| 227 |
absolute (global) maximum value
|
|
| 227 |
absolute (global) minimum value
|
|
| 227 |
absolute (global) extrema (extremum)
|
|
| 228 |
Extreme-Value Theorem for Continuous Functions
|
|
| 229 |
local (relative) maximum value
|
|
| 229 |
local (relative) minimum value
|
|
| 230 |
local (relative) extrema
|
|
| 230 |
Theorem 2
Local Extreme Values
|
|
| 230 |
critical point
|
|
| 237 |
Rolle’s Theorem
|
|
| 238 |
Mean Value Theorem
|
|
| 240 |
Corollary 1.
Functions with Zero Derivatives
|
|
| 240 |
Corollary 2.
Functions with Same Derivatives
|
|
| 241 |
differential equation (definition)
|
|
| 241 |
solution to differential equation (definition)
|
|
| 241 |
projectile motion
|
|
| 241 |
free-fall
|
|
| 242 |
projectile motion
|
|
| 246 |
increasing function
|
|
| 246 |
decreasing function
|
|
| 246 |
Corollary 3.
First Derivative Test for Increasing/Decreasing
|
|
| 247 |
First Derivative Test for Local Extrema
|
|
| 248 |
concave up
|
|
| 248 |
concave down
|
|
| 249 |
Second Derivative Test for Concavity
|
|
| 249 |
inflection point
|
|
| 250 |
stock market and inflection points
|
|
| 251 |
Second Derivative Test for Local Extrema
|
|
| 252 |
how to graph y=f(x) using y′ and y″
|
|
| 253 |
possible graphs described by
y' and y''
|
|
| 258 |
autonomous differential equation
|
|
| 258 |
equilibrium value (rest point)
|
|
| 258 |
phase line
|
|
| 258 |
how to draw phase lines and solution curves
|
|
| 259 |
stable equilibrium
|
|
| 260 |
unstable equilibrium
|
|
| 261 |
Newton’s Law of Cooling
|
|
| 261 |
falling body with resistive forces
|
|
| 262 |
terminal velocity
|
|
| 263 |
limiting population (carrying capacity)
|
|
| 263 |
logistic growth
|
|
| 263 |
sigmoid shape
|
|
| 267 |
optimization
|
|
| 269 |
How to solve min-max problems
|
|
| 272 |
Snell’s Law
|
|
| 272 |
economics
|
|
| 272 |
marginal revenue
|
|
| 272 |
marginal cost
|
|
| 272 |
marginal profit
|
|
| 273 |
Theorem 6.
Maximum Profit
At a production level yielding maximum profit,
marginal revenue equals marginal cost.
|
|
| 274 |
average daily cost
|
|
| 275 |
sensitivity of minimum cost
|
|
| 285 |
linearization
|
|
| 285 |
standard linear approximation
|
|
| 285 |
center of the approximation
|
|
| 286 |
linear approximation for roots and powers
|
|
| 287 |
common linear approximations
|
|
| 287 |
differentials
|
|
| 289 |
differential estimate of change
|
|
| 289 |
absolute change
|
|
| 289 |
relative change
|
|
| 289 |
percent change
|
|
| 291 |
sensitivity to change (using differentials)
|
|
| 291 |
error in measurement (using differentials)
|
|
| 292 |
error in differential approximation
|
|
| 292 |
energy and Einstein’s mass equation
|
|
| 298 |
Newton’s method for solving equations
|
|
| 300 |
convergence is usually assured
(Newton’s method)
|
|
| 301 |
If Newton’s method converges it converges to a root.
|
|
| 301 |
When Newton’s method converges to a root, it
might not be the one you have in mind.
|
|
| 302 |
fractal basin
|
|
| 311 |
Schwarz’s inequality
|
|
| 311 |
estimating reciprocals without division
|
|
Alex Notes