Calculus: Limits, Continuity, and Rates of Change Notes

Rates of Change, Limits, and Sensitivity in Corn Production

• The yield per acre for an irrigated field of corn in western Kansas depends on nitrogen fertilizer usage.
• The yield function is represented as $Y(F) = 130 - 65 \times 0.985^F$, where $Y(F)$ is the yield in bushels per acre and $F$ is the number of pounds of fertilizer used per acre.
• If a farmer applies $100\,lb$ per acre, the crop yield's sensitivity to small changes can be determined using rates of change and limits.

Rates of Change and Limits

• Near the earth's surface, all bodies fall with the same constant acceleration when in a vacuum. The distance a body falls from rest is a constant multiple of the square of the time fallen ($y = 16t^2$ feet in the first $t$ seconds for dense objects like rocks or ball bearings).
• Average Velocity: Divided change in distance by change in time. For an object with position function $y = f(t)$, the average rate of change over the interval from $t$ to $t + \Delta t$ is:     $\frac{\Delta y}{\Delta t} = \frac{f(t + \Delta t) - f(t)}{\Delta t}$     Alternatively, using $h$ for the elapsed time:     $\frac{f(t + h) - f(t)}{h}$
• Instantaneous Velocity: The velocity at one single instant of time. It is the limit of the average velocity as the time interval approaches zero.
• Example: Falling Rock:
  • Formula: $y = 16t^2$.
  • Average velocity from $t = 0$ to $t = 2$: $\frac{16(2)^2 - 16(0)^2}{2 - 0} = 32\,ft/sec$.
  • Instantaneous velocity at $t = 2$ is found by taking the limit of the average velocity over the interval $[2, 2 + h]$:         $\frac{16(2 + h)^2 - 16(2)^2}{h} = \frac{64 + 64h + 16h^2 - 64}{h} = 64 + 16h$
  • As $h \rightarrow 0$, the velocity approaches $64\,ft/sec$.

Definition and Properties of Limits

• Limit Notation: $\lim_{x \rightarrow c} f(x) = L$ means that we can force $f(x)$ to be as close to $L$ as we wish by restricting $x$ to be sufficiently close to $c$ (but not equal to $c$).
• The limit at $c$ does not depend on whether the function is defined at $c$ or what its value is at $c$.
• Theorem 1: Properties of Limits (If $\lim_{x \rightarrow c} f(x) = L$ and $\lim_{x \rightarrow c} g(x) = M$):
  • Sum Rule: $\lim_{x \rightarrow c} (f(x) + g(x)) = L + M$
  • Difference Rule: $\lim_{x \rightarrow c} (f(x) - g(x)) = L - M$
  • Product Rule: $\lim_{x \rightarrow c} (f(x) \times g(x)) = L \times M$
  • Constant Multiple Rule: $\lim_{x \rightarrow c} (k \times f(x)) = k \times L$
  • Quotient Rule: $\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{L}{M}$ (provided $M \neq 0$)
  • Power Rule: $\lim_{x \rightarrow c} [f(x)]^{r/s} = L^{r/s}$, provided $L^{r/s}$ is real and $L > 0$ if $s$ is even.
• Polynomial and Rational Functions:
  • For a polynomial $f(x)$, $\lim_{x \rightarrow c} f(x) = f(c)$.
  • For a rational function $f(x)/g(x)$, $\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{f(c)}{g(c)}$ if $g(c) \neq 0$.

One-Sided and Two-Sided Limits

• Right-hand Limit: $\lim_{x \rightarrow c^+} f(x) = L$ (approaching from values greater than $c$).
• Left-hand Limit: $\lim_{x \rightarrow c^-} f(x) = L$ (approaching from values less than $c$).
• Theorem 3: A function $f(x)$ has a limit as $x \rightarrow c$ if and only if the right-hand and left-hand limits at $c$ exist and are equal: $\lim_{x \rightarrow c} f(x) = L \iff \lim_{x \rightarrow c^+} f(x) = L \text{ and } \lim_{x \rightarrow c^-} f(x) = L$.
• The Greatest Integer Function: $y = \text{int } x$ (or floor function $\lfloor x \rfloor$) has different one-sided limits at integers.
  • For integer $n$: $\lim_{x \rightarrow n^+} \text{int } x = n$ and $\lim_{x \rightarrow n^-} \text{int } x = n - 1$.

