopes look like? Plug 0 in for x and y Plug 1 in for y and 0 for x 1 Plug 1 in for y and 1 for x Slope = 0 Slope = 2 Slope = 0 1 1 2 What would the graph of these slopes look like? 1 2 4 2 …and so on… 2 1 4 Revisit 2xy when doing separation of variables 2 2 8 -1 -1 1 -2

Sketch an approximate curve for y given the initial value (0,1). 1 • 1 1 2 1 2 4 Now sketch an approximate curve for y given the initial value (–1,–1). 2 • 2 1 4 Add a solution with initial value (-1, -1) 2 2 8 -1 -1 1 -2

Problems that begin with the derivative are called differential equations. When you are given an initial point, they are called initial value problems. Plotting these slopes gives you what is called a slope field. Match the slope field with the function that you think it is modeling. (#4 on worksheet) A B

Problems that begin with the derivative are called differential equations. When you are given an initial point, they are called initial value problems. Plotting these slopes gives you what is called a slope field. Match the slope field with the function that you think it is modeling. (#4 on worksheet) A B

Problems that begin with the derivative are called differential equations. When you are given an initial point, they are called initial value problems. Plotting these slopes gives you what is called a slope field. Match the slope field with the function that you think it is modeling. B (#4 on worksheet) A A B

Notice that the curves for each function follow the slopes like a boat following a river current. This is page 1 of the worksheet. After the review conducted in the previous slides, we begin the new topic here. The purpose of this slide is to give students the basic idea of what a slope field is.

Initial value problems, differential equations, and slope fields are often used to solve problems where only the rate of change (the derivative) is known. In advanced math, physics, and engineering classes, slope fields are also called direction fields or vector fields. These problems are so common and have so many applications that there are entire sequences of college courses dedicated only to different types of differential equations. Much more to come on this topic in the days to come… p

Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!

Integrals such as are called indefinite integrals because we can not find a definite value for the answer. When finding indefinite integrals, we always include the “plus C”.