MATH 200X
Review for MIDTERM II
The Basics: Midterm II will be Friday 26 October. It will cover Chapter
3 sections 1-7,9-10. The exam will be closed-book and closed-note. No
calculators are allowed.
A Review of Topics:
section 3.1: Derivative of Polynomials and Exponential Functions
This is where we are first introduced to the "easy and fast" ways of
taking derivatives. There are six crucial derivative rules in this
section. The derivative of a constant function, the power rule,
constant multiple rule, sum rule difference rule, the derivative
of ex. Implicit in this discussion is how to take
derivatives of polynomials. You do not need to memorize the definition
of e.
section 3.2: The Product and Quotient Rules
We learned the product rule and the quotient rule. All rules learned so
far are in a box at the end of this section. You should know all of
these.
section 3.3: The Derivatives of Trigonometric Functions
We learned the derivatives of all six trigonometric functions.
You should know all these. They are listed on page 193. You do not need
to know how to derive them. In particular, you do not need to know the
proof that the derivative of sin(x) is cos(x).
section 3.4: The Chain Rule
We learned the chain rule here. You should be able to use the chain
rule with the power rule, exponential function, and trig functions. We
also learned the derivative of y=a^x for bases, a, other than e.
section 3.5: Implicit Differentiation
We learned how to take the derivative of an expression implicitly. You
should know how and WHEN to do this. And when not to. In this section
we used this method to find the derivatives of the inverse
trigonometric functions. These rules are listed at the end of this
section. You do not need to memorize these. But you should know how and
WHEN to use them. The rules themselves will be listed on the last page
of your exam.
section 3.6: Derivatives of Logarithmic Functions
We learned how to take the derivative of y=loga(x) for a
general a. Also, we learned how to use the rule of the derivative of
ln(x) along with the chain rule. Finally, we put implicit
differentiation together with the rules in this section to develop a
technique called Logarithmic Differentiation. You want to know how and
when to use this technique.
section 3.7: Rates of Change in the Natural and Social Sciences
We see position, velocity, and acceleration problems reappear. We
see general applications of the derivative. The principle idea being
applied here
is that the derivative dy/dx always represents rate of change of y with
respect to x. And this principle applies no matter what variables are
being used.
section 3.9: Related Rate Problems
These problems are always asking you to find a rate of change (a
derivative with respect to time of some quantity). Solving these
problems always requires you to take the derivative of some expression
implicitly with respect to time. On an exam, you will get partial
credit for preceeding in some methodical, organized, principled way
even if you cannot solve the problem.
section 3.10: Linear Approximations and Differentials
You should know what is meant by the linearization
of f(x) at x=a, the differential
of f(x), and how (and what) to approximate using them. We also talked
about error, relative error, and percent error.
Several Notes:
1) In general you will not be asked specifically to simplify your
answer.
If you do simplify, you should do so correctly. If you do not simplify,
you MUST paranthesize correctly. You will be counted off for amiguous
or missing parantheses.
Example: y=sin x 2 . Is this y=sin (x 2) or
y=(sin x )2?
Example: y=3x*x+1. As it is
written, this means y=(3x*x)+1 as opposed to y=3x(x+1).
2) You will be expected to clearly identify what you consider to be
your answer. For example, you can circle or box it. If you write
multiple answers and do not make clear which one you want graded, you
will lose points.
3) In addition to the rule of differentiation we learned in this
chapter, I will assume you know the basic geometric and trigonometric
formulas we have used such as: Pythagorean Theorem; area and
perimeter(or circumference) of triangles, circles, rectangles; volume
and surface area of spheres and rectangular solids; volume of cylinder;
similar triangles; trig functions defined as sides of a right
triangle (ex: sin(x)= opp/hyp, etc).
Suggest Problems (starting on page 262)
#1-49,51,53,57,61,65,69,71-75,83,85,87,89,93,97,99,103,105,