Notes on Intervals, Monotonicity, Natural Domain, Binomial Theorem, and f(a+h) Expansions

Interval Types

We focus on an interval I inside the real line; typically an interval is denoted by its endpoints a and b with different bracket types indicating whether endpoints are included.
Open interval:
- $I = (a,b)$
- meaning the set {x:\ a < x < b}; endpoints a and b are not in the interval.
Closed interval:
- $I = [a,b]$
- meaning the set ${x:\ a \le x \le b}$ ; endpoints a and b are included in the interval.
Half-open (or half-closed) intervals:
- $I = [a,b)$ contains a but not b
- $I = (a,b]$ contains b but not a
Infinite intervals (Infinity is not a number, so we use parentheses around the infinite endpoint):
- $I = (a,\infty)$ represents all real x with x > a
- $I = (-\infty,b)$ represents all real x with x < b
- In practice we often consider intervals of the form $(-\infty, b)$ , $(a, \infty)$ , or a finite closed/open mix as above.
Pick two numbers a and b to form an interval; call it I for reference.

Monotonicity: Increasing and Decreasing on an Interval

Definition (in two-point form):
- If for all pairs with x1 < x2 in I we have f(x1) < f(x2), then $f$ is increasing on I.
- If for all pairs with x1 < x2 in I we have f(x1) > f(x2), then $f$ is decreasing on I.
Important: To talk about monotonicity, you need at least two points in the interval; you can't talk about a single point.
Quick graph intuition: moving to the right along the x-axis, if the y-values rise, the function is increasing on that stretch; if they fall, it is decreasing on that stretch.
Quick example descriptions:
- A graph that keeps rising as x moves to the right is increasing on that interval.
- A graph that falls as x increases is decreasing on that interval.
- A function that goes up on part of the interval and down on another part is neither increasing nor decreasing on the entire interval.
Tangent-line intuition (preview for tangent-line problems):
- If the tangent slope is positive for every point of I, the function is increasing on I.
- If the tangent slope is negative for every point of I, the function is decreasing on I.
- If the tangent slope changes sign (e.g., positive on one subinterval, negative on another), the function is not increasing or decreasing on the entire interval; there may be a horizontal tangent point that splits I into a left subinterval where it increases and a right subinterval where it decreases.

