Week 6 topics for
Calculus 1550, (5), Fall 2004.
http://www.math.lsu.edu/~verrill/teaching/calculus1550/week6.html
Topics for the sixth week, September 27 – October 1 2004:
Tangents and the meaning of a derivative.
- Monday: Tangents and secants. (sections 2.1, 2.7)
- Tuesday: Derivatives. (section 2.8)
- Wednesday: Graph of derivatives. (section 2.9)
- Thursday: Derivatives of basic functions. (section 3.1, 3.4)
- Friday: Rules for finding derivatives. (section 3.2)
This week we will have several short graph sketching quizzes
for bonus points,
and 3
homeworks.
Next week Tuesday there will be a test. Please learn everything in
the above picture, and the
following rules:
|
Derivatives of basic functions: | |
Derivatives of combinations of basic functions:
|
|---|
| d(xn)/dx = nxn – 1 | | (f+g)' = f' + g' |
| d(exp(x))/dx = exp(x) | | (f-g)' = f' – g' |
| d(sin(x))/dx = cos(x) | | (fg)' = (f')g + f(g') |
| d(cos(x))/dx = – sin(x) | | (f/g)' = f'/g – f(g')/g2 |
See last year's test at:
http://www.math.lsu.edu/~verrill/teaching/calculus1550/Spring2004/test2.pdf.
Day 23, Monday 27 September:
Tangents.
We reminded ourselves of the definition of a tangent, and of how to
find the equation for a line given its slope and a point on the line.
See Section 2.1.
We discussed one more example of horizontal asymptotes,
excerise 55, page 148.
There was a quiz on graph sketching, with solution here.
We finished up by talking about how to find average speeds over
short time intervals, from some given data.
What we discussed was similar to
section 2.1 example 2, and most of
the exericise at the end of section 2.1, where there is data, but no
formula.
Day 24, Tuesday 28 September:
Tangents, secants, rates of change and the derivative.
We looked at some data giving positions of a cat at time t; this is
similar to the homework on webworks, and similar to
section 2.1 example 2.
The slope of the secant gives the average rate of change over a time
interval.
For some pictures and animations of secants, look at:
The tangent is the limit of the secants. The slope of the
tangent is the limit of the slopes of the secants. The slope of
the tangent is the instantaneous rate of change.
The slope of the tangent is called the derivative.
What we talked about corresponds to
section 2.1 and
section 2.7.
You should read these sections to make sure you have followed
the ideas presented in class.
We mentioned the formula for the definition of a derivative, on
page 158, definition 2, section 2.8.
We will cover this in more detail tomorrow.
There was a quiz on graph sketching, with solution here.
Graph sketching is a very important part of mathematics of functions,
and is a basic requirement for doing calculus. Make sure you get
enough practice so that this all seems easy (and it will do if you do it
often enough; you just need to build up those graph sketching muscles).
Here are some useful links to help you with graph sketching:
- You need graph paper. You can print out grids from
Mathematics help central graph paper page.
Or get graph paper from
the free graph paper page.
-
For the basics of graph sketching you should already know, check out
Purplemath
Graphing Overview. They have links to several more graph sketching
pages if you find the first useful.
- For sketching quadratics, try the
intermedate algebra pages
of M. Bittinger, which has a
worksheet on graphing quadratics,
with
solutions here.
(this was found at K Schulte's
Math Worksheets resource page
, which you can look through for other worksheets on many topics.)
-
You can try the worksheets, with explanations and solutions for
curve sketching, from Hull University, (mathematics worksheet 17).
- For more practice, try working through J. Abbott's
examples of curve sketching. We've only covered (a) and (b) of the
28 questions he asks, but they all come with solutions, so they are
good for making sure you understand intercepts and asymptotes.
In a few weeks we'll get to the rest of the details about curve sketching.
Day 25, Wednesday 29 September:
Graphs of the derivative.
What did we do?
-
We recalled the definition of the derivative, as
given in
page 158, definition 2, section 2.8.
-
We sketched some graphs of derivatives, as in
section 2.9, example 1, page 165 and 166.
-
We looked at the case of |x|, as in
example 6, page 170 and 171, section 2.9.
-
We also saw one of the demonstrations of secants and tangents
on the
disc that comes with the text book.
- We looked at the motion of a pendulum, which is roughly like
cos(t), and saw that the derivative is -sin(t) (except both of these
graphs need stretching to have the right period and right max and min
values.) This is covered in
section 3.4, especially page 211 and 214.
Day 26, Tursday 30 September:
More on graphs of the derivative.
What did we do?
- Different graphs through the same set of three points (0,0),
(1,15), (2,24) and (3,77). We draw secants with slope 9 and 53.
Draw your own graphs of a function
f(x) passing through these points, and ask whether you drew a graph with
- f'(2) = 53?
- f'(2) < 53?
- f'(2) > 53?
- f'(2) = 9?
- f'(2) < 9?
- f'(2) > 9?
- f'(2) < 0?
- f'(2) infinite?
- f'(2) = 0?
- Any graph where the rate of change is zero for an interval?
- If f(t) is a distance, and t a time, what do these all mean in terms
of the instanteneous velocity at t=2, or at other times?
Try and draw graphs for each of these properties.
- Matching graphs with the graphs of the derivatives.
For practice, try:
You can find the answers for these by going to the page for
1550 (20) Fall 2003, and
1550 (6) Spring 2004.
- We wrote down the derivatives of cos(x), sin(x), exp(x) and xn.
You can find all these formula in sections
3.1 and 3.4.
Day 27, Friday 1 October
Formulae and rules for finding derivatives.
We'll fill in the details of the
table given near the top of this page.
These rules can be found in
section 3.2, box at top of page 197,
and for the sin and cosine, see
section 3.4, box in middle of page 214.
We also talked about the function A(r) of the area of a circle
in terms of its radius. This is related to
question 13 page 123, and the discussion in
section 3.11.
For practice, work through
For animations designed to help you learn the product and
quotient rule, have a look at: Calculus help's
tutorials for the calculus phobe.
Quiz solutions:
Last updated August 28 2004 by
Helena Verrill