# 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.

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### 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.**

#### Derivatives

- 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* =

Number of intervals:

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## Information

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

## Further information

See more about numerical integration in:

Trapezoidal Rule; and

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