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Lesson 70 — Consolidation Term 7: Maxima, L'Hôpital, Taylor, Newton

Integrated Differential Calculus workshop applied: optimization, curve sketching, L'Hôpital's Rule, Taylor series, concavity, marginal analysis, kinematics, and Newton-Raphson. All techniques derive from local linearization.

Used in: Year 2 High School · Equivalent Japanese Math II/III Ch. 6–7 · Equivalent German Leistungskurs Differentialrechnung

f(x) \approx \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k

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Rigorous notation, full derivation, hypotheses

Unified Theory: Linearization and Its Applications

The Mother Concept: Taylor Approximation

"The Taylor polynomial of degree $n$ centered at $x=a$ is the unique polynomial of degree $n$ that agrees with $f$ in value and in all its first $n$ derivatives at $x=a$ ." — Active Calculus §8.4

Optimization: Critical Points and the Second Derivative Test

"Finding the maximum and minimum values of a function also has practical significance because we can use this method to solve optimization problems, such as increasing profit, minimizing cost, and maximizing area." — OpenStax Calculus Vol. 1 §4.3

Concavity and Inflection Points

L'Hôpital's Rule: Ratio of Linearizations

"L'Hôpital's Rule applies whenever both $f(x) \to 0$ and $g(x) \to 0$ as $x \to a$ , or whenever $f(x) \to \pm\infty$ and $g(x) \to \pm\infty$ ." — OpenStax Calculus Vol. 1 §4.8

Idea via Taylor of Order 1. For $a = 0$ : $f(x) \approx f'(0) x$ and $g(x) \approx g'(0) x$ , so $f(x)/g(x) \approx f'(0)/g'(0)$ . L'Hôpital formalizes exactly this idea of a ratio of linearizations.

Newton-Raphson: Tangent Iteration

Newton-Raphson: the tangent to the curve at point x₀ crosses the axis at x₁, converging to the real root.

Unified Function Analysis Pipeline

Solved Examples

Example— 1· Optimization of an Open-Top Box (Application)

Problem: From a $24 \times 24$ cm sheet of cardboard, squares of side length $x$ are cut from the four corners, and the flaps are folded up to form an open-top box. Determine the value of $x$ that maximizes the volume.

Strategy: Express the volume as a function of $x$ , find the critical point via $V'(x) = 0$ , confirm it's a maximum using the second derivative test, and verify that $x$ is within the physical domain $0 < x < 12$ .

Solution:

Volume: $V(x) = x(24 - 2x)^2$ , domain: $x \in (0, 12)$ .
Expand: $V(x) = x(576 - 96x + 4x^2) = 4x^3 - 96x^2 + 576x$ .
Differentiate: $V'(x) = 12x^2 - 192x + 576 = 12(x^2 - 16x + 48) = 12(x-4)(x-12)$ .
Critical points in $(0,12)$ : $x = 4$ (the $x = 12$ is at the boundary, yielding zero volume).
Test: $V''(x) = 24x - 192$ ; $V''(4) = 96 - 192 = -96 < 0 \Rightarrow$ local maximum.
Maximum Volume: $V(4) = 4(24 - 8)^2 = 4 \cdot 256 = 1024\ \text{cm}^3$ .

Verification: $V(0) = 0$ , $V(12) = 0$ , $V(4) = 1024 > 0$ . The maximum is interior to the domain. Check: $V(4) = 4 \cdot 16^2 = 4 \cdot 256 = 1024$ .

Source. OpenStax Calculus Vol. 1 §4.7, ex. 4.41 — CC-BY-NC-SA license.

Example— 2· Indeterminate Limit via L'Hôpital's Rule (Application)

Problem: Calculate $\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$ .

Strategy: The form is $0/0$ at $x = 0$ . Apply L'Hôpital's Rule once and check if the new form is still $0/0$ ; if so, apply it again. Alternatively, use the Maclaurin series of $e^x$ to derive the result more quickly.

