*Its All about Pj Problem Strings (S _{i}P_{j}A_{jk}) -
7 Spaces Of Interest (S_{i}) and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (A_{jk})*

Expressions Of Pj Problems.

Pj Problems - Overview

Celestial Stars As Expressions Of Pj Problems

The Number Line As Expression Of Pj Problems

Geometries As Expressions Of Pj Problems

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation As Expression Of Pj Problems

The Atom As Expression Of Pj Problems

Survival As Expression Of Pj Problems

Energy As Expression Of Pj Problems

Light As Expression Of Pj Problems

Heat As Expression Of Pj Problems

Sound As Expression Of Pj Problems

Music As Expression Of Pj Problems

Language As Expression of Pj Problems

Stories As Expressions of Pj Problems

Work As Expression Of Pj Problems

States Of Matter As Expressions Of Pj Problems

Buoyancy As Expression Of Pj Problems

Nuclear Reactions As Expressions Of Pj Problems

Molecular Shapes As Expressions Of Pj Problems

Electron Configurations As Expressions Of Pj Problems

Chemical Bonds As Expressions Of Pj Problems

Energy Conversion As Expression Of Pj Problems

Chemical Reactions As Expressions Of Pj Problems

Electromagnetism As Expression Of Pj Problems

Continuity As Expression Of Pj Problems

Growth As Expression Of Pj Problems

Human-cells As Expressions Of Pj Problems

Proteins As Expressions Of Pj Problems

Nucleic Acids As Expressions Of Pj Problems

COHN - Nature's Engineering Of The Human Body

The Human-Body Systems As Expressions Of Pj Problems

Vision As Expression Of Pj Problems

Walking As Expression Of Pj Problems

Behaviors As Expressions Of Pj Problems

Sensors' Sensings As Expressions Of Pj Problems

Beauty As Expression Of Pj Problems

Faith, Love, Charity As Expressions Of Pj Problems

Photosynthesis As Expressions Of Pj Problems

Weather As Expression Of Pj Problems

Systems As Expressions Of Pj Problems

Algorithms As Expressions Of Pj Problems

Tools As Expressions Of Pj Problems

Networks As Expressions Of Pj Problems

Search As Expressions Of Pj Problems

Differential Calculus As Expression Of Pj Problems

Antiderivative As Expression Of Pj Problems

Integral Calculus As Expression Of Pj Problems

Economies As Expresions Of Pj Problems

Inflation As Expression Of Pj Problems

Markets As Expressions Of Pj Problems

Money Supply As Expression Of Pj Problems

Painting As Expressions Of Pj Problems

Single Variable Functions - Limits-Differentiability-Continuity

Calculus plays a major role in the search for solutions for myriad existential problems. *Limits*, *differentiability* and *continuity* are some of the central concepts in calculus.

1. Consider the general polynomial ** p(x) = a _{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n}**.

Show that

In general for any polynomial p,

Also, if a rational function r(x) = p(x)/q(x); where p and q are polynomials and q(x) ≠ 0;

Then,

2. Determine the following limits:

(a) **lim _{x→4} (5x^{2} - 2x + 3)**

(b)

(c)

(d)

(b) f(-2) = -3

(c) f(4) = 0

(d) Limit does not exist because as x tends to -3, x + 3 tends to 0 eventhough (x

3(a) Write the difference quotient for f(x).

(b) Write the difference quotient for f(x) at x = a.

(c) What is another name for the limit of the difference quotient for f(x) at x = a as the denominator tends to 0?

(d) What is the geometric interpretation of the answer to 3(c)?

(e) What is the interpretation of the answer to 3(c) with respect to rate of change?
**Ans:**3(a) [f(x + h) - f(x)]/h. Where h ≠ 0.

(b) [f(a + h) - f(a)]/h.

(c) **lim _{h→0} [f(a + h) - f(a)]/h** is also called the

(d) The derivative f'(a) is gometrically interpreted as the slope of the tangent to f(x) at a.

(e) The derivative f'(a) is also interpreted as the instantaneous rate of change of f with respect to x at a.

4. Use the difference quotients of the following functions to determine their derivatives:

(a) f(x) = x^{2}.

(b) f(x) = 1/x.
**Ans:**4(a) Simplification of the difference quotient results in

2x + h. So, **lim _{h→0} 2x + h = 2x**.

Therefore derivative = f'(x) = 2x

(b) Simplification of the difference quotient results in

-1/(x(x + h)). So,

Therefore derivative = f'(x) = -1/x

5(a) Relate *differentiability* with *continuity*

(b) Relate *Limits* with *continuity*.
**Ans:**5(a). If f(x) is differentiable at x = a, then f(x) is continuous at x = a. Although differentiability implies continuity, continuity does not imply differentiability. In other words, the continuity of f(x) at x = a, does not imply that f(x) is differentiable at x = a.

(b) If **lim _{x→a} f(x) = f(a)**, then f(x) is continuous at x = a.

6. f(x) = (x^{2} -x - 6)/(x - 3)

(a) Is f(x) continuous at x = 3?

(b) Is f(x) differentiable at x = 3?
**Ans:**6(a) **lim _{h→0} f(x) = 5**. f(3) = 4. So f(x) is not continuous at x = 3.

(b) If f(x) is not continuous at x = 3, it is not differentiable at x = 3.

7. If f(x) = x^{2} - x is continuous on ℜ What is the limit of g(x) = e^{f(x)} as x tends to 1?
**Ans:** If f(x) is continuous on ℜ, then g(x) is continuous on ℜ. So **lim _{x→1} g(x) = g(1) = 1**

8. Determine if f(x) is increasing or decreasing at x = a if:

(a) f'(a) > 0

(b) f'(a) < 0

(c) f'(a) = 0.
**Ans:** 8(a) if f'(a) > 0, then f(x) is increasing at x = a (first derivative rule)

(b) If f'(a) < 0, then f(x) is decreasing at x = a (first derivative rule)

(c) Not clear whether f(x) is increasing or decreasing, more information is needed.

9. Determine if f(x) is concave up or concave down at x = a if:

(a) f''(a) > 0

(b) f''(a) < 0

(c) f'(a) = 0.
**Ans:** 9(a) if f''(a) > 0, then f(x) is concave up at x = a (second derivative rule)

(b) If f''(a) < 0, then f(x) is concave down at x = a (second derivative rule)

(c) Not clear whether f(x) is convave up or concave down, more information is needed.

10. Evaluate **lim _{x→∞} sin(1/x).
Ans: 0.
**

Mind Warm Ups

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.

*Problems by Peter O. Sagay*