## 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).

### 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.