Solution:

Check form: $e^0 - 1 - 0 = 0$ and $0^2 = 0$ — $0/0$ indeterminate form.
1st L'Hôpital application: $\lim_{x \to 0} \frac{e^x - 1}{2x}$ Still $0/0$ (numerator $\to 0$ , denominator $\to 0$ ).
2nd application: $\lim_{x \to 0} \frac{e^x}{2} = \frac{e^0}{2} = \frac{1}{2}.$
Verification via Taylor: $e^x = 1 + x + x^2/2 + O(x^3)$ , so $e^x - 1 - x = x^2/2 + O(x^3)$ . Then $\frac{e^x - 1 - x}{x^2} \to \frac{x^2/2}{x^2} = \frac{1}{2}$ . Consistent.

Verification: For $x = 0.1$ : numerator $= e^{0.1} - 1.1 \approx 1.10517 - 1.1 = 0.00517$ ; denominator $= 0.01$ ; ratio $\approx 0.517 \approx 1/2$ .

Source. Active Calculus §2.6, exercise on indeterminate forms — CC-BY-NC-SA license.

Example— 3· Maclaurin Expansion and Limit (Intermediate)

Problem: Use the Maclaurin series of $\sin x$ to calculate $\displaystyle\lim_{x \to 0} \frac{\sin x - x + x^3/6}{x^5}$ .

Strategy: Expand $\sin x$ in Maclaurin series up to order 5, substitute into the numerator, and simplify — Taylor series is more elegant than applying L'Hôpital's Rule five times.

Solution:

Maclaurin series of $\sin x$ : $\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \cdots$
Numerator: $\sin x - x + \frac{x^3}{6} = \left(x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots\right) - x + \frac{x^3}{6} = \frac{x^5}{120} + O(x^7).$
Limit: $\lim_{x \to 0} \frac{x^5/120 + O(x^7)}{x^5} = \frac{1}{120}.$

Verification: For $x = 0.1$ : $\sin(0.1) \approx 0.099833417$ ; $\sin x - x + x^3/6 = 0.099833417 - 0.1 + 0.000166667 = 0.000000083$ ; $x^5 = 10^{-5}$ ; ratio $\approx 0.0083 \approx 1/120 = 0.00833$ .

Source. Active Calculus §8.4 — CC-BY-NC-SA license.

Example— 4· Complete Sketch of a Rational Function (Intermediate)

Problem: Sketch $f(x) = \dfrac{x^2}{x^2 - 1}$ , identifying domain, asymptotes, extrema, and concavity.

Strategy: Follow the 7-step function analysis pipeline. The function is rational with easily identifiable denominator zeros.

Solution:

Domain: $x^2 - 1 \neq 0 \Rightarrow x \neq \pm 1$ . Domain: $\mathbb{R} \setminus \{-1, 1\}$ .
Intercepts/Symmetry: $f(0) = 0$ (y-intercept). Zeros: $x^2 = 0 \Rightarrow x = 0$ . $f(-x) = f(x)$ : even function (symmetric about the y-axis).
Asymptotes: vertical at $x = \pm 1$ ; horizontal: $\lim_{x\to\pm\infty} f(x) = 1$ (asymptote $y = 1$ ).
$f'$ : $f'(x) = \dfrac{-2x}{(x^2-1)^2}$ . Critical point: $x = 0$ . Signs: $f' > 0$ for $x < 0$ ; $f' < 0$ for $x > 0$ — local maximum at $x = 0$ .
$f''$ : $f''(x) = \dfrac{2(3x^2+1)}{(x^2-1)^3}$ . For $x \in (-1,1)$ : denominator $< 0$ , numerator $> 0$ , thus $f'' < 0$ (concave down). For $|x| > 1$ : $f'' > 0$ (concave up). No inflection points in the domain (sign change of $f''$ occurs at points outside the domain $x = \pm 1$ ).
Extremum: $f(0) = 0$ — local (and absolute) maximum in $(-1,1)$ .

Verification: $f(0) = 0 < 1 = \lim_{x\to\infty} f(x)$ makes sense: in the interval $(-1,1)$ the function goes from $0$ down to $-\infty$ at the boundaries and back up; the maximum is $f(0)=0$ in this interval.

Source. APEX Calculus §3.5, ex. 42 — CC-BY-NC license.

