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Rates of Change
Derivatives are defined so as to correspond to rates
of change ,and so are of course useful whenever we want to compute
rates of change with respect to time, eg velocities from a known position
function, (or acceleration from velocity, and so on), but also to rates
of change with respect to variables other than time. On a graph, rate of
change of height with respect to horizontal distance (ie dy/dx) corresponds
geometrically to the slope of the tangent line to the graph of the function.
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Stressed
Out - Slope as Rate of Change
Why
Slopes -- A Calculus Preview for Algebra Students
Slopes are relevant also to problems of reflection
Derivatives
-- Introduction -- Curved Mirrors
The Coffeecup
Caustic
And the slope of a graph is undefined or zero at any extreme (high or
low) point.
So derivatives are useful for locating maxima and minima of functions.
ie for Optimization problems. See also derivCubics@ies
-
Rate of change of slope (ie derivative of derivative or "second derivative")
corresponds to curvature (See also CurvatureCircle@ies),
so knowledge of the derivative and second derivative of a function can
help us to produce a quick qualitatively correct picture of its graph.
-
Rates of change are also useful for estimating actual changes (as eg when
we estimate the actual distance travelled over a small time intercval by
multiplying instantaneous speed at some point by the length of the time
interval). This is also useful for controlling the errors involved
in various methods of approximation.
Motion
-
Trace
of Shadow@ies
-
Rolling
Ball Applet (@uPenn)
-
Projectile
Applet byDanSloughter@furman
-
DampedHarmonic
Motion (mass on spring)
-
Motion
of a piston - An application of differential calculus in robotics(@bcit)
Related Rates
Approximation
-
Affine
Approx Applet byDanSloughter@furman
-
(graphics.h...fferential)byDougArnold@pennState
-
(graphics-j...fferential)byDougArnold@pennState
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Newton's
Method Applet byDanSloughter@furman
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Newton's
Method(@ubc)
-
Calculus
Online@ubc: M100Lab 5(N'sMethod, &Euler's)
-
Calculus
Online@ubc: M100Lab 5 - Grades and Solutions
-
Calculus
Online@ubc: M100Lab 6(Lin,Quad,&Taylor)
-
Calculus
Online@ubc: M100Lab 6 - Grades and Solutions
-
Calculus
Online@ubc: M101Lab 6(TaylorPolys&Series)
-
Calculus
Online@ubc: M101Lab 6 Solutions
-
Taylor
Series @Eric'sTreasureTroves
-
Taylor
Polys Applet byDanSloughter@furman
-
Bisection
Method Tutorial
Optimization
The slope of a graph is undefined or zero at any extreme (high or low)
point.
So derivatives are useful for locating maxima and minima of functions.
ie for Optimization problems. See also derivCubics@ies
-
MathsOnlineGallery
(@uVienn.austria) SimpleMaxAreaProblem
-
f+1/f
&AGMineq@ies
-
EquilibriumTheory
for Econ byOsborne@UofT
-
Optimization
for Economics byOsborne@UofT
-
VisualFit
Applet (re LeastSquares)byTaur@JSE
-
Graphing
Since the derivative measures the slope, its value at a point tells us
how steeply to draw the graph at that point. In particular, the slope of
a graph is undefined or zero at any extreme (high or low) point, so derivatives
are useful for locating maxima and minima of functions. (ie for Optimization
problems). See, eg derivCubics@ies
Rate of change of slope (ie derivative of derivative or "second derivative")
corresponds to curvature (See also CurvatureCircle@ies),
so knowledge of the derivative and second derivative of a function can
help us to produce a quick qualitatively correct picture of its graph.
Parametric Curves
-
PrecalParamEqs@langara
Area
Inside a Parametric Curve(MathCad+QT@odu)
Polar Graphs
-
PrecalPolars@langara
Limits
L'Hopital's
rule@ies
Miscellaneous
-
Derivatives--
Introduction -- Curved Mirrors
-
The
Zebra Danio and its escape response(@ubc)
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Two
examples of differential calculus in electronics(@bcit)
-
An
Application of Differential Calculus to Food Technology(@bcit)
You might also check our 'raw list' (of links provided without comment)
to see if there are any more examples there that we haven't yet included
here.
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Langara College - Department of Mathematics and Statistics