calculus

AP Calculus Syllabus

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\vv=s'>0v=s'<
a=s''>0⤴⤷
a=s''<0Γ⤵


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

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

Integration by Parts

\[ \begin{array}{rcl} (uv)' &=& uv'+u'v \\ \displaystyle \int (uv)' &=& \displaystyle \int uv'+ \int u'v \\ \displaystyle uv &=& \displaystyle \int uv'+ \int u'v \\ \displaystyle uv-\int u'v &=& \displaystyle \int uv' \\ \displaystyle \int uv' &=& uv-\int u'v \\ \end{array} \]
uv'
\+
u'-∫
---		v

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. \(\displaystyle \frac{d}{dx}\ln{f(x)}=\frac{f'(x)}{f(x)}\)

Volumes

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
\(y=x^2, x=0, y=4\); about the \(x\)-axis
\[ \begin{array}{cl}& \displaystyle \pi\int_0^2 4^2-(x^2)^2 dx\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi\int_0^4 y\sqrt{y}dy\end{array} \] 		\(y=x^2, x=0, y=4\); about the \(y\)-axis
\[ \begin{array}{cl}& \displaystyle \pi\int_0^4 \sqrt{y}^2 dy\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi\left[\int_0^2 x\cdot 4 dx-\int_0^2 x\cdot x^2 dx\right]\end{array} \]
\(y=x^2, y=0, x=2\); about the \(x\)-axis
\[ \begin{array}{cl}& \displaystyle \pi\int_0^2 (x^2)^2 dx\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi\left[\int_0^4 y\cdot 2 dy-\int_0^4 y\cdot \sqrt{y}dy\right] \text{ (not good)}\end{array} \] 		\(y=x^2, y=0, x=2\); about the \(y\)-axis
\[ \begin{array}{cl}& \displaystyle \pi\int_0^4 \left[\int_0^4 2^2 dy-\int_0^4 \sqrt{y}^2 dy\right] dy\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi \int_0^2 x\cdot x^2 dx \text{ (good)}\end{array} \]
\(y=x^2, x=0, y=4\); about the \(y=6\)
\[ \begin{array}{cl}& \displaystyle \pi\int_0^2 (-x^2+6)^2-2^2 dx\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi \int_2^6 y\cdot \sqrt{-(y-6)}dy \end{array} \] 		\(y=x^2, x=0, y=4\); about the \(x=4\)
\[ \begin{array}{cl}& \displaystyle \pi \int_0^4 4^2-(-\sqrt{y}+4)^2 dy \\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi\int_2^4 x\cdot 4dx-2\pi \int_2^4 x(x-4)^2 dx \text{ (not good)}\end{array} \]
\(y=x^2, y=0, x=2\); about the \(y=-2\)
\[ \begin{array}{cl}& \displaystyle \pi\int_0^2 (x^2+2)^2-2^2 dx\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi\int_2^6 y\cdot 2-y\sqrt{y-2}dy\end{array} \] 		\(y=x^2, y=0, x=2\); about the \(x=-2\)
\[ \begin{array}{cl}& \displaystyle \pi\int_0^4 4^2-(\sqrt{y}-2)^2 dy\\ \stackrel{\text{shell}}{=}& \displaystyle 2\pi \int_2^4 x\cdot (x-2)^2 dx\end{array} \]

AP Calculus

AP Calculus AB and BC Course and Exam Description.

Thanks for Fiveable

Unit 1 Limits and Continuity
1.1 Introducing Calculus: Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties of Limits
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity over an Interval
1.13 Removing Discontinuities
1.14 Connecting Infinite Limits and Vertical Asymptotes
1.15 Connecting Limits at Infinity and Horizontal Asymptotes
1.16 Working with the Intermediate Value Theorem
Unit 2 Differentiation: Definition and Basic Derivative Rules
2.1 Defining Average and Instantaneous Rates of Change at a Point
2.2 Defining the Derivative of a Function and Using Derivative Notation
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
2.7 Derivatives of cos x, sinx, e^x, and ln x
2.8 The Product Rule
2.9 The Quotient Rule
2.1 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Unit 3 Differentiation: Composite, Implicit, and Inverse Functions
3.1 The Chain Rule
3.2 Implicit Differentiation
3.3 Differentiating Inverse Functions
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
3.6 Calculating Higher-Order Derivatives
Unit 4 Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.3 Rates of Change in Applied Contexts Other Than Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local Linearity and Linearization
4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
Unit 5 Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
5.3 Determining Intervals on Which a Function is Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
5.6 Determining Concavity of Functions over Their Domains
5.7 Using the Second Derivative Test to Determine Extrema
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Unit 6 Integration and Accumulation of Change
6.1 Exploring Accumulations of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
6.4 The Fundamental Theorem of Calculus and Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions Involving Area
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite Integrals
6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division and Completing the Square
6.11 Integrating Using Integration by Parts BC Only
6.12 Using Linear Partial Fractions BC Only
6.13 Evaluating Improper Integrals BC Only
6.14 Selecting Techniques for Antidifferentiation
Unit 7 Differential Equations
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Approximating Solutions Using Euler’s Method BC Only
7.6 Finding General Solutions Using Separation of Variables
7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations BC Only
Unit 8 Applications of Integration
8.1 Finding the Average Value of a Function on an Interval
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4 Finding the Area Between Curves Expressed as Functions of x
8.5 Finding the Area Between Curves Expressed as Functions of y
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
8.7 Volumes with Cross Sections: Squares and Rectangles
8.8 Volumes with Cross Sections: Triangles and Semicircles
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
8.10 Volume with Disc Method: Revolving Around Other Axes
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
8.12 Volume with Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled BC Only
Unit 9 Parametric Equations, Polar Coordinates, and Vector-Valued Functions BC Only
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Finding Arc Lengths of Curves Given by Parametric Equations
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two Polar Curves
Unit 10 Infinite Sequences and Series BC Only
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power Series
10.14 Finding Taylor or Maclaurin Series for a Function
10.15 Representing Functions as Power Series

No comments:

Post a Comment

Subscribe to: Comments (Atom)