Example— 5· Newton-Raphson for a Transcendental Equation (Real Modeling)

Problem: An investment of R$ 10,000 grows at a monthly rate $r$ to be determined. After 12 months, the amount is R$ 11,268. Solve $f(r) = (1+r)^{12} - 1.1268 = 0$ using Newton-Raphson method, with $x_0 = 0.01$ (1% per month).

Strategy: $f'(r) = 12(1+r)^{11}$ . Iterate $x_{n+1} = x_n - f(x_n)/f'(x_n)$ until stabilization.

Solution:

Iteration 1: $f(0.01) = (1.01)^{12} - 1.1268 = 1.12683 - 1.1268 = 0.00003$ . $f'(0.01) = 12 \cdot (1.01)^{11} = 12 \cdot 1.11567 = 13.388$ . $x_1 = 0.01 - 0.00003/13.388 \approx 0.009998$ .

After 2 iterations, $r^* \approx 1.00\%$ per month (annual nominal rate $\approx 12.68\%$ p.a.).

Verification: $(1.01)^{12} = 1.12683 \approx 1.1268$ . Consistent with the problem statement.

Source. APEX Calculus §6.1, Newton-Raphson application — CC-BY-NC license.

Exercise list

40 exercises · 10 with worked solution (25%)

15 15 2 5 3

Ex. 70.1Answer key
Find the critical points, local extrema, and inflection point of $f(x) = x^3 - 6x^2 + 9x + 1$ .
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Ex. 70.2
Maximize $f(x) = xe^{-x}$ on $[0, +\infty)$ . What is the absolute maximum?
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Ex. 70.3
Sketch $f(x) = (x^2 - 4)/x$ . Identify asymptotes, monotonicity, and concavity.
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Ex. 70.4
Find the absolute minimum of $f(x) = x + 4/x$ on $(0, +\infty)$ .
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Ex. 70.5Answer key
A cylindrical can with $V = 1000\ \text{cm}^3$ must be constructed using minimal material. Determine the optimal radius and height.
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Ex. 70.6
From a $20 \times 30\ \text{cm}$ piece of cardboard, squares of side $x$ are cut from the corners, and the flaps are folded up. What value of $x$ maximizes the volume of the box?
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Ex. 70.7
Find the point on the parabola $y = x^2$ closest to the point $(3, 0)$ .
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Ex. 70.8Answer key
A fence of $200\ \text{m}$ is to enclose a rectangular area against a wall (the wall forms one side). Maximize the area.
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Ex. 70.9
Determine the inflection points and concavity of $f(x) = e^{-x^2}$ .
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Ex. 70.10
Perform a complete sketch of $f(x) = \ln(1 + x^2)$ : domain, symmetry, extrema, inflection points, asymptotic behavior.
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Ex. 70.11Answer key
L'Hôpital's Rule applies directly to $\lim_{x \to 0} \sin x / x$ (form $0/0$ ). Which alternative describes a case where the rule does not directly apply?
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Ex. 70.12
Calculate $\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}$ .
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Ex. 70.13
Calculate $\lim_{x \to +\infty} \dfrac{x^2}{e^x}$ .
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Ex. 70.14
Calculate $\lim_{x \to 0^+} x^{\sin x}$ (indeterminate form $0^0$ ).
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Ex. 70.15
Write the Maclaurin polynomial of $\sin x$ of order 5 ( $P_5$ ).
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Ex. 70.16
Use Taylor expansion to calculate $\lim_{x \to 0} \dfrac{\tan x - \sin x}{x^3}$ .
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Ex. 70.17Answer key
Approximate $\sqrt{1.1}$ using the Maclaurin series of $(1+x)^{1/2}$ up to order 3.
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Ex. 70.18
Approximate $e^{-0.1}$ with the Maclaurin polynomial of $e^x$ of order 3. Calculate the error.
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Ex. 70.19
Calculate $\lim_{x \to 0} \dfrac{x - \sin x}{x^3}$ using Taylor series.
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Ex. 70.20
Write the Maclaurin series for $\ln(1+x)$ up to the $x^5$ term.
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Ex. 70.21