Natural Domain of a Formula: Example with a Rational Function

Definition: The natural domain of a function given by a formula is the set of inputs x for which the formula makes sense (typically real-valued outputs).
We usually restrict to real numbers unless stated otherwise.
Example:
- Consider $f(x) = \frac{x+4}{x^2-9}$
- Denominator is zero when $x^2-9=0\Rightarrow x=\pm 3$ ; these inputs are not allowed.
- Therefore the domain is all real numbers except $x=\pm 3$ .
- Interval notation form: $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$
- Alternative form (explicit): ${x\in\mathbb{R}: x \neq -3, x \neq 3}$
Another related example (domain with cancellation):
- Suppose we have a simplified form after factoring, e.g. something like $\dfrac{t-2}{t-2}$ .
- The original domain excludes the point where the denominator is zero, i.e. $t=2$ , even though for all other t the expression equals 1.
- Thus the domain is $(-\infty, 2) \cup (2, \infty)$ , and the simplified expression is valid only for $t\neq 2).</li></ul></li><li>Important caution:<ul><li>Even if cancellation appears to remove a restriction in the simplified form, remember the domain is determined by the original formula before simplification.</li></ul></li><li>Quick practical takeaway:<ul><li>When describing a domain, you can describe it as a set, as a list, or as a union of intervals; the union notation is often preferred for WebAssign/online systems.</li></ul></li></ul><h3 collapsed="false" seolevelmigrated="true">Binomial Expansion and the Binomial Theorem</h3><ul><li>Binomial theorem statement (for integers$ n\ge 0 $):<ul><li>$ (a+b)^n = \displaystyle\sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k} $</li><li>Here the exponents on a and b add up to$ n $in each term.</li></ul></li><li>Binomial coefficients:<ul><li>$ {n \choose k} = \frac{n!}{k!(n-k)!} $</li><li>These are the coefficients that appear in the expansion.</li></ul></li><li>How to visualize coefficients quickly:<ul><li>Pascal's triangle is a convenient tool to generate the coefficients: row n contains the coefficients for$ (a+b)^n $.</li><li>Construction rule: start with 1 at the top; each entry below is the sum of the two entries above it; the outer edges are always 1.</li></ul></li><li>Example: Expand$ (a+b)^5 $using binomial coefficients:<ul><li>$ (a+b)^5 = {5 \choose 0} a^5 + {5 \choose 1} a^4 b + {5 \choose 2} a^3 b^2 + {5 \choose 3} a^2 b^3 + {5 \choose 4} a b^4 + {5 \choose 5} b^5 $</li><li>Numerically:$ a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a b^4 + b^5
In general, the coefficient pattern along row n is: 1, n, (\dfrac{n(n-1)}{2}), …, {n \choose k}, …, 1.
Alternative viewpoint (combinatorial intuition): ${n \choose k}$ counts the number of ways to pick k factors of b from n factors of (a+b).
Practical note on exponents: in each term the exponents satisfy (n-k) + k = n $.</li></ul><h3 collapsed="false" seolevelmigrated="true">Expansion of f(a+h) for Polynomial Functions: A Concrete Example</h3><ul><li>Goal: illustrate how to compute the difference quotient using a binomial-like expansion.</li><li>Example 1: Take$ f(x) = x^3 $.<ul><li>By the binomial theorem:$ f(a+h) = (a+h)^3 = a^3 + 3 a^2 h + 3 a h^2 + h^3. $</li><li>Since$ f(a) = a^3 $, we have</li><li>$ f(a+h) - f(a) = (a^3 + 3 a^2 h + 3 a h^2 + h^3) - a^3 = 3 a^2 h + 3 a h^2 + h^3. $</li><li>Factor out h:$ f(a+h) - f(a) = h \big( 3 a^2 + 3 a h + h^2 \big). $</li><li>If$ h \neq 0 $, divide by h to obtain</li><li>$ \frac{f(a+h) - f(a)}{h} = 3 a^2 + 3 a h + h^2. $</li><li>This demonstrates the pattern that the difference quotient follows from the binomial expansion and that the linear approximation as$ h\to 0 $tends toward the derivative value$ f'(a) = 3 a^2 $(the leading term of the expansion with h approaching 0).</li></ul></li><li>Example 2: A quadratic function to illustrate cancellation and a concrete quotient:<ul><li>Let$ f(x) = 4 + 3x - x^2 $and evaluate around$ x=3 $.</li><li>Compute the value at 3:$ f(3) = 4 + 3\cdot 3 - 3^2 = 4 + 9 - 9 = 4. $</li><li>Compute at 3+h:$ f(3+h) = 4 + 3(3+h) - (3+h)^2 = 4 + 9 + 3h - (9 + 6h + h^2) = 4 - 3h - h^2. $</li><li>Difference quotient:$ \frac{f(3+h) - f(3)}{h} = \frac{ (4 - 3h - h^2) - 4 }{h} = \frac{-3h - h^2}{h} = -3 - h, \quad h \neq 0.
This illustrates cancellation and simplification of the quotient to a simple linear expression in h (which would tend to -3 as h -> 0).

Important caution about expansions:

