AP Calculus BC

MCV4UP Course Overview

AP Calc Classroom

Calc Textbook

1. Limits and Continuity

Definition of the Derivative:

$$\frac{dy}{dx}=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$

Symmetric Difference Quotient:

$$f^{\prime }(a)\approx \frac{f(a+h)-f(a-h)}{(a+h)-(a-h)}$$ $$f^{\prime }(a)=\frac{f(a+h)-f(a-h)}{2h}$$

Continuity: A function $f$ is continuous at $a$ if

$$\underset{x\to a}{\lim }f(x)=f(a)$$

But continuity does not imply differentiability - counterexample: $f(x)=|x|$

2. Derivative Rules

Power Rule:

$$f^{\prime }(x)=n\cdot x^{n-1}$$

Constant Rule: Let $f(x)=c$

$$f^{\prime }(x)=0$$

Constant Multiple Rule: Let $f(x)=c\cdot g(x)$

$$f^{\prime }(x)=c\cdot g^{\prime }(x)$$

Product / Sum Rule: Let $f(x)=g(x)\pm h(x)$

$$f^{\prime }(x)=g^{\prime }(x)\pm h^{\prime }(x)$$

Power Rule:

$$f^{\prime }(x)=n\cdot x^{n-1}$$

Definition of $e$

$$e=\underset{n\to \infty }{\lim }(1+\frac{1}{n}{)}^n$$

Product Rule: Let $f(x)=g(x)\cdot h(x)$

$$f^{\prime }(x)=g^{\prime }(x)\cdot h(x)+g(x)\cdot h^{\prime }(x)$$

Quotient Rule: Let $F(x)=\frac{f(x)}{g(x)}$

$$F^{\prime }(x)=\frac{f^{\prime }(x)\cdot g(x)-f(x)\cdot g^{\prime }(x)}{(g(x){)}^2}$$

Logs and Exponents

1. $\frac{d}{dx}{e}^x=e^x$
2. $\frac{d}{dx}ln(x)=\frac{1}{x}$
3. $\frac{d}{dx}{b}^x=b^x\cdot ln(b)$
4. $\frac{d}{dx}{\log }_b(x)=\frac{1}{x\cdot ln(b)}$

Trig Functions

1. $\frac{d}{dx}\sin (x)=\cos (x)$
2. $\frac{d}{dx}\cos (x)=-\sin (x)$
3. $\frac{d}{dx}\tan (x)={\sec }^2(x)$
4. $\frac{d}{dx}\cot (x)=-{\csc }^2(x)$
5. $\frac{d}{dx}\sec (x)=\sec (x)\cdot \tan (x)$
6. $\frac{d}{dx}\csc (x)=-\csc (x)\cdot \cot (x)$

3. Chain, Inverse, Implicit differentiation

Implicit Differentiation: Differentiate $y$ with respect to $x$

• Explicitly Defined: $y=f(x)$
• Implicitly Defined: $y=x$

Inverse Function Theorem: For $f^{-1}(x)=g(x)$

$$g^{\prime }(x)=\frac{1}{f^{\prime }(g(x))}$$ $$g^{\prime }(x)=\frac{1}{2(x^{\frac{1}{2}})}=\frac{1}{2}{x}^{-1/2}$$

Inverse Trig Functions

$$\frac{dy}{dx}\arcsin (x)=\frac{1}{\sqrt{1-x^2}}$$ $$\frac{dy}{dx}\arccos (x)=-\frac{1}{\sqrt{1-x^2}}$$ $$\frac{dy}{dx}\arctan (x)=\frac{1}{1+x^2}$$ $$\frac{dy}{dx}arccot(x)=-\frac{1}{1+x^2}$$ $$\frac{dy}{dx}arccsc(x)=\frac{1}{cs{c}^2(x)}$$

4. Contextual Applications of differentiation

Related Rates

• Just relate the rates

Local Linear Approximation

$$f(a+h)=f(a)+f^{\prime }(a)\ast h$$

L’Hôpital’s Rule

$$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

5. Analytical Applications of differentiation

MVT: If f is continuous and differentiable, there exists a $c\in (a,b)$ such that

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$$

EVT: If f is continuous and differentiable, it has an absolute min and max

Optimization

• Set the desired $\frac{dy}{dx}=0$

Derivative Tests

• First derivative = increase / decrease
• Second derivative = concave up / down
• Critical points = when $\frac{dy}{dx}=0$
• Inflection points = when $\frac{d^{2}y}{d{x}^2}=0$

