Langara College - Department of Mathematics and Statistics

## Applications of Derivatives

#### 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.
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
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)

#### Approximation

Affine Approx Applet byDanSloughter@furman
(graphics.h...fferential)byDougArnold@pennState
(graphics-j...fferential)byDougArnold@pennState
Newton's Method Applet byDanSloughter@furman
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 - 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

##### Polar Graphs
PrecalPolars@langara

#### Limits

L'Hopital's rule@ies

#### Miscellaneous

Derivatives-- Introduction -- Curved Mirrors
The Zebra Danio and its escape response(@ubc)
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.

If you have come across any good web-based illustrations of these and related concepts,