MATH 211, Spring 1999, Section C

Review for Exam I, Friday, 2/5/99

Note: You must bring a valid picture I.D. to the exam.

The exam will cover Sections 1.1 -- 1.4 of Chapter 1 and all of Chapter 0. Sections 1.3 and 1.4 are the most important and will have the most emphasis on the test. In studying you should go back over the textbook carefully, review all of the homework problems, then go over your quizzes (solutions are posted on the course WEB page). There are also some sample tests on the internet, but you must be careful in using them, since they include Sections 1.6 and 1.7. As the last step in studying you can go over the list of sample test questions. Most test questions will be similar to homework problems in the text, so be sure you understand how to solve all of the problems on the list of assigned homework from the textbook. You might also be asked to state a definition or formula, such as the definition of the derivative, the quadratic formula, or the power rule for derivatives.

Chapter 1

Section 1.1: The point-slope form of a line,
y = m (x - x1) + y1,
gives the equation of the line through the point (x1,y1) with slope m. The slope-intercept form of a line,
y = mx + b,
is a special case of the first form, and gives the equation of the line through the point (0,b) with slope m. The number b is called the y-intercept of the line. To find the equation of the line through two given points, first find the slope, and then use the point-slope form.

Section 1.2: This is just a preview of the next two sections, but it is still important to read it again.

Section 1.3: For the function f(x), the slope of the graph of y = f(x) at the point (x,y) is given by the derivative f'(x). This slope is the same as the slope of the line tangent to the graph at (x,y).
The simplest case is that of a straight line f(x) = mx + b, since then the slope is f'(x) = m, for all possible values of x. The power rule gives the derivative in some more complicated cases:

If f(x) = xr, then f'(x) = rxr-1.

The notation dy/dx is also used for the derivative of y = f(x). Using this notation, the power rule is
d/dx (xr) = rxr-1.
Finally, in this section study pages 82--84 very carefully.

Section 1.4: You must know the limit definition of a derivative, and how to use it:

              f(x+h) - f(x)
f'(x) = lim   -------------
        h->0        h
In the text, the authors usually give the definition in the special case when x = a.
You should know the rules for finding limits (given on page 90). In particular, to find the limit as x -> a of a polynomial, just substitute a. This works for a rational function too, provided substituting a into the denominator does not lead to division by zero. Study Examples 4--6 very carefully. Finally, finding the limit as x approaches infinity of the rational function f(x)/g(x) is the same as finding the horizontal asymptote (if there is one).

Chapter 0

You need to know how to graph some basic functions by hand, since you are not allowed to use a calculator. Some important functions:
linear functions (see Section 1.1 as well as Sections 0.1 and 0.2);
quadratic functions y = ax2 + bx +c;
f(x) = x3 (see page 8);
f(x) = 1/x (see page 9);
the absolute value function (see page 23);
piecewise-defined functions (see Example 4 on page 20).
You need to know how to find sums, products, and compositions of functions (Section 0.3). In Section 0.4, be sure you have memorized the quadratic formula. Sections 0.4 and 0.5 From section 0.6 you should know the formulas for some areas an volumes:
the area of a rectangle is its length times its width;
the area of a triangle is one half its base times its height;
the area of a circle is pi times the radius squared;
the volume of a rectangular box is its length times its width times its height;
the volume of a cylinder is the area of its base times its height
(this formula applies to a box just as well as to a circular cylinder).

Sample test questions:

p.30 #15, 17, 27, 29;
p.39 #33;
p.56 #13, 25;
p.70 #17, 21, 39;
p.85 #3, 7, 14, 15, 35, 36, 39, 47, 56;
p.96 #15, 23, 25, 33, 52

Answers:
p.85 #14 f'(x)=-1/3x-4/3;
p.85 #36 the tangent line is y = 1/10(x-25) + 5
p.96 #52 the limit is 1