Functions and Models Study Notes

The foundational concept in calculus involves understanding functions, which are relations that assign exactly one output for each input.
Example of a real-world phenomenon modeled by functions includes population size, product demand, speed of falling objects, etc.
Types of mathematical models involved in functions encompass various essential forms.

Definition of Mathematical Model: A mathematical description of a real-world phenomenon such as:
- Size of a population
- Demand for a product
- Speed of a falling object
- Concentration of a product in a chemical reaction
- Life expectancy of a person at birth
- Cost of emission reductions
Process of Mathematical Modeling: Illustrated with figures in the material.

Linear Function Definition: If $y$ is a linear function of $x$ , the graph is a line represented algebraically as: $y = f(x) = mx + b$ where:
- $m$ = slope of the line
- $b$ = y-intercept.

Scenario: As dry air moves upward, it expands and cools.
- Ground temperature: $20^ ext{C}$
- Temperature at height $1$ km: $10^ ext{C}$
- Formulation of temperature function $T(h)$ based on height:
  - Linear assumption: $T = mh + b$
Calculation:
- Given: $T = 20$ when $h = 0$ , thus $b = 20$ .
- Next, $T = 10$ when $h = 1$ , leading to the finding of slope $m$ :
  $10 = m(1) + 20 \implies m = -10$ .
- Conclusively, the function becomes:
  $T = -10h + 20$
Graph Representation:
- Slope: $m = -10^ ext{C/km}$ ; representing the rate of temperature change with respect to height.
- For $h = 2.5$ km, the temperature calculates as:
  $T = -10(2.5) + 20 = -5^ ext{C}$ .

Definition: If no physical law exists to form a model, one can use an empirical model, which captures data trends via fitted curves.

Definition of Polynomial Function: Expressed in the form: $P(x) = an x^n + a{n-1} x^{n-1} + ext{…} + a_0$ where:
- $n$ = nonnegative integer
- $a0, a1, …, a_n$ = coefficients of the polynomial
Domain: The set of all real numbers.
Degree of Polynomial: If leading coefficient $a_n eq 0$ , then the degree is $n$ .
Example Degree: Given function of degree 6, $P(x) = x^6 + 3x^5 + 2$ implies the leading term is $x^6$ with leading coefficient 1.

Linear Function:
- Form: $P(x) = mx + b$ (degree 1).
Quadratic Function:
- Form: $P(x) = ax^2 + bx + c$ (where $a eq 0$ ; degree 2).

Situation: A ball dropped from 450 m height.

Height over time data:

Time (s)	Height (m)
0	450
1	445
2	431
3	408
4	375
5	332
6	279
7	216
8	143
9	61
Approach:

A scatter plot reveals a potential parabola, hence suggesting a quadratic model.
Use of computational tools (graphing calculator) to fit a quadratic model through point set.
Ground Impact Timing:

The ball hits at height $h = 0$ , leading to quadratic solution for $t$ using the quadratic formula.
Result: $t ext{ (positive root)} <br></p></li></ul></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td></tr><tr><td colspan="1" rowspan="1"><p>ightarrow t ext{ approximately } 9.67 ext{ seconds}.$

Function Types Summary

Sine: Defined via $f(x) = ext{sin}(x)$ , where the domain is $ext{all real numbers}$ and the range is $[-1, 1]$ . Periodic with period $2 ext{π}$ .
Cosine: Similar properties as sine; $ext{cos}(x)$ .
Tangent: Defined via relation to sine and cosine, undefined at integers of $ext{π/2}$ . Range is all reals, periodic with period $ext{π}$ .
Other Trigonometric Functions: Include cosecant, secant, and cotangent as reciprocals of the fundamental functions.

Exponential Function Definition:
- Form: $f(x) = a^x$ with positive constant base $a$ .
Logarithmic Function Definition:
- Form: $f(x) = ext{log}_a(x)$ , where a > 0 and $a eq 1$ ; inversely related to exponential functions.

A comprehensive understanding of functions encompasses various types and applications crucial for solving calculus problems. Functions model typically real-world situations, further emphasizing their significance in mathematical analysis and development.