1. Functions and Limits
Functions and their Representations.
A Catalog of Essential Functions.
The Limit of a Function
Epsilon and delta Explained
Calculating Limits
Limits Part I. Covering 10 examples and the Squeeze Theorem.
Limits Part II. Infinite Limits. See the Description below for details.
Limits Part III. Limits at Infinity. See the Description below for details.
The Limit from Right and Left using a graph.
Limit sin 3x over x
Evaluate the Limit, or state that it does not exist.
Limit as h approaches 0 of [ SQRT (9+h) - 3] / h.
Limit with Conjugate
Limits at infinity.
Continuity
Continuity I
Sketch the graph of a continuous function except for the stated discontinuity: Discontinuous, but continuous from the right at 2.
Continuity II
How can you eliminate the discontinuity of f? In other words, how would you set f(5) to ensure f is continuous at 5?
Continuity III
Use the Intermediate Value Theorem to show that there is a root of the given equation x^2+x-9=0 in the interval (1,2).
Continuity IV
Find the x-value at which f is discontinuous and determine whether f is continuous from the right, or from the left, or neither.
Limits Part IV. Continuity.
Limits Part V. Precise Definition of Continuity.
Limits Involving Infinity
Limit involving infinity
Vertical Asymptote and Horizontal Asymptote.
2. Derivatives
Derivatives and Rates of Change
Introduction to Derivative.https://youtu.be/3EgXYLH6eUI
Four Examples.
Use the Definition of the Derivative to find f ' (x).
Average Velocity. A ball is thrown in the air....
The Derivative of a Function
Basic Differentiation Formulas
Rules of Differentiation
Derivatives of Trigonometric Functions. Ten Examples.
The Product and Quotient Rules
The Product and the Quotient Rules. Eight Examples.
The Quotient Rule Proof in Calculus.
The Chain Rule
Two Examples.
Implicit Differentiation
Find dy/ dx for 2x^3+x^2y-xy^3=2.
Related Rates
Related Rates. Five Examples.
Linear Approximations and Differentials
3. Inverse Functions. Exponential. Logarithmic, and Inverse Trigonometric Functions.
Exponential Functions
Find the limit of e to [3/(5-x ) ] As x—-> 5-
Inverse Functions and Logarithms
Derivatives of Logarithmic and Exponential Functions
Exponential Growth and Decay
Inverse Trigonometric Functions
Hyperbolic Functions
Indeterminate Forms and L’Hospital’s Rule
L'hospital's Rule
4. Applications of Differentiation
Maximum and Minimum Values
Maxima and Minima.
The Mean Value Theorem
Let f(x) = -3x^2+2x+4. Find the value(s) of x that satisfy the Mean Value Theorem on the interval [ -1, 1].
Graph the Secant Line, the Tangent Line, and f(x).
The Mean Value Theorem "MVT".
Derivatives and the Shapes of Graphs
What Derivatives Tell Us.
Curve Sketching
Graphing Functions using Derivatives.
Optimization Problems
Optimization. What is the Maximum Vertical Distance between y=x+2 and y=x^2 for x in [-1, 2].
Optimization. Find two positive numbers whose product is 100 and whose Sum is a Minimum.
Optimization Problems.
Antiderivatives
5. Integrals
Areas and Distances
The Definite Integral
Definite Integral and General Riemann Sum.
Riemann Sums and Approximating Area under Curves.
Evaluating Definite Integrals
Integrals of Even and Odd Functions and the Function Average.
The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus Part I and Part II with Examples.
The Substitution Rule
Substitution Rule for Indefinite and Definite Integrals.