| |
| Limits |
| Definition
of a Limit |
| Limits:
Formal definition and proof
using epsilon - delta syntax |
| Calculate
Limits |
| Explanation
of pi using limit of ratio
of circumference to radius |
| Indeterminate
Forms - L'Hopital Rule |
| Limit
of indeterminate form using
L'hopital's rule with both
x and y variables |
| Derivatives |
| Related
Rates |
| Word
problem revenue rate changing
using related rates and chain
rule |
| Word
problem involving sand dumped
using derivatives and rates |
| 1991
AP: Tightrope walker - find
speed of shadow on ground
and on side wall of building
-- related rates |
| Related
rate problem man walking
away from light what is speed
of shadow |
| Related
rate problem bucket on pulley
man walking with rope find
speed of bucket |
| Great
word problem - minimize cost
given variable labor and
ops costs and fuel efficiency
as fcn of speed |
| Word
problem funnel water being
added find dh/dt given dV/dt
-- a related rates problem
in effect |
| Integrals |
| How
to Calculate Integrals |
| Integration
Jungle Guide |
| Difficult
Integrals |
| Integral
of 1/(x * sqrt(4x+1)) |
| Integral of 1 over sqrt of a polynomial: Is it a real
problem or is it BS just
to torture calculus students? |
| Gnarly
integral of (2x + 1) / sqrt(2
-3x - 2x^2) using inverse
trig and complete the square |
| Integral
with sin and cos terms in
denom using algebraic tricks
and trig identities |
| Integral
using partial fractions and
other tricks |
| Fundamental
Theorem Of Calculus |
| Fundamental
Theorem Part 1: Is it a remnant
from the last UFO to visit
earth? |
| Fundamental
Theorem Part II Proof |
| d/dx
of integral from 0 to x of
a function of f(t)dt |
| d/dx
of integral from -x to x
of e^-t^2 dt |
| Contrived
AP calculus problem 12 +
integral from a to x of f(t)/t^3
= 4 * sqrt(x) solve for f(x)
and a |
| d/dx
of integral from 1 to 2x
of sqrt(1+t^2) dt |
| d/dx
of the integral from 0 to
x^3-1 of f(t) dt |
| Riemann
Sums |
| Tutorial:
Riemann sums, finite sums,
integrals part 1 |
| Tutorial:
Riemann sums, finite sums,
integrals part 2 |
| Classic
AP Calculus Riemann Sum problems
explained "soup to nuts" |
| Riemann
sum of ln(x-2) using the
average of left and right
for each interval |
| Area
Under Curve |
| Integral
to find the area of a triangle |
| Polar
coordinate function r = 1
- 2sin(theta) used to calculate
area using integral |
| Find
area bounded by sqrt(x) +
sqrt(y) = 1, y=0, x=0 Substitution
in definite integral |
| Volume
of Revolution |
| Volume
of revolution between x^2
and x^3 around y axis |
| Area
and volume free response
problem from the 2002 AP
exam |
| Surface
Area of Revolution |
| Surface
area of a doughnut using
volume of revolution |
| Curve
Length |
| Substitution |
| Integral
of tan^n(x) sec^2(x) dx |
| Integral
of 1/(x * sqrt(4x+1)) --Shrewd
Manipulation Required to
find Inverse Trig Solution |
| Find
area bounded by sqrt(x) +
sqrt(y) = 1, y=0, x=0 Substitution
in definite integral |
| Integral
involving sqrt(a^2 - x^2)
-- trigonometric substitution |
| Integral
of [ln(x-2)]/x |
| Integral
involving substitution of
lnx |
| Integral
of (sinx)^3cosxdx |
| Integration
by Parts |
| Integration
to find reduction formula
for integral of tan^n(x)
dx -- interesting problem |
| Integral
of x^2 * sinx dx using multiple
rounds of integration by
parts |
| Integral
of e^5x sin(3x) dx -- Integration
by parts used successively |
| Inverse
Trig - Complete the Square
First |
| Gnarly
integral of (2x + 1) / sqrt(2
-3x - 2x^2) using inverse
trig and complete the square |
| Partial
Fractions |
| Integral
using partial fractions and
other tricks |
| Partial
Fractions: Soup to Nuts Part
4 of 4 |
| Partial
Fractions: Soup to Nuts Part
3 of 4 |
| Partial