6. Integration

Riemann Sums

$$F(f,C,P)=\sum _{i=1}^{n}f(c_i)\cdot (x_i-x_{i-1})$$

• $f(c_i)$ = height
• $(x_i-x_{i-1})$ = width

FTC I:

$$\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)$$

FTC II:

$${\int }_a^{b}f(t)dt=f^{\prime }(b)-f^{\prime }(a)$$

Integral Rules

Order of Integration

$${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$$

Zero

$${\int }_a^{a}f(x)dx=0$$

Constant Multiple

$${\int }_a^{b}k\ast f(x)dx=k\ast {\int }_a^{b}f(x)dx$$

Sum / Difference

$${\int }_a^b(f(x)\pm g(x))dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx$$

Additivity

$${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx$$

Integration by parts (for products)

$$\int udv=uv-\int vdu$$

7. Differential Equations

Modelling with differential equations

• Same thing as related rates

Slope fields

• Are literally just a bunch of lines

Solving separable differential equations

• cross-multiply $\frac{dy}{dx}$ and solve

8. Applications of Integration

Average value of a function

$$=\frac{{\int }_a^{b}f(x)}{b-a}$$

Net change

• Just a regular integral

Area between curves

• Subtract two integrals

Volumes formed by revolving regions

• I still don’t understand this

Volumes by cross-section

• Area = $\int$ volume of shapes
• Area = $\int$ area * thickness (dx)
• Semicircles
  • Area of a semicircle = $\frac{1}{2}\pi (4-y)$
  • Thickness = $dy$
  • Volume = ${\int }_0^4\frac{1}{2}\pi (4-y)dx$
• Equilateral Triangles
  • Area = $\frac{\sqrt{3}}{4}{b}^2$
  • Volume = $\int \frac{\sqrt{3}}{4}{b}^2\times dx$
• Hemispheres
  • Area = $\pi r^2$
  • Volume = $\int \pi r^2\times dx$

9. Vectors

Parametric Equations

Take a value t that’s defined on an interval l. Parametric equations in t:

$$x=x(t),y=y(t)$$

Differentiating parametric equations:

$$x=2sin(1+3t)\to \frac{dx}{dt}2cos(1+3t)\ast t$$ $$y=2t^3\to \frac{dy}{dt}6t^2$$ $$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{6t^2}{6cos(1+3t)}=\frac{t^2}{cos(1+3t)}$$

Derivative at $t=-\frac{1}{3}$

$$\frac{dy}{dx}=\frac{{\left(-\frac{1}{3}\right)}^2}{cos(0)}=\frac{1}{9}$$

Second Derivatives

$$\frac{d^{2}y}{d{x}^2}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$$

Parametric Curve Length

$$dx=\frac{dx}{dt}\ast dt=x^{\prime }(t)dt$$ $$dy=\frac{dy}{dt}\ast dt=y^{\prime }(t)dt$$

Curve Length:

$${\int }_{t_a}^{t_b}\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}\ast dt$$

Differentiating Vector-Valued Functions

$$\vec{r}(a)=x(a)\hat{i}+y(a)\hat{j}$$ $$\vec{r}(a+b)=x(a+b)\hat{i}+y(a+b)\hat{j}$$

Differentiating Vector Functions

$$\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}$$ $$\underset{h\to 0}{\lim }\frac{\vec{r}(t+h)-\vec{r}(t)}{h}=\underset{h\to 0}{\lim }\frac{(x(t+h)-x(t))\hat{i}}{h}+\underset{h\to 0}{\lim }\frac{(y(t+h)-y(t))\hat{j}}{h}$$ $${\vec{r}}^{\prime }(t)=\vec{x}(t)\hat{i}+y^{\prime }(t)\hat{j}$$

First and second derivatives are just normal

Planar Motion

Distance → Velocity’ → Acceleration”

Motion along a curve (derivatives):

$$\frac{d}{dt}[x^2{y}^2]=\frac{d}{dt}[16]$$

Magnitude of Velocity

$$\vec{V}(t)=(\frac{dx}{dt},\frac{dy}{dt})$$ $$|\vec{V}(t)|=\sqrt{{\frac{dx}{dt}}^2+{\frac{dy}{dt}}^2}$$

Motion along the curve (integrals):

$$\Delta x={\int }_1^3\frac{1}{t+7}dt$$ $$\Delta y={\int }_1^3{t}^{4}dt$$

Polar Functions

$$x=r\cos \theta$$ $$y=r\sin \theta$$

Area bounded by Polar Curves = Integrals

10. Vector Algebra

1-4: Vector Properties

Given $\vec{v}=(v_x,v_y),\vec{w}=(w_x,w_y)$

1. Sum: $\vec{v}+\vec{u}=(u_x+v_x,u_y+v_y)$
2. Constant product: $k\cdot \vec{v}=(k\cdot v_x,k\cdot v_y$)
3. Magnitude: $||\vec{v}||=\sqrt{v_x^2+v_y^2}$

