Unit 06 · Term 1

Applications of Derivatives & Limits

Putting the derivative to work: shape of graphs, optimization, and L'Hôpital's rule for the indeterminate limits the algebra of limits can't reach.

By the end you can

Use the first and second derivatives to describe a graph.
Solve optimization problems with justification.
Apply L'Hôpital's rule correctly.
Find limits at infinity and asymptotes.

Why it matters & where it comes from

Why it exists

A huge share of real problems are "find the best" — biggest, cheapest, fastest. Derivatives locate those optima. L'Hôpital's rule rescues limits that collapse to the meaningless $0/0$.

In the real world

Optimization is the engine of operations research, logistics, engineering design, and machine learning (training a model is minimizing a loss function). The same calculus that sketches a curve also routes delivery trucks and tunes models.

Story & open questions

"L'Hôpital's rule" was not his. Johann Bernoulli discovered it, and the Marquis de l'Hôpital paid for the right to publish it under his own name in 1696 — one of mathematics' most famous misattributions, now impossible to undo.

Problem-solving workflow

This unit is where calculus becomes a practical tool. The hard part is often not differentiating; it is translating the problem into a one-variable function with a justified domain.

Optimization routine

Draw or name the quantities. State what is being maximized or minimized.

Use constraints to write that quantity as a function of one variable.

State the domain, including endpoints if they are allowed.

Find critical points, compare endpoint behavior, and explain why the candidate is global, not merely local.

Using L'Hôpital responsibly

L'Hôpital's rule is powerful because it converts a limit of function values into a limit of rates of change. It is also easy to misuse. Before applying it, confirm the expression is a quotient and that the quotient has form $0/0$ or $\infty/\infty$.

For $0\cdot\infty$, rewrite one factor into the denominator.
For $\infty-\infty$, combine terms into a single quotient first.
For $1^\infty$, $0^0$, or $\infty^0$, take logarithms and analyze the exponentiated limit.
If repeated differentiation gets more complicated, switch to algebra, known limits, or series when available.

Monotonicity and concavity

Tests

On an interval: $f'>0\Rightarrow$ increasing, $f'<0\Rightarrow$ decreasing. $f''>0\Rightarrow$ concave up, $f''<0\Rightarrow$ concave down. An inflection point is where concavity changes.

The first derivative test classifies a critical point by the sign change of $f'$; the second derivative test uses $f''(c)$: positive means local min, negative means local max.

Curve sketching

A reliable routine: domain → intercepts and symmetry → asymptotes → $f'$ for increase/decrease and extrema → $f''$ for concavity and inflections → plot.

Example

For $f(x)=\dfrac{x}{1+x^2}$: $f'(x)=\dfrac{1-x^2}{(1+x^2)^2}$, so $f$ increases on $(-1,1)$ and decreases outside; max at $x=1$, min at $x=-1$; horizontal asymptote $y=0$.

Optimization

Closed-interval method

A continuous $f$ on $[a,b]$ attains its extremes at critical points or endpoints. Evaluate $f$ at all of them and compare.

Example

Maximize the area of a rectangle with perimeter $20$.

Sides $x$ and $10-x$; area $A(x)=x(10-x)=10x-x^2$.

$A'(x)=10-2x=0\Rightarrow x=5$; $A''<0$, so a max. The square $5\times5$ wins, area $25$.

L'Hôpital's rule

Theorem L'Hôpital

Suppose $f$ and $g$ are differentiable on a punctured interval near the limit point, $g'(x)\ne0$ there, and $f,g$ both tend to $0$ or both diverge in magnitude. If $\lim\frac{f'}{g'}=L$ exists (possibly infinite), then, under these hypotheses, $\lim\frac{f}{g}=L$.

Example

$\displaystyle\lim_{x\to1}\frac{\ln x}{x-1}\overset{\frac00}{=} \lim_{x\to1}\frac{1/x}{1}=1.$ And $\displaystyle\lim_{x\to\infty} \frac{\ln x}{x}\overset{\frac{\infty}{\infty}}{=}\lim_{x\to\infty} \frac{1/x}{1}=0.$

Limits at infinity and asymptotes

$\lim_{x\to\infty}f(x)=L$ gives a horizontal asymptote $y=L$. For rational functions, compare leading degrees. Vertical asymptotes occur where the function blows up (often a zero denominator with nonzero numerator).

Common pitfalls

• L'Hôpital applies only to $\tfrac00$ and $\tfrac{\infty}{\infty}$. Rewrite other forms ($0\cdot\infty$, $\infty-\infty$, $1^\infty$) first.
• Don't differentiate the quotient with the quotient rule — differentiate top and bottom separately.
• Always justify a max/min (endpoints, sign of $f'$ or $f''$), don't just find a critical point.