F(a+h) is not equal to F(a) + h. Expanding F(a+h) requires applying the binomial expansion to the term(s) involving h, not simply replacing h by 1 or adding h to F(a).
Always compute the expansion correctly (e.g., for a power n, use (a+h)^n = \sum_{k=0}^n {n \choose k} a^{n-k} h^{k} $) and then simplify.</li></ul></li><li>Practical takeaway for problems:<ul><li>When asked to compute a derivative-like quotient, write out f(a+h) using the binomial expansion when appropriate, subtract f(a), factor out h, and cancel if possible (keeping in mind the case h = 0 is excluded for the cancellation step).</li><li>Recognize the pattern for polynomials and use it to simplify quickly rather than multiplying out by hand.</li></ul></li><li>Quick summary of the derivative intuition:<ul><li>The expression$ \frac{f(a+h)-f(a)}{h} $represents the slope of the secant line between x = a and x = a+h.</li><li>In the limit as$ h\to 0 $, this tends to the derivative$ f'(a) $.</li><li>The algebraic expansion makes this limit process explicit for polynomials.</li></ul></li></ul><h3 collapsed="false" seolevelmigrated="true">Quick Reference and Tips</h3><ul><li>Interval notation vs set notation for domains:<ul><li>Use interval notation when describing a domain that is a connected union of intervals; or express as a set description like$ {x:\ x\neq -3, x\neq 3} $.</li></ul></li><li>When describing monotonicity, remember you need two distinct points to compare; monotonicity is about how f behaves as x increases across an interval.</li><li>For functions given by formulas, always identify the natural domain by checking where the formula is defined (e.g., avoid divisions by zero or square roots of negative numbers in the real setting).</li><li>Do not confuse a simplified expression with the domain of the original expression; the original expression dictates the domain.</li><li>Practice tip for WebAssign-type problems: express domain in interval notation as a union of intervals, e.g.,<ul><li>For the domain of$ \frac{x+4}{x^2-9} $, answer:$ (-\infty,-3)\cup(-3,3)\cup(3,\infty) $.</li></ul></li><li>Remember the binomial coefficients follow the pattern in Pascal’s triangle and can be read as$ {n \choose k} = \frac{n!}{k!(n-k)!} $.</li><li>The binomial expansion pattern generalizes to any integer n, providing a fast way to expand sums of powers without long multiplication.</li></ul><h3 collapsed="false" seolevelmigrated="true">Summary of Key Formulas Introduced</h3><ul><li>Interval types and endpoints:<ul><li>Open:$ (a,b) $</li><li>Closed:$ [a,b] $</li><li>Half-open:$ [a,b), (a,b] $</li><li>Infinite endpoints:$ (a,\infty), (-\infty,b) $</li></ul></li><li>Monotonicity (two-point condition):<ul><li>Increasing:$ x1 < x2 \Rightarrow f(x1) < f(x2) $</li><li>Decreasing:$ x1 < x2 \Rightarrow f(x1) > f(x2) $</li></ul></li><li>Natural domain of a rational function example:<ul><li>$ f(x) = \frac{x+4}{x^2-9} \Rightarrow x \neq \pm 3 $</li><li>In interval notation:$ (-\infty,-3) \cup (-3,3) \cup (3,\infty) $</li></ul></li><li>Cancellation caveat example:<ul><li>If$ \frac{t-2}{t-2} $, domain still excludes$ t = 2 $; domain:$ (-\infty, 2) \cup (2, \infty) $.</li></ul></li><li>Binomial theorem:<ul><li>$ (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k} $</li><li>Coefficients from Pascal’s triangle / factorial form:$ {n \choose k} = \dfrac{n!}{k! (n-k)!} $</li></ul></li><li>Example expansions:<ul><li>$ (a+b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a b^4 + b^5 $</li></ul></li><li>Cubic expansion example:<ul><li>$ f(a+h) = (a+h)^3 = a^3 + 3 a^2 h + 3 a h^2 + h^3 $</li><li>$ f(a+h) - f(a) = 3 a^2 h + 3 a h^2 + h^3 = h(3 a^2 + 3 a h + h^2) $</li><li>If$ h \neq 0 $,$ \frac{f(a+h) - f(a)}{h} = 3 a^2 + 3 a h + h^2 $</li></ul></li><li>Quadratic example around x = 3:<ul><li>$ f(x) = 4 + 3x - x^2 $</li><li>$ f(3) = 4 $</li><li>$ f(3+h) = 4 - 3h - h^2 $</li><li>$ \frac{f(3+h) - f(3)}{h} = -3 - h $$

These notes connect to broader ideas: monotonicity and derivative intuition, domain restrictions, and efficient algebraic tools (binomial theorem and Pascal’s triangle) that will recur in tangent-line and derivative problems.