Видео

- Using rectangles to approximate the area under a curve - How to set up a sum of areas - How to identify a given sum as an area under a curve

- We derive Euler's formula and use it to prove Euler's identity - How e^(it) is the unit circle - How all complex numbers have polar form re^(it) for some r and t - Why multiplying two negative numbers gives us a positive number - Why radians are the "natural" units for the trig functions

- How to use known Taylor series expressions to derive new ones - Other odds & ends about Taylor series

- We find the cubic approximation of e^x at x0 = 0 - We find the nth Taylor Polynomial of e^x at 0 - We derive the general formula for the Taylor series of a function f(x) at some point x0 - We find the Taylor series of cos(x) based at 0 and write it in both expanded form and with summation notation

- Motivating example: estimate sqrt(4.01) - How to find the linear approximation to a given function at a given point two different ways - How to find the quadratic approximation to a given function at a given point two different ways - We show how these two ways yield the same result, though they may look different - We stress that the motivating example is not what we're really after: there w...

- Brief review of steps for solving optimization problems from Algebra II Lecture 45 - ruclips.net/video/5QFu8WQr9M4/видео.html - Example problems - Extra credit challenge problem

- Logarithmic differentiation - Implicit differentiation - Finding the tangent line of an object at a point using implicit differentiation - A related rates problem - Solutions to a differential equation - Finding extrema (maxima and minima) of a function - Using derivatives to determine where a function is increasing/decreasing and concave up/down

- We investigate how the first and second derivative of a function affect the shape of a graph - First and Second Derivative Tests - The sign of f'(x) relates to f(x) increasing or decreasing - The sign of f''(x) relates to f(x) being concave up or concave down - Using these facts to determine whether a critical number of f is a local maximum or minimum

- Definitions of global/local maximum/minimum - Extreme Value Theorem - Fermat's Theorem (not the last one) - Method for finding the absolute minimum and maximum of a function on a finite domain - Critical numbers

- Derivative of the natural log - Derivatives of exponential and log functions of other bases - Logarithmic differentiation

- Proof that the derivative of the exponential function is itself - Brief discussion of differential equations

- Various related rates problems and solutions

- Discussion and example of explicit vs. implicit functions - Examples of using implicit differentiation to find derivatives/tangent lines

- Finding tangent lines to a given function at a given point - Given the graph of a function, draw the graph of its derivative - Examples of using the chain rule

Lecture 13 - The Chain Rule

Lecture 12 - Derivatives of Trig Functions

Lecture 11 - Product Rule and Quotient Rule

Lecture 10 - Differentiation Formulas

Lecture 9 - Derivatives and Differentiability

Lecture 8 - The Derivative of a Function

Lecture 7 - Quiz Review

Lecture 6 - Continuity

Lecture 5 - Rigorous Definition of a Limit

Lecture 4 - Limit Laws

Lecture 3 - Infinite Limits

Lecture 2 - Intro to Limits

Lecture 1 - Intro to Calculus

The Existence of Euler's Number (Part 2)

The Existence of Euler's Number (Part 1)