Constructing Vectors: tip - tail Given $\vec{p}=(p_x,p_y,p_z),\vec{q}=(q_x,q_y,q_z)$

$$\overrightarrow{PQ}=(q_x-p_x,q_y-p_y,q_z-p_z)$$

5: Dot Product

”the distance from the vector to the shadow”

$$\vec{v}\cdot \vec{w}=||v_x||||w_x||\cos \theta$$ $$\cos \theta =\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||||\vec{v}||}$$

Properties

1. Square: $\vec{v}\cdot \vec{v}=||\vec{v}|{|}^2$
2. Commutative: $\vec{v}\cdot \vec{w}=\vec{w}\cdot \vec{v}$
3. Distributive: $\vec{v}\cdot (\vec{w}+\vec{u})=\vec{v}\cdot \vec{w}+\vec{v}\cdot \vec{u}$
4. Associative: $(k\vec{v})\cdot \vec{w}=k(\vec{v}\cdot \vec{w})$
5. Orthogonal: $\vec{v}\cdot \vec{w}=0$ if $\theta =90^o$

6: Projections

”the shadow of v cast onto w”

$$pro{j}_{\vec{w}}\vec{v}=\frac{||\vec{v}||\cos \theta }{||\vec{w}||}\vec{w}=\frac{\vec{v}\cdot \vec{w}}{||\vec{w}|{|}^2}\vec{w}$$

Unit Vectors $\hat{i}=(1,0,0),\hat{j}=(0,1,0),\hat{k}=(0,0,1)$

$$\hat{k}=\hat{i}\times \hat{j}$$

7-8: Cross Product

Finding a vector that is orthogonal to both vectors

$$\vec{v}\times \vec{w}=\overrightarrow{(v_z{w}_z-v_z{w}_y,v_z{w}_x-v_x{w}_z,v_x{w}_z-v_y{w}_x)}$$

Properties

1. Orthogonal
2. $||\vec{v}\times \vec{w}||=||\vec{v}||\vec{w}||sin\theta$
3. Right Hand Rule
4. $v\times w=-(w\times v)$
5. Distributive across addition
6. Distributive across multiplication

9: 3D Lines

Given:

• $A(x_0,y_0,z_0)$: Point
• $(x_0,y_0,z_0)$: Position Vector
• $\vec{d}=\overrightarrow{(a,b,c)}$: Direction Vector
• $P(x,y,z)$: Any point Vector Equation of a line:

$$\overrightarrow{(x.y.z)}=\overrightarrow{(x_0,y_0,z_0)}+t\overrightarrow{(a,b,c)}$$

Parametric Form:

$$x=z_0+at$$ $$y=y_0+bt$$ $$z=z_0+ct$$

Symmetric Form

$$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$

Example: Finding the shortest distance between

$$\frac{x-2}{7}=\frac{y-1}{3}=\frac{z-5}{-2},\frac{x+3}{4}=\frac{y-8}{2}=\frac{z+14}{-1}$$

point on A: $A(2,1,-5)$ direction of A: $\overrightarrow{d_1}=(7,3,-2)$ point on B: $B(-3,8,-14)$ direction of B: $\overrightarrow{d_2}=(4,2,-1)$

Vector connecting $l_1$ and $l_2$ is $\vec{v}=\overrightarrow{BA}=\overrightarrow{(5,-7,9)}$ Vector orthogonal to both $l_1$ and $l_2$ is $\vec{h}=\overrightarrow{d_1}\times \overrightarrow{d_2}=\overrightarrow{(1,-1,2)}$ Shortest distance is $||\vec{v}\downarrow \vec{n}||=\frac{\vec{v}\cdot \vec{n}}{||\vec{n}||}=|\frac{30}{\sqrt{6}}|=5\sqrt{6}$

10: Planes

Cartesian / Scalar equation of a plane:

$$ax+by+cz=d$$

• $\overrightarrow{(a,b,c)}$ is a vector normal to the plane
• $d=a{x}_0+b{y}_0+c{z}_0$ with point $(x_0,y_0,z_0)$ on the plane

Vector Equation of a plane:

$$\overrightarrow{OP}=\overrightarrow{O{P}_0}+s\vec{a}+t\vec{b}$$

• $P(x,y,z)$ is a point
• $P_0(x_0,y_0,z_0)$ is a fixed point
• $\vec{a},\vec{b}$ are non-collinear vectors

Parametric Equation

$$x=x_0+s{a}_x+t{b}_x$$ $$y=y_0+s{a}_y+t{b}_y$$ $$z=z_0+s{a}_z+t{b}_z$$

11. Vector Geometry

12. Infinite Series

12.1 Defining Infinite Series

Converges: has a limit as n approaches infinity Diverges: has no limit (infinity, oscillates)

