Write-ups of the problems below are due Monday Oct. 16

Monday Oct 9

Topics: A.3: Theorem 4 (no proof),  Theorem 3 (deduced from Theorem 4), and 5 (with proof).

Solve, write up and turn in: Sec. 2.6 (p. 80) 30, 34, 36,37,39 and
Calculate the following limits as x->0:
and
Tuesday Oct 10
Topics: Differentiability and the definition of the derivative Sec.2.3
Solve, write up and turn in: Note: in the first limit above take either x->0+  or  x->1.
and explain why the problem as given is incorrectly formulated.
Sec. 2.3 (p. 62) Use the definition of the derivative to calculate
the derivatives in the following 4, 7, 11, 24, 32, 38
1. Show that the function f(x)=|x| is not differentiable at x=0.
2. Is the function g(x)=x|x| differentiable at x=0?

Wednesday Oct 11
Topics: tangent line Sec. 2.2, Calculation of derivatives 3.1
Solve: Sec. 2.2 (p.57) 1, 11
Solve, write up and turn in: Sec. 2.2 (p.57) 2, 7, 8 (Note that the derivatives need to be calculated using the definition,
exccept for c,x,x^2,...x^n)
and
3. Deduce the formula  a^n-b^n = (a-b) (a^(n-1) + a^(n-2)b +...+ ab^(n-2) + b^(n-1))
by replacing r=b/a in the formula
1-r^n = (1-r) (1+ r +...+ r^(n-2) + r^(n-1))
4. Prove that the derivative of x^n is nx^(n-1)  Hint: use the above factorization.

Thursday Oct 12
Topics: 3.1, 3.2

Solve: Sec. 3.1, 3.2  solve as many problems as needed to ensure you master the calculation of derivatives.
Here are a few suggestions. Sec. 3.1 - 1a,b,f , 7, 9
Sec. 3.2 - 1, 8, 9, 11, 13, 23, 29, 31

Solve, write up and turn in: Sec. 3.1 (p.87) 6, 10, 11, 14, 15, 17, 22, 23
Sec. 3.2 (p. 92) 38, 44, 46

If you erred on the first problem of the quiz on Wednesday:
practice the trick of amplifying the fraction with the "conjugate" sum of radical in the following problem.
Use the definition of the derivative to calculate the derivatives of the functions
f(x)=square root of x (see the solution in Sec. 2.3 Example 3, p. 60),  g(x)= square root of (x+2).

Friday Oct 13
Topics: 3.3
Solve: Sec. 3.3  solve as many problems as needed to ensure you master the calculation of derivatives.
Here are a few suggestions. Sec. 3.3- 1, 3, 5,7, 11, 13, 33, 35

Solve, write up and turn in: Sec. 3.3- 43, 45b,c, 46 (in problem 46 also check your answer!).
Also
5. Derive the formula for the chain rule by using the approximation:  f(x+h) ~ f(x)+h f '(x) .
(What you have to do is follow the calculations we did in class, erase o(1) and replace therefore the equality sign "="
with the approximately equal sign "~")
Motivation: a very important skill to develop for applications is working approximately.

Review the basic formulas of trigonometry.

Bonus problem.
Show (for extra 5 points!) that
if a function f(x) is continuous and has a right derivative at a, and a left derivative at a, and the derivatives are equal,
then the function is differentiable at a.