calculus

Section	Question Type	Part	Calculator	Number of Questions	Timing
I	Multiple-choice questions	A	No	30	60 min
		B	Yes	15	45 min
II	Free-response questions	A	Yes	2	30 min
		B	No	4	60 min

Limit

Methods

• 代入
• 變形
• 左右
• 漸進

Four Special Limits

• \(\displaystyle \lim_{x\to 0}\frac{\sin{x}}{x}=1\)
• \(\displaystyle \lim_{x\to 0}\frac{\cos{x}-1}{x}=0\)
• \(\displaystyle \lim_{x\to 0}\frac{e^x-1}{x}=1\)
• \(\displaystyle \lim_{x\to 0}\frac{\ln{(1+x)}}{x}=1\)

The Squeeze Theorem

Continuity

Definition

\[ \lim_{x\to a} f(x)=f(a) \]

The Intermediate Value Theorem

Derivative

\[ \begin{array}{rcl} f'(a) &=& \displaystyle \lim_{x\to a}\frac{f(x)-f(a)}{x-a} \\ f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \end{array} \]

Differentiation Rules

The Derivative of a Constant Function

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle c \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{c-c}{h}=\lim_{h\to 0}\frac{0}{h}=0\\ \end{array} \]

The Constant Multiple Rule

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle cg(x) \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{cg(x+h)-cg(x)}{h}=c\left(\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}\right)=cg'(x)\\ \end{array} \]

The Sum Rule

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle g(x)+k(x) \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{[g(x+h)+k(x+h)]-[g(x)+k(x)]}{h} \\ &=& \displaystyle \lim_{h\to 0}\frac{g(x+h)-g(x)}{h}+\lim_{h\to 0}\frac{k(x+h)-k(x)}{h}=g'(x)+k'(x) \end{array} \]

The Power Rule

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle x^m \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{(x+h)^m-x^m}{h} \\ &=& \displaystyle \lim_{h\to 0}\frac{\binom{m}{m}x^m+\binom{m}{m-1}x^{m-1}h+\binom{m}{m-2}x^{m-2}h^2+\cdots +\binom{m}{2}x^2 h^{m-2}+\binom{m}{1}x h^{m-1}+\binom{m}{0}h^m-x^m}{h} \\ &=& \displaystyle \lim_{h\to 0}\binom{m}{m-1}x^{m-1}+\binom{m}{m-2}x^{m-2}h+\cdots +\binom{m}{2}x^2 h^{m-3}+\binom{m}{1}x h^{m-2}+\binom{m}{0}h^{m-1} \\ &=& \displaystyle mx^{m-1} \end{array} \]

Polynomial

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle a_n x^n+a_{n-1}x^{n-1}+\cdots+a_2 x^2+a_1 x+a_0 \\ \frac{dy}{dx} =f'(x) &=& \displaystyle a_n n x^{n-1}+a_{n-1} (n-1) x^{n-2}+\cdots +a_2\cdot 2 x+a_1 \end{array} \]

The Product Rule

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle g(x)\cdot k(x) \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{g(x+h)\cdot k(x+h)-g(x)\cdot k(x)}{h} \\ &=& \displaystyle \lim_{h\to 0}\frac{g(x+h)k(x+h)-g(x+h)k(x)+g(x+h)k(x)-g(x)k(x)}{h} \\ &=& \displaystyle \lim_{h\to 0}g(x+h)\frac{k(x+h)-k(x)}{h}+\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}k(x) \\ &=& \displaystyle g(x)k'(x)+g'(x)k(x) \end{array} \]

The Quotient Rule

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle \frac{g(x)}{k(x)} \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{\frac{g(x+h)}{k(x+h)}-\frac{g(x)}{k(x)}}{h} \\ &=& \displaystyle \lim_{h\to 0}\frac{k(x)g(x+h)-g(x)k(x+h)}{k(x+h)k(x)h} \\ &=& \displaystyle \lim_{h\to 0}\frac{1}{k(x+h)k(x)}\cdot \frac{k(x)g(x+h)-k(x)g(x)+g(x)k(x)-g(x)k(x+h)}{h} \\ &=& \displaystyle \lim_{h\to 0}\frac{1}{k(x+h)k(x)}\left[k(x)\frac{g(x+h)-g(x)}{h}-g(x)\frac{k(x+h)-k(x)}{h}\right] \\ &=& \displaystyle \frac{k(x)g'(x)-g(x)k'(x)}{[k(x)]^2} \end{array} \]

