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Langara College - Department of Mathematics and Statistics
Internet Resources for the Calculus Student - Rates of Change
Slope of a Curve - Tangents and Secant Lines
One interpretation of the "slope" at a point on a curve is as the slope
of the "tangent line" which just touches the curve at that point. This
may give a value of the slope for each point on the graph of a function,
and so the slope value is just another function of the position (this new
function is what is called the "derivative" of the function you started
with). The following pages may help you to see how this new function
is related to the one that it is derived from.
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This shows
the tangent line to a curve, varying as the contact point moves along the
curve.
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Guessing
the Derivative from a graph
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Pick
the derivative!
When given the graph of a function, it is not hard to estimate the
tangent slope at any point by just drawing the tangent and measuring rise
and run between any two points on it. But the accuracy of this will be
limited by that of our measurements - which cannot be made arbitrarily
precise.
It is easy to determine the exact slope of a straight line through two
given points with known coordinates, but for the tangent line we start
with the knowledge of just the one contact point, so how can we
actually compute its slope? One approach is to approximate the tangent's
slope by that of a nearby secant line (ie a line which passes through two
points on the curve). This may still be an approximation, but if we have
a formula for the given function, then we can take the second point as
close as we like to the first and so may be able to achieve arbitrarily
close approximations to the "true" tangent slope. The following applets
allow you to see how the secant line approaches a tangent as the two points
get close together.
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This page includes
a simple movie of secant lines approaching a tangent
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This is another
motion-picture version of the secant lines approaching a tangent
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These animations
are nicely designed but slow to load (they
are also available in a non-Java "gif animation" version)
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This
one allows you to zoom in or out and to vary the graph used for the demo
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Explore
the construction of tangent lines as limits of secant lines (with
a MathView notebook illustrating the slope calculations and input that
you can edit to change the function or contact point)
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This
lab at UBC deals with the same issue, and
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This
lesson from another on-line course also includes an example of a "secant
applet"
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and here's another Derivative
Definition Java Applet
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.
If you have come across any good web-based illustrations of these and
related concepts,
please do let
us know and we will add them here.
Derivatives
Intro to Concept and Definition
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Introduction
to Derivative Concept (@ubc)
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Intro
to Derivative from Hofstra project
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Derivatives
Intro @PWS
Tangent Slopes
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Surfing(Derivatives)@ies
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VisualCalculus
- Tangent Lines
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Animation:all
the tangent lines to a curve
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Deriv
from TanSlope(MathCad+QT@odu)
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(graphics.html#tangent)byDougArnold@pennState
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(graphics-j.html#tangent)byDougArnold@pennState
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MathsOnlineGallery
(@uVienn.austria) derivative(asTanSlope)
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Why
Slopes -- A Calculus Preview for Algebra Students
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Derivatives--
Introduction -- Curved Mirrors
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Stressed
Out - Slope as Rate of Change
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Secant Approximation to Tangent Lines
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derivDef@ies
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Sec&TanLines@ies
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DerivativeDefinition
Java Applet
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JavaDiff(by
kThompson@OregonState)
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Animation:Secant
approaching the Tangent
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Tangent
Line Applet byDanSloughter@furman
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SecantGrapher
byJohnOrr@uNebraskaLincoln
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Visual
Calculus - Tangent as Limit of Secants
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(graphics.html#secants)byDougArnold@pennState
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(graphics-j.html#secants)byDougArnold@pennState
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Our
Own Version of the Secant -> Tangent thing
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Affine
Approx Applet byDanSloughter@furman
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derivOneSide@ies
Avg&Inst Rates
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Rolling
Ball Applet (@uPenn)
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Velocity@upenn(withQTmovies)
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Relative
Motion@upenn
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(graphics.html#bounce)byDougArnold@pennState
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(graphics-j.html#bounce)byDougArnold@pennState
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Derivatives:
Rate of change/Learn
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Avg&InstRates
- from PWS
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Calculus
Online@ubc: Lab 2(avg&inst rates)
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Lab
2 Grades and Solutions
Differentiability
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Continuity
and Differentiability(@ubc)
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MathsOnlineGallery
(@uVienn.austria) NowhereDiffble
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(graphics-j.html#jagged)byDougArnold@pennState
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(graphics.html#jagged)byDougArnold@pennState
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Calc
1 pathologies(byTVogel@TexA&M)
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Interpretation
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What
does the derivative tell us about a function?(@ubc)
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Pickthe
correct derivative! by David Yuen
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CurvatureCircle@ies
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LnGraphCircle@ies
Some Basic Rules
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ProdRule@ies
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derivCubics@ies
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Introduction
to Derivatives Computation(@ubc)
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Differentiating
Linear Functions
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Differentiating
Sums(@ubc)
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Differentiating
Products(@ubc)
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Differentiating
Quotients(@ubc)
Exp&Log
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PrecalcExp&Logs@langara
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Karl's
Calculus Tutor - 6.1 Be Fruitful and Multiply
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S.O.S.
Math - Calculus(starts with exp&log)
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e(1)@ies
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TheNumber
e (def by exp'(0)=1)
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Derivatives
of Exponentials(@ubc)
Trig
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PrecalcTrigFunctions@langara
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DerivSin@ies
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TheDerivetaive
of the Sine
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derivSinAnimation@odu
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Derivatives
of Trigonometric functions(@ubc)
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Inverse
Trigonometric Functions(@ubc)
Chain Rule
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Relative
Motion@upenn
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Composite
Functions@ies
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ChainRule@ies
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The
Chain Rule(@ubc)
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Implicit
Differentiation(@ubc)
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Applications
of The Chain Rule(@ubc)
Mean Value Theorem
MVT@ies