## Differential Calculus in 1 dimension and higher dimensions

Informal minicourse, January 2014.

Thursday, January 9, 16, and 23, 6-8 PM, WWH 312.

No credit, no grades, no assignments graded.

Instructor: Ernest Davis
davise@cs.nyu.edu.
If you are interested in taking this, please email me in advance.

Textbook:
Online Calculus Textbook (MIT Open Courseware).

Email List

### List of Topics

(Note: The references to the text may be inaccurate, since I'm just working
off the table of contents.)
#### Meeting 1: January 9

- Definition of derivative. Basic applications. Text: 1.1-1.3, 1.7, 2.1
- Numerical differentiation
(i.e. (f(x+e)- f(x))/e). First-order approximation (i.e. f(x+e) =
f(x) + e*f'(x) +o(e).) Text: 3.1.
- Basic rules (powers, f+g. f*g, f/g). Text: 2.2, 2.3, 2.5.
- Chain rule f(g(x)). Text: 4.1
- Exponential and logarithm (Note: omitting trigonometric functions).
Text: 6.1, 6.2

**Suggested exercises** Note: I've listed odd-numbered exercises because
the solutions are online. Somewhat annoyingly, a lot of the more interesting
problems are even-numbered.

Section 2.1, p. 49: 5, 7, 71, 21

Section 2.2, p. 56: 3, 5, 7, 9, 13

Section 2.5, p. 77: 1, 5, 19, 23, 29, 31, 39 (except d)

Section 3.1, p. 96: 11, 19, 23, 25

Section 4.1, p. 158: 1

Section 6.2, p. 241: 1, 3, 5, 9

#### Meeting 2: January 16

- Maximum and minimum as zeros of derivative. Text: 3.2.
- Higher-order derivatives and their significance. Text: 3.3
- Definition of an integral, fundamental theorem of calculus. Text: 5.1, 5.2

**Selected exercises**

Section 3.2 p. 103: 1, 5, 9, 17, 23, 25, 39

Section 3.3 p. 111: 11, 13, 27, 45

Section 5.2 p. 186: 1, 3, 5
#### Meeting 3: January 23

- Partial derivative. Text: 13.1-13.3
- Chain rule. Text: 13.5
- Tangent to a curve. Text: 12.1
- Directional derivative and Gradient. 13.4
- Max and min as points of zero gradient. Text: 13.6
- Jacobian, time permitting (ties in nicely with linear algebra, for those
who have taken Math Techniques). Not in text.
- Gradient descent, time permitting. Not in text.

**Selected exercises**

Section 13.2 p. 479: 1, 3, 5, 7, 13, 17

Section 13.5 p. 503: 13, 15, 19, 21

Section 13.3 p. 488: 1, 3, 5

Section 12.1 p. 452: 1, 5
Section 13.4 p. 495: 1, 13, 15

Section 13.6 p. 512: 1, 5, 13.