Chain Rule

Trigonometric Function

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle \sin{x} \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{\sin{(x+h)}-\sin{x}}{h} =\lim_{h\to 0}\frac{\sin{x}\cos{h}+\sin{h}\cos{x}-\sin{x}}{h} \\ &=& \displaystyle \sin{x}\left(\lim_{h\to 0}\frac{\cos{h}-1}{h}\right)+\cos{x}\left(\lim_{h\to 0}\frac{\sin{h}}{h}\right)=\cos{x} \end{array} \]

Implicit Differentiation

Exponential Logarithm---Approach 1

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle \ln{x} \\ \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{\ln{\left(1+\frac{h}{x}\right)}}{h} \\ &\stackrel{u=\frac{h}{x}, h=ux}{=}& \displaystyle \lim_{h\to 0}\frac{\ln{(1+u)}}{ux}=\frac{1}{x}\left(\lim_{h\to 0}\frac{\ln{(1+u)}}{u}\right)=\frac{1}{x} \end{array} \] \[ \begin{array}{rcl} y &=& e^x \\ \ln{y} &=& x \\ \displaystyle \frac{dy}{dx}\frac{1}{y} &=& 1 \\ \displaystyle \frac{dy}{dx} &=& y=e^x \end{array} \]

Exponential Logarithm---Approach 2

\[ \begin{array}{rcl} y=f(x) &=& \displaystyle e^x \\ \displaystyle \frac{dy}{dx} =f'(x) &=& \displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{e^{x+h}-e^x}{h}=e^x\left(\lim_{h\to 0}\frac{e^h-1}{h}\right)=e^x \end{array} \] \[ \begin{array}{rcl} y &=& \ln{x} \\ e^y &=& x \\ \displaystyle \frac{dy}{dx}e^y &=& 1 \\ \displaystyle \frac{dy}{dx} &=& \displaystyle \frac{1}{e^y}=\frac{1}{x} \end{array} \]

Derivative of Inverse Trigonometric Function

\[ \begin{array}{rcl} y=f(x) &=& \sin^{-1}(x) \\ \sin{y} &=& x \\ \displaystyle \frac{dy}{dx}\cos{y} &=& 1 \\ \displaystyle \frac{dy}{dx} &=& \displaystyle \frac{1}{\cos{y}} \\ \displaystyle \frac{dy}{dx} &=& \displaystyle \frac{1}{\sqrt{1-x^2}} \\ \end{array} \]

\(1\)
	⊿	\(x\)
	\(\sqrt{1-x^2}\)

The Derivative of the Inverse Function

\[ \begin{array}{rcl} y=f(x) &=& \\ \displaystyle \left.\frac{dy}{dx}\right]_{x=a} =(f^{-1})'(a) &=& \displaystyle \lim_{x\to a}\frac{f^{-1}(x)-f^{-1}(a)}{x-a} \\ f^{-1}(x)=y && x=f(y) \\ f^{-1}(a)=b && a=f(b) \\ &=& \displaystyle \lim_{y\to b}\frac{y-b}{f(y)-f(b)}=\frac{1}{\lim_{y\to b}\frac{f(y)-f(b)}{y-b}}=\frac{1}{f'(b)}=\frac{1}{f'(f^{-1}(a))} \\ \end{array} \]

Application

Higher Derivatives

\[ \frac{dy}{dx}=f'(x), \frac{d^2y}{dx^2}=f''(x), \frac{d^3y}{dx^3}=f'''(x), ..., \frac{d^ny}{dx^n}=f^{(n)}(x), \]

Linear Approximation

We have seen that a curve lies very close to its tangent line near the point of tangency. In fact, by zooming in toward a point on the graph of a differentiable function, we noticed that the graph looks more and more like its tangent line. (See Figure 2.1.2.) This observation is the basis for a method of finding approximate values of functions.

The idea is that it might be easy to calculate a value \(f(a)\) of a function, but difficult (or even impossible) to compute nearby values of \(f\). So we settle for the easily computed values of the linear function \(L\) whose graph is the tangent line of (f\) at \((a, f(a))\). (See Figure 1.)

