PjProblemStrings Sequencing And The Derivative.

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PjProblemsStrings Sequencing Of The Derivative

The *derivatives* and *anti-derivatives* of functions are very important analytical tools. They form the branch of mathematics generally known as *calculus*. *Calculus* has been and continue to be an indispensable tool in the analysis of existential problems. The great Isaac Newton (1642-1727 A.D.) formulated the *calculus*. He called his formulation the *method of Fluxions*.

(a) What *work* does the *derivative* do?

(b) Use PjProblemStrings Sequencing to broadly describe this *work* of the *derivative*.

(c) What is the geometric interpretation of the *derivative* of a single variable function and a multi-variable function.

(d) Is the geometric interpretation consistent with *PjProblemStrings Sequencing*?

(e) What are some of the applications of the *derivative*?

**The strings**:
S_{7}P_{3}A_{3k} (force. k =1, 2), S_{7}P_{4}A_{4k} (motion. k =1, 2, 3,4), S_{7}P_{5}A_{5k} (change. k =1, 2). Where k is not a multiplicand.
**The math**:

All Pj Problems are at play. However, Pj Problems of Interest are of types *force*, *motion* and *change*.
**(a)**The *derivative* as an analytic tool, measures *relational dynamism*. In other words, a *variable change* (change in a dependent variable) that is caused by another *variable change* (change in an independent variable). The process of deriving a *derivative* is called *differentiation*.
**(b)***Space dynamism* is a fundamental existential phenomenon. It is caused by *forces*. A *force* is either a *pull* or a *push*. The PjProblemStrings representing these forces are:

- S_{7}P_{3}A_{31} (a multi-matter dynamic space in which the dynamism of interest is caused by *pulls*)

- S_{7}P_{3}A_{32} (a multi-matter dynamic space in which the dynamism of interest is caused by *pushes*).

The *dynamism* of a space may be caused by *pushes* and *pulls* acting concurrently.
*PjProblemStrings sequencing* describes *dynamism* as *motion sandwiched between two forces*. The PjProblemStrings representation is:

- **S _{i}P_{3}A_{3k}S_{i}P_{4}A_{4k}S_{i}P_{3}A_{3k}**. Where

The first force is the

In figure 1a, there is no change in the dependent variable relative to the change in the independent variable. So, the slope is zero and by definition, the

In figure 1b, the change in the dependent variable relative to the change in the independent variable is constant. So, the slope is a constant and so is the

In figure 1c, the dependent variable changes at every point relative to the change in the independent variable. So the slope varies at every point of the curve. Therefore, the slope at a point is the slope of a line tangent to the point. The slope of the tangent line is the slope of the secant line as the change in the independent variable tends to zero.

Figure 1d is a geometric interpretation of the

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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