6.4 Understand that the average value on the interval [a,b] of a function f(x) is the height of a rectangle with base [a,b] and area equal to the integral of f from a to b. (Example 1, Exercises 1, 2)
6.5 This test will only cover WORK. A constant force F displacing its point of application a distance L does work W=FL. Be able to set up and solve problems which apply calculus to situations where the force varies with distance. The simplest involve stretching a spring, where F(x)=kx (Example 2, Exercises 3, 4, 5, 6); you integrate dW=F(x)dx. Example 1 is analogous, as are Exercises 1, 2. More complicated problems involve slicing: typically emptying out tanks of various shapes by pumping the liquid over the top. Choose a vertical coordinate x and let dW be the work done in lifting to the top the "slice" of liquid lying between x and x+dx, assuming for the calculation that this liquid is all at height x. In this case the force is the weight of the slice: you need to calculate its volume (area times dx) and multiply that by its density: the force will involve x AND dx, while the displacement is the distance from x to the top (Example 3, Exercises 11, 12, 13, 14). A similar analysis is necessary for "rope" problems (Exercises 7, 8, 9) and their variants (Exercise 11). Check also review problems 15-18 on p. 498.
7.2 Understand that a first-order equation y'=f(x,y) corresponds to a slope field or "direction field" in the x,y plane (Exercises 3-6). And that solutions to the equation correspond to curves that are tangent to the slope field at each of their points (Figures 3 and 4 on p. 510). Given a slope field and an initial condition, be able to sketch the corresponding solution (Example 1, Exercises 1, 2(a), 7, 8).
7.3 Understand Euler's method and be able to apply it by hand with a small number of steps (Exercise 1(a), 2, 3, 4). You may use your calculator's Euler program to check your work, but you will have to be able to show all the steps explicitly.
7.4 Be able to solve a separable equation y'=f(x,y) by separating x and y, integrating separately, and solving the resulting equation for y. Be able to evaluate the constant of integration from an initial condition (Examples 1, 2, 3, Exercises 1-12, 13, 14).
7.5 Understand that a general exponential function involves
two constants: the C and k in
y=Cekt. Typically in problems you are
given one data point which you can use to find k.
Then you are given C and t and asked to
find y (Exercise 2(c), 3(b), 8(b), 9(b)),
or you are given C and y
and asked to find t (Exercise 2(d), 3(c), 4(d),11).
The data point is often given in terms of the half-life or the doubling time: the t such that y(t)=.5y(0) or y(t)=2y(0). Know how to use this piece of information to find k (Exercises 2(a), 8(a), 9(a)). Be able to calculate the half-life or doubling time from k (Exercises 4(c), 10, 18). Be able to use other data points (Exercises 3, 4, 10).
7.6 Understand the role of the growth constant k and the carrying capacity K in the logistic equation. Be able to use Euler's method to find approximate solutions given k, K and an initial condition (Example 2, Exercises 1, 2(abc), 4, 5.
Appendix H Be able to do basic arithmetic with complex
numbers: addition, subtraction, multiplication, division.
Understand the complex exponential ex+iy
= ex(cos y + isin y).
Be able to convert back and forth between the x+iy
and the reit forms for
representing a complex number, and understand how they are
related to cartesian and polar coordinates in the plane.
Understand multiplication and division geometrically in terms
of the reit form.
November 7, 2000