In other words, we use the tangent line at \((a, f(a))\) as an approximation to the curve \(y=f(x)\) when \(x\) is near \(a\). An equation of this tangent line is \[ y=f(a)+f'(a)(x-a) \] and the approximation \[ f(x)\approx f(a)+f'(a)(x-a) \] is called the linear approximation or tangent line approximation of \(f\) at \(a\). The linear function whose graph is this tangent line, that is, \[ L(x)=f(a)+f'(a)(x-a) \] is called the linearization of \(f\) at \(a\).

Indeterminate Forms and L'Hospital's Rule

Antiderivatives

Rate of Change

Related Rate

畫圖

Maxima and Minima (See OpenStax)

Definition
Let \(f\) be a function defined over an interval \(I\) and let \(c\in I\). We say \(f\) has an absolute maximum on \(I\) at \(c\) if \(f(c)\geq f(x)\) for all \(x\in I\). We say \(f\) has an absolute minimum on \(I\) at \(c\) if \(f(c)\leq f(x)\) for all \(x\in I\). If \(f\) has an absolute maximum on \(I\) at \(c\) or an absolute minimum on \(I\) at \(c\), we say \(f\) has an absolute extremum on \(I\) at \(c\).

Extreme Value Theorem
If \(f\) is a continuous function over the closed, bounded interval \([a, b]\), then there is a point in \([a, b]\) at which \(f\) has an absolute maximum over \([a, b]\) and there is a point in \([a, b]\) at which \(f\) has an absolute minimum over \([a, b]\).

Definition
A function \(f\) has a local maximum at \(c\) if there exists an open interval \(I\) containing \(c\) such that \(I\) is contained in the domain of \(f\) and \(f(c)\geq f(x)\) for all \(x\in I\). A function \(f\) has a local minimum at \(c\) if there exists an open interval \(I\) containing \(c\) such that \(I\) is contained in the domain of \(f\) and \(f(c)\leq f(x)\) for all \(x\in I\). A function \(f\) has a local extremum at \(c\) if \(f\) has a local maximum at \(c\) or \(f\) has a local minimum at \(c\)

Note that the endpoints are not able to be local extrema.

extreme形容詞
extremum單數
extrema複數

Definition
Let \(c\) be an interior point in the domain of \(f\). We say that \(c\) is a critical number of \(f\) if \(f'(c)=0\) or \(f'(c)\) is undefined. We call the point \((c, f(c))\) a critical point of \(f\). Note that these two terms are often used interchangeably in this text and elsewhere.

Fermat's Theorem
If \(f\) has a local extremum at \(c\) and \(f\) is differentiable at \(c\), then \(f'(c)=0\).

Location of Absolute Extrema
Let \(f\) be a continuous function over a closed, bounded interval \(I\). The absolute maximum of \(f\) over \(I\) and the absolute minimum of \(f\) over \(I\) must occur at endpoints of \(I\) or at critical points of \(f\) in \(I\).

The Mean Value Theorem (See OpenStax)

不要講Rolle's Theorem，AP Calculus不考，學生常搞混。

Mean Value Theorem
Let \(f\) be continuous over the closed interval \([a, b]\) and differentiable over the open interval \((a, b)\). Then, there exists at least one point \(c\in (a, b)\) such that \[ f'(c)=\frac{f(b)-f(a)}{b-a}. \]

Definition (See Section 1.1)
We say that a function \(f\) is increasing on the interval \(I\) if for all \(x_1, x_2\in I\), \[ f(x_1)\leq f(x_2)\text{ when }x_1\lt x_2. \] We say \(f\) is strictly increasing on the interval \(I\) if for all \(x_1, x_2\in I\), \[ f(x_1)\lt f(x_2)\text{ when }x_1\lt x_2. \] We say that a function \(f\) is decreasing on the interval \(I\) if for all \(x_1, x_2\in I\), \[ f(x_1)\geq f(x_2)\text{ if }x_1\lt x_2. \] We say that a function \(f\) is strictly decreasing on the interval \(I\) if for all \(x_1, x_2\in I\), \[ f(x_1)\gt f(x_2)\text{ if }x_1\lt x_2. \]

Increasing and Decreasing Functions
Let \(f\) be continuous over the closed interval \([a, b]\) and differentiable over the open interval \((a, b)\).
i. If \(f'(x)\gt 0\) for all \(x\in (a, b)\), then \(f\) is an increasing function over \([a, b]\).
ii. If \(f'(x)\lt 0\) for all \(x\in (a, b)\), then \(f\) is a decreasing function over \([a, b]\).

