DIFFERENTIATION
· Definition of the Derivative and the Tangent Line Problem; Differentiability
o Tangent Line Introduction to the definition of the derivative
§ More on the definition of the derivative
§ Definition of the derivative at a point
§ Definition of the derivative
§ Even more on the definition of the derivative
§ Definition of Derivative applet
§ Another definition of derivative applet
§ Algebraic definition of the derivative
§ Problems, solutions, and explanations on the definition of the derivative
§ Slopes, Tangents, and Derivatives
o Difference Quotient (slide show)
§ Derivative at a point applet
o Tabular view of the derivative
o Relationship of the graph of f and f '
§ Graphing the derivative -- good problems under Exercises for this Topic
o Second derivative applet
§ A tabular view of the second derivative
§ Twice differentiable function applet
o Geometric interpretation of the derivative
§ The derivative illustrated by a surfer
§ A function and its derivative plotter
o Animated view of a secant line approaching a tangent line
§ Animated View of zooming in on a tangent line -- linearization of a curve
§ Secant line approaching a tangent line applet
§ Secant and Tangent line animation for y = x3, y = x sin (1/x), and y = x2 sin (1/x)
§ Secants approaching tangent demonstration
§ Secant and tangent lines plotter
o Average rate of change and the derivative
§ Average rate of change
§ Instantaneous rates of change
§ Instantaneous rate of change applet
§ Rates of change problems, solutions, and explanations
o One-sided derivatives applet
o Derivative drawn from slopes demonstration
o Derivative Calculator applet
o Graphs the Derivative with the function applet
o The derivative as a function: algebraic approach
§ The derivative function applet
o Differentiability implies Continuity Theorem
§ A continuous nowhere differentiable function
§ Another nowhere differentiable function applet
§ Continuity and differentiability
§ Problems on the importance of differentiation
§ Finding derivatives numerically on your calculator (instructions)
o Differentiability and what it means
o Making a piecewise function continuous and differentiable
· Differentiation Rules: Power Rule, Product Rule, & Quotient Rule
o Power Rule
§ Some differentiation rules
§ More on Differentiation Rules
§ Proof of the power rule
§ Derivative of a cubic applet
§ Constant. line, and power functions applet
§ Constant multiple applet
§ Combination: sum, and difference applet
§ Power Rule (slide show)
§ Problems to work
o Product Rule
§ Rules to use in calculating derivatives of functions
§ Differentiation using the product rule
§ Applet illustrating and explaining the Product Rule
§ Product Rule (slide show)
§ Drill problems
o Quotient Rule
§ Differentiation using the quotient rule
§ Product and Quotient rules
§ Product and Quotient rules problems, solutions, and explanations
§ Proof of the quotient rule
§ Quotient Rule (slide show)
§ Product and Quotient rule applet
o Position, velocity, acceleration
§ More on velocity, acceleration, & other rates of change
§ Average Rates of Change
§ The derivative as a rate of change: numerical approach
§ Animated bouncing ball and problem
§ Moving Man applet
§ Motion on a line applet
§ Instructions for using your calculator to simulate particle motion problem
§ Review Problems on displacement -- Go to Calculus Book I, then Applications of the Derivative, then Rate of Change, then Displacement
§ Problems to work
· Differentiation Rules for Trigonometric Functions
o Derivative of sin x applet
o More on derivatives of trigonometric functions -- good problems under Section 3 Exercises
§ Finding derivatives of trigonometric functions
§ Formulas for derivatives of trigonometric functions
§ Derivatives of Trigonometric Functions
§ Derivatives of Trigonometric Functions problems, solutions, and explanations
o Trigonometric functions derivatives applet
o Proof that p is irrational
· Chain Rule
o More on the chain rule
§ Chain Rule (slide show)
§ More on the chain rule
§ Chain rule applet
o Differentiation using the chain rule
§ Chain rule applet
§ Using the chain rule
§ Chain Rule problems, solutions, and explanations
o Problems--decomposing a composed function
o Proof of the chain rule
o Graphical Differentiation Worksheet
o Transformations of functions and derivatives applet
· Implicit Differentiation
o Implicit differentiation
§ Even more on implicit differentiation
o Graph a sine function of your choosing
o Graph derivative with implicit function applet
§ Implicit function plotter
o Problems to work
§ More problems to work
§ Implicit Differentiation problems, solutions, and explanations
· Related Rates
o Related Rates Explained
§ Related Rates problems, solutions, and explanations
o Balloon Problem
§ Melting snowball problem applet
o Related Rates Animation, three problems, video explanation, & exercises
§ Related Rates airplane problem demonstration
o Find the error
o Review problems on related rates--Go to Calculus Book I, then Applications of the Derivative, then Related Rates
§ Related Rates problems
§ Related Rates - solve the Turvey by doing the problems
§ Related Rates problems & solutions
§ More Related Rates problems & solutions
· Summaries and Review Problems: Derivative Rules through Chain Rule
o Find the error
o Summary of introduction to the derivative
§ A list of differentiation formulas
§ Techniques of Differentiation summary
o Proofs of Various Derivative Formulas
o Interpretation of the Derivative problems, solutions, and explanations (rates of change)
§ Differentiation Formulas problems, solutions, and explanations
o Review problems for finding the slope and equation of a tangent line -- Go to Calculus Book I, then Derivatives, then Slope and Tangents, then Tangent Line Slope or Tangent Line Equation
§ Review problems for linearizing a function--Go to Calculus Book I, then Derivatives, then Linearization, then Linearization again
§ Review problems for finding derivatives with the above methods -- Go to Calculus Book I, then Techniques and Theory of Differentiation, then anything in sections 1 through 4
§ Review problems on velocity -- Go to Calculus Book I, then Applications of the Derivative, then Rate of Change, then Velocity
o Problems involving trigonometric functions
§ Lots of differentiation problems to do
§ More problems to do
§ More derivative problems
§ Introduction to derivative problems
o Quiz on differentiating functions
o Review quiz for related rates problems (do quiz 327 and quiz 341)