Fractions: Soup to Nuts Part
2 of 4 |
| Partial
Fractions: Soup to Nuts Part
1 of 4 |
| Trig
Functions |
| Integral
of (sinx)^3cosxdx |
| Integral
of tan^n(x) sec^2(x) dx |
| Integration
to find reduction formula
for integral of tan^n(x)
dx -- interesting problem |
| Integral
of e^5x sin(3x) dx -- Integration
by parts used successively |
| Exponential
and Logarithmic Functions |
| Exponential
decay problem using carbon
14 dating |
| Slope
Fields and Differential Equations |
| Slope
fields and differential equations:
explained "soup to nuts" |
| Differential
Equation solved using separation
of variables -- excellent
Partial Fractions Tutorial |
| Differential
Equation x' - xcos(t) = cos(t)
solution using separation
of variables |
| Integral
of 1 over sqrt of a polynomial:
Is it a real problem or is
it BS just to torture calculus
students? |
| Series |
| Find
total distance traveled by
bouncing ball |
| Word
Problems |
| How
to Analyze and Solve Word
Problems |
| How
to solve calculus word problems |
| Word
Problems involving Rates
of Change |
| Classic
algebra 1 word problem -
Dan and Ken both mowing the
lawn -- how long does it
take? |
| Related
Rates |
| Word
problem revenue rate changing
using related rates and chain
rule |
| Word
problem involving sand dumped
using derivatives and rates |
| Word
Problems involving Summation
or Accumulation of Change |
| Word
problem - calculate ice thickness
as function of time given
rate information |
| Exponential
decay problem using carbon
14 dating |
| Position,
Velocity, and Acceleration |
| Classic
AP Calculus test free response
problem almost guaranteed
to be on the test |
| Vector
Velocity and Position --
Calculate if/when skaters
intersect |
| Word
Problems involving Distance |
| 1991
AP: Tightrope walker - find
speed of shadow on ground
and on side wall of building
-- related rates |
| Related
rate problem bucket on pulley
man walking with rope find
speed of bucket |
| Related
rate problem man walking
away from light what is speed
of shadow |
| Word
Problems involving Cones |
| Word
problem funnel water being
added find dh/dt given dV/dt
-- a related rates problem
in effect |
| Word
problem involving sand dumped
using derivatives and rates |
| Word
Problems involving Ladders
and Shadows |
| 1991
AP: Tightrope walker - find
speed of shadow on ground
and on side wall of building
-- related rates |
| Related
rate problem man walking
away from light what is speed
of shadow |
| Word
Problems involving Economics,
Profit, Revenue, Costs |
| Great
word problem - minimize cost
given variable labor and
ops costs and fuel efficiency
as fcn of speed |
| Word
problem revenue rate changing
using related rates and chain
rule |
| AP
Calculus Test Classic Problems |
| d/dx
of integral from -x to x
of e^-t^2 dt |
| 2005
AP Calculus AB Free Response
- Rod heated from one end,
find a number of attributes |
| Contrived
AP calculus problem 12 +
integral from a to x of f(t)/t^3
= 4 * sqrt(x) solve for f(x)
and a |
| Area
and volume free response
problem from the 2002 AP
exam |
| Classic
AP Calculus test free response
problem almost guaranteed
to be on the test |
| 1991
AP: Tightrope walker - find
speed of shadow on ground
and on side wall of building
-- related rates |
| Polar
Coordinates |
| Polar
coordinate function r = 1
- 2sin(theta) used to calculate
area using integral |
| Multi-variable
Calculus |
| My
favorite calculus problem
ever - Maximize Utility as
function of buying several
goods |
| Lagrange
Multipliers |
| Lagrange
Multiplier Solution to Multi-variable
Utility Maximization Problem |
| Gradient
and Partial Derivatives |
| My
favorite calculus problem
ever - Maximize Utility as
function of buying several
goods |