Derivatives and the Shape of a Graph (See OpenStax)

First Derivative Test
Suppose that \(f\) is a continuous function over an interval \(I\) containing a critical point \(c\). If \(f\) is differentiable over \(I\), except possibly at point \(c\), then \(f(c)\) satisfies one of the following descriptions:
i. If \(f'\) changes sign from positive when \(x\lt c\) to negative when \(x\gt c\), then \(f(c)\) is a local maximum of \(f\).
ii. If \(f'\) changes sign from negative when \(x\lt c\) to positive when \(x\gt c\), then \(f(c)\) is a local minimum of \(f\).
iii. If \(f'\) has the same sign for \(x\lt c\) and \(x\gt c\), then \(f(c)\) is neither a local maximum nor a local minimum of \(f\).

Definition
Let \(f\) be a function that is differentiable over an open interval \(I\). If \(f'\) is increasing over \(I\), we say \(f\) is concave up over \(I\). If \(f'\) is decreasing over \(I\), we say \(f\) is concave down over \(I\).

Test for Concavity
Let \(f\) be a function that is twice differentiable over an interval \(I\).
i. If \(f''(x)\gt 0\) for all \(x\in I\), then \(f\) is concave up over \(I\).
ii. If \(f''(x)\lt 0\) for all \(x\in I\), then \(f\) is concave down over \(I\).

Definition
If \(f\) is continuous at \(a\) and \(f\) changes concavity at \(a\), the point \((a, f(a))\) is an inflection point of \(f\).

Second Derivative Test
Suppose \(f'(c)=0\), \(f''\) is continuous over an interval containing \(c\).
i. If \(f''(c)\gt 0\), then \(f\) has a local minimum at \(c\).
ii. If \(f''(c)\lt 0\), then \(f\) has a local maximum at \(c\).
iii. If \(f''(c)=0\), then the test is inconclusive.

Summary

f'>0	⇒	f↗,	f'<0	⇒	f↘
☐'>0	⇒	☐↗,	☐'<0	⇒	☐↘
f''>0	⇒	f'↗,	f''<0	⇒	f'↘

	f'>0⇒f↗	f'<0⇒f↘
f''>0⇒f'↗	⤴	⤷
f''<0⇒f'↘	Γ	⤵

a\s\v	v=s'>0	v=s'<
a=s''>0	⤴	⤷
a=s''<0	Γ	⤵

x³   x=0   f ↗   ↗ f' + 0 + f'' - 0 +	-x³   x=0   f ↘   ↘ f' - 0 - f'' + 0 -	x⁴   x=0   f ↘   ↗ f' - 0 + f'' + 0 +	-x⁴   x=0   f ↗   ↘ f' + 0 - f'' - 0 -
(sgn x)√|x|   x=0   f ↗   ↗ f' + X + f'' + X -	-(sgn x)√|x|   x=0   f ↘   ↘ f' - X - f'' - X +	√|x|   x=0   f ↘   ↗ f' - X + f'' - X -	-√|x|   x=0   f ↗   ↘ f' + X - f'' + X +

⇒ x² |x|   local extrema   f'(a)=0 f'(a) DNE critical point   ⇍ x³ (sgn x)√|x|

⇒ x³ (sgn x)√|x|   inflection point   f''(a)=0 f''(a) DNE     ⇍ x⁴ √|x|

Optimization Problems

Integrals

The Fundamental Theorem of Calculus, Part 1

If \(f(x)=\int_{a}^{x} h(t)dt\), then \(f'(x)=h(x)\).

