Applications of Differential and Integral Calculus

date2026-07-15document_iddoc_a8836c05a3377b058a8821f1a194227edescription局所的な変化、区間上の累積、導関数の総変化を区別し、応用問題で微分と積分を選択する基準を整理する。prerequisites微分法の基本 / 局所線型近似と微分可能性 / 微分公式と計算法 / 積分法の基本 / 積分公式と計算法 / 微分積分学の基本定理 / 関数のグラフtype講義content_typelecturestatusactive

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1Introduction

This lecture classifies whether a problem asks for local change, accumulation over an interval, or total change obtained from a derivative, and explains how that classification determines the choice between differentiation and integration. Use differentiation for tangent lines, instantaneous velocity, monotonicity, and extrema. Use integration for area, mass, average value, and displacement. The fundamental theorem of calculus connects a local rate of change to the change over an entire interval.

2Local Quantities and Differentiation

If $f$ is differentiable at $a$ , then as $h \to 0$ ,

f (a + h) = f (a) + f^{'} (a) h + r (h), \frac{r (h)}{h} \to 0.

This linear approximation identifies $f^{'} (a)$ as the local rate of change at $a$ and shows that the error term $r (h)$ is of higher order than $h$ . Consequently, tangent lines, instantaneous velocity, and local approximation require a derivative.

If $f$ is differentiable on an interval, then $f$ increases where $f^{'} (x) > 0$ and decreases where $f^{'} (x) < 0$ . In this lecture, an interior point $c$ of the domain is called a critical point if $f^{'} (c) = 0$ or if $f$ is not differentiable at $c$ . If $f$ has a local extremum at an interior point $c$ and is differentiable there, Fermat's stationary-point theorem gives $f^{'} (c) = 0$ . The converse is false: one must also check a sign change of the derivative or use another valid test.

2.1Example: Local Extrema

For $f (x) = x^{3} - 3 x$ ,

f^{'} (x) = 3 (x - 1) (x + 1) .

The sign of $f^{'}$ changes from positive to negative at $x = - 1$ and from negative to positive at $x = 1$ . Thus $f$ has a local maximum at $x = - 1$ and a local minimum at $x = 1$ .

3Accumulated Quantities and Integration

Integration partitions an interval, sums local contributions, and takes a limit. If a continuous function $f$ is nonnegative on $[a, b]$ , then

\int_{a}^{b} f (x) d x

is the area between its graph and the $x$ -axis. Integrating the linear density of a rod with respect to position gives its mass, and integrating a flow rate with respect to time gives the total amount transferred. Multiplying the unit of the integrand by the unit of the integration variable gives the unit of the accumulated quantity.

3.1Average Value of a Function

Let $a < b$ and let $f$ be continuous on $[a, b]$ . The average value of $f$ is the constant that produces the same total accumulation:

f_{a v g} = \frac{1}{b - a} \int_{a}^{b} f (x) d x .

For example, the average value of $f (x) = x^{2}$ on $[0, 3]$ is

\frac{1}{3} \int_{0}^{3} x^{2} d x = 3.

The definite integral is the total accumulation; dividing it by the interval length gives the average value.

4Total Change and the Fundamental Theorem

Let $s$ denote position and let $v = s^{'}$ denote velocity. If $v$ is continuous on $[a, b]$ , the fundamental theorem gives

\int_{a}^{b} v (t) d t = s (b) - s (a) .

The left-hand side accumulates an instantaneous rate, while the right-hand side is the total change in position.

Displacement and distance traveled are different quantities. Displacement is

\int_{a}^{b} v (t) d t,

whereas distance traveled is

\int_{a}^{b} | v (t) | d t .

When velocity changes sign, motion in opposite directions cancels in displacement but not in distance traveled.

4.1Example: Displacement and Distance

Let $v (t) = t - 1$ for $0 [PARSE ERROR: Undefined("Command(\"le\")")] t [PARSE ERROR: Undefined("Command(\"le\")")] 2$ . The displacement is

\int_{0}^{2} (t - 1) d t = 0.

Since velocity changes sign at $t = 1$ , the distance traveled is

\int_{0}^{1} (1 - t) d t + \int_{1}^{2} (t - 1) d t = 1.

5Selection Criteria

Requested quantity	Primary method	Required check
Tangent line, instantaneous velocity, local approximation	Differentiation	Is the function differentiable at the point?
Monotonicity	Sign of the derivative	What is the sign of $f^{'}$ on each subinterval?
Local extremum	Behavior on both sides of a critical point	Does the sign of $f^{'}$ change?
Area, mass, or total amount	Integration	What are the local contribution and interval?
Average value	Integral divided by interval length	Is $a < b$ ?
Total change	Integral of a derivative	Do the hypotheses of the fundamental theorem hold?
Distance traveled	Integral of speed	Has the interval been split where velocity changes sign?

6Conditions and Common Errors

The equation $f^{'} (c) = 0$ alone does not imply that $c$ is an extremum. Points where $f$ is not differentiable and endpoints of the domain may also be candidates. If $f$ is continuous on a closed interval $[a, b]$ , find an absolute maximum or minimum by comparing its values at all interior critical points and both endpoints.

To calculate geometric area, split the interval wherever the ordering of the curves or the sign of a function changes. A definite integral is a signed accumulation and does not always represent geometric area.

7Exercises

The following exercise set also includes partial derivatives, multiple integrals, and differential equations. Use it after completing the corresponding later lectures.

data/exercise/math/calculus/advanced-calculus-applications.exercise.n.md

8Related Material

data/lecture/math/calculus/differentiation-rules-and-computation.lecture.n.md data/lecture/math/calculus/local-linear-approximation-and-differentiability.lecture.n.md data/lecture/math/calculus/integration-rules-and-computation.lecture.n.md data/lecture/math/calculus/fundamental-theorem-of-calculus.lecture.n.md