Monday Oct 16

Topics: 3.4, 3.5

Solve: Sec. 3.4 problems 5 , 9, 11
           Sec. 3.5  problems 1, 8, 23
Solve, write up and turn in: Sec. 3.4  problems  15,  20, 22, 27
                                                Sec. 3.5  problems  30, 32, 34, 35

Bonus problem. (Friday's bonus problem was a no brainer. Here is what it should have said.)
                          Show (for extra 5 points!) that
                          if a function f(x) is continuous at a and the left limit  x->a-  of  f ' (x) exists, and the right limit  x->a+  of  f '(x)  exists
                          and
the limits are equal,
                          then the function is differentiable at a.


Tuesday Oct 17

Topics: 4.1 (part), A.3 (part)

Solve, write up and turn in: Sec. 4.1- 18, 22, 23, 24, 25, 27, 29

Problems 18, 22 need (for easier solutions) the use of intervals of increase/decrease or the second derivative test. Solve them after we discuss these issues.

 

FYI - solving polynomials-web resources: Here is how  The cubic formula looks  like.  This web article  is very nice, please read it.

                         If after reading the article above you are still curious about the formula for solving polynomials of fourth degree,
                                              you can check this website:
  Solving Quartic Equation

                         Finally, for polynomials degree 5 and higher there are no general formulas for roots which involve radicals only.   

Wednesday Oct 18
Topics:  A.3, 4.1
Solve, write up and turn in: Sec. 4.1- Now it is a good time to solve 18, 22 (see below also)
                                            1. In problem 18 it is easy to find the zeroes of the derivative.
                                               The derivative does not change the sign between its zeroes. Explain why!
                                            2. Find the intervals of increase, decrease, local maxima, minima and graph the function
                                                y=x(x-1)^(1/3)  Caution: you have a radical, therefore there is a point where the derivative does not exist.
                                                                                                     Show on the graph what happens there.
                                                      
Sec. 4.1: 7, 9, 11, 13, 30

Thursday Oct 19
Topics: 
3.6, 4.2,
Review trigonometry:
                         Web resource:  Referring angles to an acute one(preferred)                                                                                                               
                            and another one:   Trig functions for any angle
                            see also Sec. 1.7
                                                               
Solve: Sec. 1.7  3, 7
           Find the period of sin(3x+4), and of tan(5x).

<>Solve, write up and turn in:
                                            Sec. 4.2
1, 7, 9, 10, 15, 18, 21, 27
                                               
                                            Sec. 3.6  3a, 5

                                            3. Show that if a function is concave up, then the graph is above its tangent.
                                                Hint: the y on the graph is y=f(x). The y on the tangent at (x0,y0=f(x0)) is y=f(x0)+f'(x0)(x-x0).
                                                         So you need to show that f(x)>f(x0)+f'(x0)(x-x0) for all x.
                                                         Define the function g(x)=f(x)-f(x0)-f'(x0)(x-x0) and show that this function g(x) has a minimum at x=x0.
                                                         (How? Use the derivative g'(x), of course.)

Midterm exams are during the regular class hour and room.
Review guide for Exam 1

Friday Oct 20
Topics: 
2.4 velocity, acceleration, 4.3

Read Example 3 Sec. 4.3

Solve, write-up and turn in:  Sec. 2.4:  9
                                              Sec. 4.3: 1, 3, 5, 11, 28, 31


Information for Exam 1:


Review guide for Exam 1

A few review problems:

Use the epsilon-delta definition of the limit to show that the limit of (3x-1), as x goes to 1, is 2.

Additional problems for Chapter 2 (Page 81):

1, 3.

Use the definition of the derivative in 6a,b.

Where is the function continuous, where is it differentiable: 7b.

13, 15

19, 25, 29, 41

Additional problems for Chapter 3 (Page 111):

1, 5, 7, 33b,g,o, 38, 40 a

The Homework problems in Chapter 4.

Show that if f(x) is continuous on [0,5] and f(0)=3, f(5)=4, then there is a c between 0 and 5 so that f(c)=pi.