The following is an application. If \(f(x)=\int_{a}^{g(x)} h(t)dt\), then \[ \begin{array}{rcl} f'(x) &=& \displaystyle \frac{d}{dx}\int_a^{g(x)}h(t)dt \\ &\stackrel{u=g(x)}{=}& \displaystyle \frac{d}{dx} \int_a^u h(t)dt \\ &\stackrel{\square=\int_a^u h(t)dt}{=}& \displaystyle \frac{d\square}{dx} \\ &\stackrel{\text{chain rule}}{=}& \displaystyle \frac{d\square}{du}\cdot \frac{du}{dx} \\ &=& \displaystyle \frac{d}{du}\int_a^u h(t)dt \cdot \frac{du}{dx} \\ &\stackrel{\text{The Fundamental Theorem}}{=}& \displaystyle h(u)\cdot \frac{du}{dx} \\ &=& \displaystyle h(g(x))\cdot \frac{du}{dx} \end{array} \]

The Fundamental Theorem of Calculus, Part 2

\[ \begin{array}{rcl} \displaystyle \text{By the Fundamental Theorem }\frac{d}{dx}\boxed{\int_c^x f(t)dt} &=& \displaystyle f(x) \\ \displaystyle \text{Suppose that }\frac{d}{dx} \boxed{F(x)} &=& \displaystyle f(x) \\ \displaystyle \Rightarrow F(x) &=& \displaystyle \int_c^x f(t)dt \\ \displaystyle \Rightarrow F(a) &=& \displaystyle \int_c^a f(t)dt \\ \displaystyle F(b) &=& \displaystyle \int_c^b f(t)dt \\ \displaystyle \Rightarrow \text{Area} &=& \displaystyle \int_a^b f(t)dt \\ &=& \displaystyle \int_c^b f(t)dt-\int_c^a f(t)dt \\ &=& \displaystyle F(b)-F(a) \end{array} \]

Formula Sheet

General Formulas

\(\displaystyle \frac{d}{dx}(c)=0\)

\(\displaystyle \frac{d}{dx}[cf(x)]=cf'(x)\)

\(\displaystyle \frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)\)

\(\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}\) (Power Rule)

\(\displaystyle \frac{d}{dx}(a_n x^n+a_{n-1}x^{n-1}+\cdots+a_2 x^2+a_1 x+a_0)=a_n n x^{n-1}+a_{n-1} (n-1) x^{n-2}+\cdots +a_2\cdot 2 x+a_1\)

\(\displaystyle \frac{d}{dx}[f(x)g(x)]=f(x)g'(x)+g(x)f'(x)\) (Product Rule)

\(\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}\) (Quotient Rule)

\(\displaystyle \frac{d}{dx}f(g(x))=f'(g(x))g'(x)\) (Chain Rule)

Trigonometric Functions

\(\displaystyle \frac{d}{dx}(\sin{x})=\cos{x}\)	\(\displaystyle \frac{d}{dx}(\cos{x})=-\sin{x}\)
\(\displaystyle \frac{d}{dx}(\tan{x})=\sec^2{x}\)	\(\displaystyle \frac{d}{dx}(\cot{x})=-\csc^2{x}\)
\(\displaystyle \frac{d}{dx}(\sec{x})=\sec{x}\tan{x}\)	\(\displaystyle \frac{d}{dx}(\csc{x})=-\csc{x}\cot{x}\)

Inverse Trigonometric Functions

\(\displaystyle \frac{d}{dx}(\sin^{-1}{x})=\frac{1}{\sqrt{1-x^2}}\)	\(\displaystyle \frac{d}{dx}(\cos^{-1}{x})=\frac{-1}{\sqrt{1-x^2}}\)
\(\displaystyle \frac{d}{dx}(\tan^{-1}{x})=\frac{1}{1+x^2}\)	\(\displaystyle \frac{d}{dx}(\cot^{-1}{x})=\frac{-1}{1+x^2}\)
\(\displaystyle \frac{d}{dx}(\sec^{-1}{x})=\frac{1}{x\sqrt{x^2-1}}\)	\(\displaystyle \frac{d}{dx}(\csc^{-1}{x})=\frac{-1}{x\sqrt{x^2-1}}\)

Exponential and Logarithmic Functions

\(\displaystyle \frac{d}{dx}(e^x)=e^x\)	\(\displaystyle \frac{d}{dx}(b^x)=b^x\ln{b}\)
\(\displaystyle \frac{d}{dx}\ln{|x|}=\frac{1}{x}\)	\(\displaystyle \frac{d}{dx}(\log_b{x})=\frac{1}{x\ln{b}}\)
補充. \(\displaystyle \frac{d}{dx}e^{f(x)}=f'(x)e^{f(x)}\)
補充. \(\displaystyle \frac{d}{dx}\ln{f(x)}=\frac{f'(x)}{f(x)}\)