Calc 1A
Assignment Three
We now have our preliminaries out
of the way.
The heart of calculus (at this
level) is the introduction of the concept of the derivative. Until now, most of
the mathematics you have dealt with regarding changing quantities has been
limited to either functions which are linear or quadratic OR have been
approximations of more complicated functions by use of linear functions. While
we will continue that latter behavior at times, we will now be given the tools
to study functions which are changing at a variable rate (ie: nonlinearly) by
defining an instantaneous rate of change. We will accomplish this by taking a
limit of a difference quotient, which in plain English means we will be
imagining a linear approximation over a smaller and smaller interval near a
point.
Much notation will be introduced.
It will be crucial that we get clear on its meaning fast because the concepts
will be thrown at us left and right and the notation is the best means we have
for quickly communicating all that is going on.
We will study functions which
describe other functions. A derivative function f' (read "f prime"),
in effect, will provide us with the instantaneous rate of change of f at any
point. By studying the derivative function we can learn about the function
whose derivative it is. (The derivative function is derived from the original).
As we have seen before when we
described functions as continuous at a point, or simply continuous, we can now
describe functions as differentiable at a point, or differentiable. Many
similarities and patterns will be noted from our earlier work. Among these will
be formal rules of derivatives, which will closely mirror the formal rules
we've seen for limits and continuity. There will however be some profound
differences particularly when we get to products and quotients.
Initially our work will be
algebraic and challenging (although when you are good at taking a limit
derivative, 'challenging' may graduate to 'tedious') but then we will learn
simple ways to do that which was long. Once we are good at taking derivatives
quickly, we will be ready, in our next Assignment to USE THEM. (Can't wait,
can't wait.....). We will build a large repertoire of functions which we can
study the derivatives of.
Finally, we will discuss local
linear approximation, which merely means we will use tangent lines to
approximate functions near a point and we will learn IMPLICIT differentiation
which is a technique for taking derivatives of implicitly defined functions.
This technique becomes necessary when we can't explicitly write a function in
y=f(x) form.
If all goes according to plan, we
will be following the daily schedule below:
|
Day Number |
Class Topic |
Homework-Reading Class notes and Examples is part of EVERY NIGHT'S homework. |
|
One |
Bill Out West: Average rate of change over an interval. Definition of Derivative as instantaneous rate of change. |
Pg 175 #1-15 odd (omit 7). |
|
Two |
Notational fun and finding derivatives. Writing equations of tangent lines. |
Pg 187 # 7,9,13,17,21,29; Pg 175
#7, 17 QUIZ, DAY AFTER TOMORROW |
|
Three |
Graphs of derivative functions and what they tell us about their parent functions. |
Pg 187 #3,5,19 (see ex. 4 pg 180), 25,27,31,33 (use 'help' from 'Bill out West', 35. QUIZ, DAY AFTER TOMORROW |
|
Four |
Definition of differentiability. |
Pg 187 #23,29,37,40,43, read ex. 5 on pg 182 and then calculate nderiv of absval at 0, QUIZ TOMORROW |
|
Five |
QUIZ covering days 1, 2,
and 3 (max 15 minutes). Theorem: Differentiability implies continuity. The operator d/dx. |
Pg. 189 #41 (use def.2 of derivative and show limit DNE. Go back to cty theorem with roots to evaluate limit), 43 (use limits). Read section 3.2 with care. Pg 199 #79. |
|
Six |
Formal properties of derivatives. Finding critical values of a derivative function. Using the 'Power rule'. |
Pg 197 # 1-11 odd, 29,31, 29, 51, 53. |
|
Seven |
Product and Quotient rules. |
Pg. 197 #13, 15 (Don't multiply out) , 17 (same comment), 19, 23,25,27,59, 75 (see instructions on bottom of preceding column) QUIZ, DAY AFTER TOMORROW |
|
Eight |
Practice. Higher derivatives and notation. |
Pg 199 #77, 41,43,45,69. QUIZ
TOMORROW.
|
|
Nine |
QUIZ covering days 3
through 6 (max 15 minutes). Derivatives of trig functions. |
Pg 199#47, 37, 83; Pg 202
#1-13 odd, 27,29
|
|
Ten |
Chain rule. |
Pg 202 #41, 37, 35, 19, 21;
Pg 208 #1, 3, 5 (write with negative exponent),7,9,11,25, 45
|
|
Eleven |
More chain rule.
|
Pg 208 # 13,17,19,27, 41. |
|
Twelve |
Differentials and Local linear approximation. |
Pg 210# 61,63; Pg 217 #1,3,5,27; QUIZ, DAY AFTER TOMORROW |
|
Thirteen |
Implicit Differentiation |
Pg 217 #11,29, Pg 253 # 11,13,19 . Exponential /Log sheet (Not collecting but a lesson's understanding will depend on your doing it) QUIZ, TOMORROW |
|
Fourteen |
QUIZ (max 10 minutes)
covering days 7-12. More Implicit Differentiation.
Intro to Log differentiation.
|
Pg 253#23,27,29, Pg
260#35,45. Begin organizing your study notes for exam.
|
|
Fifteen |
Derivatives of Log and
Exponential Functions.
|
Pg 260 #1,3,7,11,19,31, 49,
59 . Read Page 231 middle: "Increasing or Decreasing....", Read
middle of pg 232 through ex. 9; Pg 234 #11
|
|
Sixteen |
.Derivatives of Inverse
Functions including Trig inverses
|
Pg 268 # 21,25,29, Sheet of inverses. Pg 260 # 5, 9, 13, 15, 17, 23, 25, 27 |
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Exam |
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