Practice

Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.

Practice 1 · Closed interval

Find the absolute max and min of $f(x)=x^3-3x$ on $[-2,2]$.

Solution

Critical points $x=\pm1$. Values: $f(-2)=-2$, $f(-1)=2$, $f(1)=-2$, $f(2)=2$. Max $2$, min $-2$.

Practice 2 · L'Hôpital

Evaluate $\lim_{x\to0}\dfrac{e^x-1-x}{x^2}$.

Answer

Apply L'Hôpital twice: $\to\lim\frac{e^x-1}{2x}\to\lim\frac{e^x}{2}=\tfrac12$.

Practice 3 · Optimization

A box with square base and no top has volume $32$. Minimize its surface area.

Solution

$S=x^2+4xh$ with $x^2h=32$, so $S(x)=x^2+128/x$ for $x>0$. $S'=2x-128/x^2=0\Rightarrow x=4$, $h=2$. Since $S''=2+256/x^3>0$ and $S\to\infty$ at both ends of its domain, this is the global minimum.

Practice 4 · Indeterminate form

Evaluate $\lim_{x\to0^+}x\ln x$.

Answer

Rewrite as $\dfrac{\ln x}{1/x}$ ($\tfrac{-\infty}{\infty}$); L'Hôpital gives $\dfrac{1/x}{-1/x^2}=-x\to0$.

Practice 5 · Related rates

A pebble drops into a lake, making a circular ripple. When the radius is $2$ m and growing at $10$ cm/s, how fast is the enclosed area growing?

Solution

$A=\pi r^2$, so $\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}=2\pi(2)(0.1)=0.4\pi\approx1.26$ m²/s.

Practice 6 · Related rates

A $10$ m ladder leans on a wall and slides. When the top is $8$ m up and dropping at $1$ m/s, how fast is the foot sliding out?

Solution

$x^2+y^2=100$; at $y=8$, $x=6$. Differentiate: $x\,x'+y\,y'=0$, so $6x'+8(-1)=0\Rightarrow x'=\tfrac43$ m/s.

Practice 7 · Optimization

A farmer has $300$ m of fence for a rectangular field plus one fence dividing it equally down the middle. What is the largest possible area?

Solution

With the divider parallel to the width $W$: $2L+3W=300$, area $A=LW=\tfrac{W(300-3W)}2$. Then $A'=150-3W=0\Rightarrow W=50$, $L=75$, so $A=3750$ m².

Practice 8 · Indeterminate power

Evaluate $\lim_{x\to\infty}x^{1/x}$.

Answer

Take logs: $\ln\!\left(x^{1/x}\right)=\dfrac{\ln x}{x}\to0$ (L'Hôpital), so the limit is $e^0=1$.

Expanded topic checklist

Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:

Optimization. Identify candidates from endpoints, critical points, and non-differentiable points before comparing values.
Curve sketching. Use first derivative, second derivative, concavity, asymptotes, and limits as one coherent diagnostic process.
LHopital. Apply LHopital only to valid indeterminate forms and only after checking the hypotheses.
Approximation. Use derivative information to estimate local change and control error when possible.

Textbook/source alignment

Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.

Core theorem map

The spine is: derivative signs locate monotonicity, second derivatives locate curvature, and limits describe endpoint or asymptotic behavior.

Study routine. Make every application problem pass through a model-building step: variable, constraint, objective, interval, candidates, conclusion.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Optimization setup

A rectangle has perimeter $20$. Maximize its area.

Guide

Let sides be $x$ and $10-x$. Area $A=x(10-x)$ on $[0,10]$, with critical point $x=5$, so the maximum area is $25$.

Extension 2 · LHopital check

Evaluate $\lim_{x\to0}\sin x/x$ and say whether LHopital is legally applicable.

Guide

It is a $0/0$ form and the derivative quotient is $\cos x/1$, whose limit is $1$. LHopital is legal here, though the limit is often proved earlier geometrically.

Extension 3 · Concavity

Find intervals of concavity for $f(x)=x^4-2x^2$.

Guide

Use $f\prime\prime(x)=12x^2-4=4(3x^2-1)$. Concave down on $(-1/\sqrt3,1/\sqrt3)$ and up outside.

Extension 4 · Pitfall

Give an example where a critical point is not a local maximum or minimum.

Guide

For $f(x)=x^3$, $f\prime(0)=0$, but the function increases through $0$, so there is no local extremum.

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