12.2 Geometric Series

Sum of infinite geometric series for $|r|<1$

$$S=\frac{a}{1-r}$$

Diverges if $|r|>1$

12.3 n-th term divergence test

If ${\lim }_{n\to \infty }{a}_n\ne 0$, then ${\sum }_{n=1}^{\infty }{a}_n$ will diverge

• This is because every subsequent term should get smaller and smaller
• The inverse is not true

12.4 Integral Test

$$\sum _{n=1}^{\infty }\frac{1}{n^2}\to f(x)=\frac{1}{x^2}$$ $$\sum _{n=1}^{\infty }\frac{1}{n^2}=1+\sum _{n=2}^{\infty }\frac{1}{n^2}<1+{\int }_1^{\infty }\frac{1}{x^2}dx$$

1. If ${\int }_k^{\infty }f(x)dx$ converges, then ${\sum }_{n=k}^{\infty }f(n)$ converges
   1. $\frac{1}{x^2}$ converges because integral = $-\frac{1}{x}$
2. If ${\int }_k^{\infty }f(x)dx$ diverges, then ${\sum }_{n=k}^{\infty }f(n)$ diverges
   1. $\frac{1}{x}$ diverges because integral = $ln(x)$

12.5 Harmonic and p-series

Harmonic Series

$$\sum _{n=1}^{\infty }\frac{1}{n}=\frac{1}{1}+\frac{1}{2}+\dots$$

p-series

$$\sum _{n=1}^{\infty }\frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\dots$$ $$\sum _{n=1}^{\infty }\frac{1}{n^p}=1+\frac{1}{2^p}+\frac{1}{3^p}+\dots$$

• $p>1$ converge
• $p\le 1$ diverge

12.6 Comparison Tests

Direct Comparison Test: For any $a_n\le b_n$

• If $b_n$ converges, then any smaller series $a_n$ converges
• If $a_n$ diverges, $b_n$ also diverges

Limit Comparison Test $a_n\ge 0$, $b_n>0$ If $li{m}_{n\to \infty }\frac{a_n}{b_n}>0$ and finite, either both converge or both diverge

12.7 Alternating Series Test

${\sum }_{n\to k}^{\infty }{a}_n$ converges if:

1. $a_n=(-1{)}^n{b}_n=(-1{)}^{n+1}{b}_n$
2. ${\lim }_{n\to \infty }{b}_k=0$
3. $b_n$ is decreasing Proves the inverse of the n-th term divergence test (if limit = 0)

12.8 Ratio Test for convergence

$$\sum _{n=k}^{\infty }{a}_k$$ $$\underset{n\to \infty }{\lim }|\frac{a_{n+1}}{a_n}|=L$$

• if L<1 → converges
• if L>1 → diverges
• if L=1 → inconclusive

12.9 Absolute or Conditional Convergence

• Absolutely: Even if you take the absolute value
• Conditionally: not

12.10 Alternating Series Error Bound

$$\sum _{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n^2}=1-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+\frac{1}{25}-\dots$$ $$S=S_4+R_4$$ $$S=\frac{115}{144}+R_4$$ $$\frac{115}{114}<S<\frac{115}{144}+\frac{1}{25}$$

12.11 Taylor and Maclaurin Approximation

Maclaurin Series = Taylor series centered at 0 Approximating p(x) using f(x), but we want the first, second, etc. derivatives to match

$$p(x)=f(0)+f^{\prime }(0)x+\frac{1}{2}{f}^{\prime \prime }(0)x^2+\frac{1}{2\ast 3}{f}^{\prime \prime \prime }(0)x^3+\cdots f^n(0)\frac{\ast x_n}{n!}$$

Taylor Series

$$p(x)=f(c)+\frac{f^{\prime }(c)}{1!}(x-c)+\frac{f^{\prime \prime }(c)}{2!}(x-c{)}^2+\cdots +\frac{f^n(c)}{n!}(x-c{)}^n$$

12.12 Lagrange Error Bound

Trying to bound the Taylor approximation

$$R_{N,a}(x)=E_{N,a}(x)=f(x)-P_{N,a}(x)$$ $$E(a)=f(a)-P(a)=0$$ $$E^{\prime }(A)=f^{\prime }(a)-P^{\prime }(a)=0$$ $$E^{(N)}(a)=f^{(N)}(a)-P^{(N)}(a)=0$$ $$E^{n+1}(x)=f^{n+1}(x)\le M$$ $$E(b)\le K$$

Formula:

$$|R_n(x)|\le |\frac{M}{(n+1!)}(x-c{)}^{N+1}|$$