The Squeeze Theorem

• If $g(x) \le f(x) \le h(x)$ for all $x \neq c$ in an interval about $c$, and $\lim_{x \rightarrow c} g(x) = \lim_{x \rightarrow c} h(x) = L$, then $\lim_{x \rightarrow c} f(x) = L$.
• Example: Show $\lim_{x \rightarrow 0} [x^2 \sin(1/x)] = 0$.
  • $-1 \le \sin(1/x) \le 1$
  • $-x^2 \le x^2 \sin(1/x) \le x^2$
  • Since $\lim_{x \rightarrow 0} (-x^2) = 0$ and $\lim_{x \rightarrow 0} (x^2) = 0$, then $\lim_{x \rightarrow 0} x^2 \sin(1/x) = 0$.

Limits Involving Infinity

• Horizontal Asymptotes: The line $y = b$ is a horizontal asymptote if $\lim_{x \rightarrow \infty} f(x) = b$ or $\lim_{x \rightarrow -\infty} f(x) = b$.
• Infinite Limits (Vertical Asymptotes): The line $x = a$ is a vertical asymptote if $\lim_{x \rightarrow a^+} f(x) = \pm \infty$ or $\lim_{x \rightarrow a^-} f(x) = \pm \infty$.
• End Behavior Models: For numerically large values of $x$, complex functions can be modeled by simpler ones ($g$ is an end behavior model for $f$ if $\lim_{x \rightarrow \infty} f(x)/g(x) = 1$).
  • A right end behavior model applies as $x \rightarrow \infty$.
  • A left end behavior model applies as $x \rightarrow -\infty$.
  • For any polynomial $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$, the dominant term $g(x) = a_n x^n$ is the end behavior model.

Continuity

• Continuity at a Point:
  • Interior Point: Continuous at $c$ if $\lim_{x \rightarrow c} f(x) = f(c)$.
  • Endpoints: Continuous at left endpoint $a$ if $\lim_{x \rightarrow a^+} f(x) = f(a)$; continuous at right endpoint $b$ if $\lim_{x \rightarrow b^-} f(x) = f(b)$.
• Types of Discontinuity:
  • Removable: The limit exists, but it does not equal the function value (or the function is undefined at that point).
  • Jump: Left and right-hand limits exist but are unequal.
  • Infinite: The function approaches infinity on one or both sides.
  • Oscillating: The function oscillates infinitely many times as it approaches the point.
• Theorem 7: Composites of Continuous Functions: If $f$ is continuous at $c$ and $g$ is continuous at $f(c)$, then the composite $g \circ f$ is continuous at $c$.
• Intermediate Value Theorem (IVT): A function $f$ that is continuous on a closed interval $[a, b]$ takes on every value between $f(a)$ and $f(b)$. If $y_0$ is between $f(a)$ and $f(b)$, then $y_0 = f(c)$ for some $c$ in $[a, b]$.

Rates of Change, Tangent Lines, and Sensitivity

• Secant Line: A line through two points on a curve. Its slope represents the average rate of change.
• Tangent Line: The limiting position of secant lines as the interval width approaches zero.
• Slope of a Curve: The slope of the curve $y = f(x)$ at point $P(a, f(a))$ is:     $m = \lim_{h \rightarrow 0} \frac{f(a + h) - f(a)}{h}$
• Normal Line: The line perpendicular to the tangent line at a given point. Its slope is the negative reciprocal of the tangent's slope.
• Sensitivity: Describes how one variable responds to small changes in another ($AT \approx \text{sensitivity} \times AD$). If $T$ is temperature and $D$ is dosage, sensitivity is:     $\lim_{\Delta D \rightarrow 0} \frac{\Delta T}{\Delta D}$
• Case Study: Fruit Fly Population:
  • Average growth rate from day 23 to day 45: $\frac{340 - 150}{45 - 23} = \frac{190}{22} \approx 8.6\,flies/day$.
  • Instantaneous growth rate at day 23: Approximately $17.5\,flies/day$ (the slope of the tangent at that point).

Questions & Discussion

• Question: Can a function have more than one horizontal asymptote?
• Response: Yes. For example, $f(x) = \frac{x}{\sqrt{x^2 + 1}}$ has horizontal asymptotes at $y = 1$ and $y = -1$.
• Question: Is the function $y = |x|$ continuous and does it have a tangent at the origin?
• Response: It is continuous everywhere, including $x = 0$. However, it does not have a tangent at $x = 0$ because the left-hand limit of the slope ($-1$) and right-hand limit of the slope ($1$) are not equal.
• Question: Is it possible to determine $\lim_{x \rightarrow c} (f(x) + g(x))$ if $\lim_{x \rightarrow c} f(x) = L$ and $\lim_{x \rightarrow c} g(x) = \infty$?
• Response: Yes, the sum will approach infinity.