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Calculus Concepts by First Principles Applet

You can use this applet to explore the following concepts from first principles:

  • Derivatives (slope of a curve);
  • Numerical integration (area under a curve); and
  • Curve length

Before calculus was developed in the 17th century, the only way to find the slopes, areas under a curve and curve lengths was to draw rectangles or trapezoids with increasingly smaller widths to get a good approximation.

You can get an idea how this works in the following applet.

Continues below

Things to Do

In this applet, you start with a pre-defined function that has been drawn for you. You can see calculation changes in the information section as you vary the parameters using the sliders.

You can enter your own function, but make sure the function values are (mostly) above the x-axis.


  • Use the slider below the curve to move point P closer to point Q. Observe the changes to the slope in the information section underneath the graph sliders. to the right of the graph.
  • The slope is found using the formula "vertical rise over horizontal run", or:
    `"gradient" = (y_2 - y_1)/(x_2 - x_1)`
  • The exact slope is given at each of the points P and Q.

Area under a curve

  • Use the first slider below the curve to change the domain (the x-values) of the graph.
  • Use the second slider below the curve to increase the number of intervals (try n = 20 and see the accuracy increase).
  • The total area of the trapezoids is given, as well as the exact area under the curve. The area of each trapezoid is found using the formula
    `A = ((p+q)h)/2,`
    where p and q are the lengths of the parallel sides, and h is the distance between them.

Length of a curve

  • Use the sliders as before to change the domain and the number of intervals.
  • Observe how the curve length approximation improves as we use more intervals.
  • The total length of the magenta (pink) segments is given (each one found using Pythagoras' Theorem:
    `c=sqrt(a^2 + b^2),`
    as well as the exact length of the curve.

Enter function: y =

Choose type: derivative area curve length


Number of intervals:

Copyright ©


The function:

Credits: From an idea by PiPo.

Some graphs to try

The grapher will accept any of the following functions (use the notation shown). You can copy from the examples below if you wish.

  • Straight lines: (like 3x - 2)
  • Polynomials: (like x^3 + 3x^2 - 5x + 2)
  • Any of the trigonometric functions: 1 + sin(x), 3+cos(x/2), tan(2x), csc(3x), sec(x/4), cot(x)
  • Exponential (e^x) and logarithm (ln(x) for natural log and log(x) for log base 10)
  • Absolute value: use "abs" like this: abs(x)
  • The hyperbolic functions and their inverses: sinh(x), cosh(x), tanh(x), arcsinh(x), arccosh(x), arctanh(x)

You can also use any combinations of the above, like ln(abs(x)).

More applets

You can see related applets with further examples here:

Derivatives Graphs - polynomials

Derivatives Graphs - involving discontinuities

Riemann Sums Applet

Riemann Sums applet - negatives and discontinuities.

Further information

See more about numerical integration in:

Trapezoidal Rule; and

Simpson's Rule.

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