Comprehensive Exercises

These problems reinforce core skills: derivative definitions, rule fluency, composite and implicit cases, mean value theorems, L’Hôpital’s rule, and applications such as extrema and curvature.

基础概念

练习 1

Find $f'(1)$ for $f(x) = x^2 + 2x + 1$ .

Reference Answer (2 个标签)

derivatives differentials

$f'(x) = 2x + 2$ , so $f'(1) = 4$ . (Using the limit definition yields the same result.)

练习 2

Check continuity and differentiability of $f(x) = |x|$ at $x=0$ .

Reference Answer (2 个标签)

derivatives differentials

$\lim_{x\to 0}|x| = 0 = f(0)$ ⇒ continuous.
Left derivative $=-1$ , right derivative $=1$ ⇒ not differentiable.

Differentiation Rules

练习 3

Differentiate $f(x) = x^3 \sin x$ .

Reference Answer (2 个标签)

derivatives differentials

Product rule: $f'(x) = 3x^2\sin x + x^3\cos x$ .

练习 4

Differentiate $f(x) = \dfrac{x^2 + 1}{x + 1}$ .

Reference Answer (2 个标签)

derivatives differentials

Quotient rule: $f'(x) = \dfrac{x^2 + 2x - 1}{(x+1)^2}$ .

练习 5

Differentiate $f(x) = \sin(e^{x^2})$ .

Reference Answer (2 个标签)

derivatives differentials

Chain rule (three layers): $f'(x) = 2x\,e^{x^2}\cos(e^{x^2})$ .

复合与隐函数

练习 6

Differentiate $f(x) = \arcsin(x^2)$ .

Reference Answer (2 个标签)

derivatives differentials

Let $y=\arcsin(x^2)$ , so $x^2=\sin y$ . Then $2x = \cos y\;y'$ and $y' = \dfrac{2x}{\sqrt{1 - x^4}}$ .

练习 7

For $x^3 + y^3 = 3xy$ , find $dy/dx$ .

Reference Answer (2 个标签)

derivatives differentials

$3x^2 + 3y^2y' = 3y + 3xy'$ ⇒ $y' = \dfrac{y - x^2}{y^2 - x}$ .

练习 8

For $x=t^2,\ y=t^3$ , compute $\dfrac{d^2 y}{dx^2}$ .

Reference Answer (2 个标签)

derivatives differentials

$\dfrac{dy}{dx} = \dfrac{3t^2}{2t} = \dfrac{3t}{2}$ , and $\dfrac{d^2y}{dx^2} = \dfrac{3}{4t}$ .

Mean Value Theorems

$\xi$ （Xi）：中值定理中出现的内部点，读作“克西”。

练习 9

Show that if $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a)=f(b)$ , then $\exists \xi\in(a,b)$ with $f'(\xi)=0$ .

Reference Answer (2 个标签)

derivatives differentials

Directly by Rolle’s theorem.

练习 10

Prove $|\sin x - \sin y| \le |x - y|$ .

Reference Answer (2 个标签)

derivatives differentials

MVT on $f(t)=\sin t$ gives $\cos \xi = \dfrac{\sin y - \sin x}{y-x}$ with $|\cos \xi|\le 1$ .

L’Hopital’s Rule

练习 11

Compute $\lim_{x\to 0} \dfrac{\sin x}{x}$ .

Reference Answer (2 个标签)

derivatives differentials

$\tfrac{0}{0}$ type → $\lim \dfrac{\cos x}{1} = 1$ .

练习 12

Compute $\lim_{x\to \infty} \dfrac{x^2}{e^x}$ .

Reference Answer (2 个标签)

derivatives differentials

Two rounds: $\dfrac{2x}{e^x} \to \dfrac{2}{e^x} \to 0$ .

练习 13

Compute $\lim_{x\to 0^+} x\ln x$ .

Reference Answer (2 个标签)

derivatives differentials

Rewrite $\dfrac{\ln x}{1/x}$ ( $\tfrac{\infty}{\infty}$ ). Derivatives give $-x \to 0$ .

Applications of Derivatives

练习 14

Find extrema of $f(x) = x^3 - 3x^2 + 2$ .

Reference Answer (2 个标签)

derivatives differentials

$f'(x)=3x(x-2)$ ⇒ critical points $0,2$ .
$f''(0)<0$ ⇒ local max at $(0,2)$ .
$f''(2)>0$ ⇒ local min at $(2,-2)$ .

练习 15

Tangent to $y=\ln x$ at $(1,0)$ .

Reference Answer (2 个标签)

derivatives differentials

$f'(1)=1$ ⇒ $y-0 = 1(x-1)$ ⇒ $y = x-1$ .

练习 16

Max/min of $f(x) = x^3 - 3x^2 + 2$ on $[0,3]$ .

Reference Answer (2 个标签)

derivatives differentials

Check $x=0,2,3$ : $f(0)=2$ , $f(2)=-2$ , $f(3)=2$ . Max $2$ , min $-2$ .

综合应用

练习 17

If $y = x^x$ , find $y'$ .

Reference Answer (2 个标签)

derivatives differentials

Log-differentiate: $\ln y = x\ln x$ ⇒ $\dfrac{y'}{y} = \ln x + 1$ ⇒ $y' = x^x(\ln x + 1)$ .

练习 18

For $f(x) = x^3 - 3x^2 + 2$ , find concavity intervals and inflection point.

Reference Answer (2 个标签)

derivatives differentials

$f''(x)=6(x-1)$ .
Concave down on $(-\infty,1)$ , up on $(1,\infty)$ .
Inflection at $(1,0)$ .

练习 19

Verify Cauchy MVT for $f(x)=x^2$ , $g(x)=x^3$ on $[1,2]$ .

Reference Answer (2 个标签)

derivatives differentials

$\dfrac{2\xi}{3\xi^2} = \dfrac{3}{7}$ ⇒ $\xi = \dfrac{14}{9}$ .

练习 20

Curvature of $y = x^2$ at $(1,1)$ .

Reference Answer (2 个标签)

derivatives differentials

$f'(1)=2,\ f''(1)=2$ . Curvature $\kappa = \dfrac{|2|}{[1+2^2]^{3/2}} = \dfrac{2}{5\sqrt{5}}$ .

$\kappa$ （kappa）：希腊字母，读作“卡帕”，用于表示曲率。

Summary

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\xi$	希腊字母	Xi（克西）	中值定理中的内部点
$\kappa$	希腊字母	Kappa（卡帕）	曲率符号

中英对照

中文术语	英文术语	音标	说明
练习题	exercises	/ˈeksəsaɪzɪz/	巩固与自测的题目
曲率	curvature	/ˈkɜːvətʃə/	曲线弯曲程度的度量

Previous导数的应用学习导数在研究函数性质中的应用，包括单调性、极值、凹凸性、拐点、最大最小值等。

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