CALCULUS 2
by Peter J. Ponzo
TABLE OF CONTENTS
EXAMPLE PROBLEMS
ASSORTED PROBLEMS
LECTURE 1
INTRODUCTION TO DIFFERENTIAL EQUATIONS
POPULATION GROWTH
Logistic Equation
The World's Simplest DE
velocity and position
SEPARABLE DIFFERENTIAL EQUATIONS
population of a species
a certain species of clam
LECTURE 2
MORE ON DIFFERENTIAL EQUATIONS
a Little Partial Fractions
A spherical mothball
population of clams
EXPONENTIAL DECAY
An Egyptian scroll
the half-life
Newton's Law of Cooling
LECTURE 3
MORE ON DIFFERENTIAL EQUATIONS
Direction of Solutions
following the direction
the DE PORTRAIT
LINEAR FIRST ORDER DEs
INTEGRATING FACTOR
the temperature of the object
a nice trig identity!
LECTURE 4
SEQUENCES AND SERIES
Sequences
PARTIAL SUMS
SERIES
the HARMONIC SERIES
LECTURE 5
CONVERGENCE of SERIES
A Test for Convergence of an Infinite Series
series where every term is positive
the nth term test
LECTURE 6
ALTERNATING SERIES and ABSOLUTE CONVERGENCE
ALTERNATING SERIES
the terms must get smaller fast!
the Alternating Harmonic Series
"decrease to zero" means two things
Estimating the Sum of a Convergent Alternating Series
ABSOLUTE CONVERGENCE
LECTURE 7
TAYLOR POLYNOMIALS and TAYLOR SERIES
TAYLOR POLYNOMIALS
LECTURE 8
INFINITE POWER SERIES
TAYLOR SERIES
the RATIO TEST
RADIUS OF CONVERGENCE
Maclaurin Series
POWER SERIES
LECTURE 9
MORE ON SERIES
Estimating the Sum of an Alternating Taylor Series
the error is less than the first neglected term
magic of brackets
Estimating the Sum of "Other" Taylor Series
LECTURE 10
CURVES and PARAMETRIC EQUATIONS
Plotting Parametric Curves
Slope of the Tangent Line to a Parametric Curve
think of the parameter "t" as time
the velocity vector
LECTURE 11
MORE ON PARAMETRIC REPRESENTATION OF CURVES
the Tangent and Position Vectors
the derivative of a vector
Length of a Curve
End of Part 1
LECTURE 12
SOME APPLICATIONS
Polar Curves, revisited
TRAPEZOIDAL RULE
the CYCLOID
xxxx
the Straight Line
that terrible curve
an Astroid
LECTURE 13
FUNCTIONS OF TWO VARIABLES
LEVEL CURVES
an Orthogonal Trajectory
3 Dimensional Surfaces
a cylinder, one variable is missing
Revolving 2-D Curves to get 3-D Surfaces
LECTURE 14
DERIVATIVES OF FUNCTIONS OF TWO VARIABLES
the PARTIAL DERIVATIVE
pressure of a gas
A square box
The profit per hat
HIGHER PARTIAL DERIVATIVES
LECTURE 15
DIRECTIONAL DERIVATIVES
LEVEL CURVES again
The density of ants
In what direction
The temperature of a plate
LECTURE 16
the GRADIENT
VECTORS
unit vectors
Direction Cosines
the gradient vector
Is that an accident?
A little More About Vectors
the DOT Product
the angle between vectors
If P•Q = 0 then they must be perpendicular
LECTURE 17
more on the GRADIENT, and the CHAIN RULES
The distribution of a certain type of plant
let do it
the CHAIN RULES
this dimensional stuff
LECTURE 18
another CHAIN RULE
thermodynamics is cover-to-cover partial derivatives
a COLLECTION of CHAIN RULES
Directional Derivatives, revisited
Implicit Differentiation, revisited
the GRADIENT vector is normal to the level curve
LECTURE 19
the TANGENT PLANE
LECTURE 20
OPTIMIZATION
Least Squares Fit
SOLUTIONS TO "ASSORTED PROBLEMS"
End of Calculus 2
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