Calculate $\lim_{x \to 0} \dfrac{e^x - \cos x - x}{x^2}$ via Taylor series.
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Ex. 70.22
What is the interval of convergence for the Maclaurin series of $\ln(1 + x)$ ?
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Ex. 70.23
Write the Maclaurin polynomial $P_6(x)$ for $\cos x$ .
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Ex. 70.24
Calculate $\lim_{x \to +\infty} \left(1 + \dfrac{2}{x}\right)^x$ (form $1^\infty$ ).
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Ex. 70.25
Cost $C(q) = q^2/4 + 5q + 100$ (Reais), price $p = 50$ Reais/unit. Find the quantity $q^*$ and maximum profit $\pi^*$ .
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Ex. 70.26Answer key
A monopolist faces demand $q = 100 - 2p$ and has cost $C(q) = 10q$ . Find $q^*$ , $p^*$ , and the maximum profit.
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Ex. 70.27
A ball is thrown vertically upward with an initial velocity $v_0 = 20\ \text{m/s}$ ( $g = 10\ \text{m/s}^2$ ). Calculate the maximum height and the time of flight.
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Ex. 70.28Answer key
Given $s(t) = t^3 - 9t^2 + 24t$ . When is $v(t) = 0$ ? Calculate the total distance traveled over $0 \leq t \leq 5$ .
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Ex. 70.29
Mass-spring system: $m = 0.5\ \text{kg}$ , $k = 50\ \text{N/m}$ , amplitude $A = 0.1\ \text{m}$ . Calculate the period and the maximum velocity.
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Ex. 70.30
Use Newton-Raphson on $f(x) = x^2 - 7$ with $x_0 = 3$ to calculate $\sqrt{7}$ to 5 decimal places.
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Ex. 70.31
Use Newton-Raphson on $f(x) = x^3 - 2$ with $x_0 = 1$ to approximate $\sqrt[3]{2}$ . Perform 3 iterations.
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Ex. 70.32
Apply Newton-Raphson to $f(x) = e^x - 3x$ to find both real roots. Use $x_0 = 0$ for one and $x_0 = 1.5$ for the other.
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Ex. 70.33
Kepler's equation is $E - 0.3\sin E = 1$ . Use Newton-Raphson with $E_0 = 1$ and perform 4 iterations.
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Ex. 70.34
Show that a strictly convex ( $f'' > 0$ ) $C^2$ function on $\mathbb{R}$ has at most one minimum point.
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Ex. 70.35Answer key
Use the Maclaurin series of $\sin x$ to prove that $\sin x < x$ for all $x > 0$ .
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Ex. 70.36
Sketch $f(x) = x^x$ on $(0, +\infty)$ . Find the minimum and analyze the behavior at the domain's extremes.
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Ex. 70.37
Newton-Raphson applied to $f(x) = \arctan x$ with $x_0 = 2$ diverges. Explain geometrically why, and show numerically.
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Ex. 70.38Answer key
Prove via Taylor series: if $f'(a) = 0$ and $f^{(k)}(a)$ is the first non-zero derivative at $a$ , then $a$ is an extremum if $k$ is even, and a saddle point/inflection point if $k$ is odd.
Ex. 70.39
Show, via Taylor of order 1, that $\lim_{x\to a} f(x)/g(x) = f'(a)/g'(a)$ when $f(a) = g(a) = 0$ and $g'(a) \neq 0$ .
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Ex. 70.40Answer key
Formally derive the $MR = MC$ condition for maximum profit. Explain why a monopolist produces less than a competitive firm.
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Sources

Active Calculus — Matt Boelkins, David Austin, Steve Schlicker · Grand Valley State University · 2024 · CC-BY-NC-SA. Sections §2.6 (L'Hôpital's Rule), §3.1–3.4 (optimization), §8.4–8.5 (Taylor series).
APEX Calculus — Gregory Hartman et al. · Virginia Military Institute · 2024 · CC-BY-NC. Chapters 3 (function analysis), 4 (applications), 6 (Newton's Method and applications), 8 (Taylor series).
Calculus Volume 1 — OpenStax (Strang, Herman et al.) · 2023 · CC-BY-NC-SA. Sections §4.3 (max-min), §4.7 (applied optimization), §4.8 (L'Hôpital's Rule), §4.9 (Newton's Method).