6.3 Be able to calculate the length of the curve

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=Ce ^{kt}`. Typically in problems you are
given one data point which you can use to find

The data point is often given in terms of the half-life or the doubling time: the

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 `e ^{x+iy}
= e^{x}(cos y + isin y)`.
Be able to convert back and forth between the

November 7, 2000