# Preclass Notes

MATH 126

## Week 1 Monday (12.1)

• Three-dimensional spaces
• We need an origin - a reference point. Begin with two axes that are perpendicular. The third axis must extend perpendicular to both of the existing axes.
• Right hand rule: helps you determine in which direction a third axis points.
• We can describe every point space with some set of coordinates $$(a, b, c)$$/$$(x, y, z)$$.

• $$\mathbb{R} = (-\infty, \infty)$$ - the set of real unmbers.
• Set notation. All points in 3D form:
$\mathbb{R} \times \mathbb{R} \times \mathbb{R} = \mathbb{R}^3$ $= {(x, y, z) | x, y, z \in \mathbb{R}$

## Week 4 Monday (13.3b)

• Unit tangent vector for a smooth curve - $$\vec{T}(t)$$.
• Principal unit normal vector/normal vector - $$\vec{N}(t)$$.
$\vec{N}(t) = \frac{\vec{T}'(t)}{\| \vec{T} '(t) \| }$
• $$\vec{T}(t)$$ and $$\vec{T}'(t)$$ are orthogonal.
• Binormal vector - $$\vec{B}(t)$$$$\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$ • We can create a time-dependent coordinate system with three orthogonal axes using these vectors. This is a TNB frame. $\| \vec{B}(t) \| = \| \vec{T}(t) \| \cdot \| \vec{N} (t) \| \cdot \sin \theta \implies 1 = 1 \cdot 1 \cdot 1$ • $$\vec{B}(t)$$ is a unit vector. • Normal plane - plane spanned by $$\vec{N}(t)$$ and $$\vec{B}(t)$$. • The normal plane is orthogonal to the unit tangent vector; thus, some point on a curve $$P$$ and the derivative of the curve at $$P$$ can be plugged into the plane equation to define the normal plane. ## Week 4 Friday (13.4) • Velocity vector $$\vec{v}$$. Speed $$s = \| \vec{v} \|$$. • Derivative of velocity - acceleration. $\vec{F} = m \cdot \vec{a}$ Problem. An object of mass 10kg is being acted on by a force $$\vec{F}(t) = \langle 130t, 10e^t, 10e^{-t} \rangle$$. You are given $$\vec{v}(0) = \langle 0, 0, 1 \rangle$$ and $$\vec{r}(0) = \langle 0, 1, 1 \rangle$$. Find the position function $$\vec{r}(t)$$. Solution. If we integrate both sides, we have that $$\vec{a} = \vec{r}'' \implies \langle \frac{130}{2}t^2, 10e^t - 10, -10e^-t + 20 \rangle = \int 10 \cdot \vec{r}'(t)$$. Integrating again gives $$\langle \frac{65}{3}t^3, 10e^t - 10t, 10e^{-t} + 20t \rangle = 10 \vec{r}(t)$$. Dividing everything by 10 yields the curve. • Reference to acceleration and velocity in that very moment. • Two components: a tangent and normal component of acceleration. $\nu = \text{speed}$ $\vec{T}(t) = \frac{\vec{v}(t)}{\nu} \implies \vec{T} \nu = \vec{v}(t)$ Differentiating both sides: $\vec{T}' \nu + \vec{T} \nu' = \vec{v}'(t)$ Using the formula for curvature, we obtain $\kappa = \frac{\| \vec{T}' \|}{v} \implies \kappa \nu = \| \vec{T}' \|$ $\vec{N} = \frac{\vec{T}'}{\| \vec{T}' \| \implies \vec{T}' = \kappa \nu \vec{N}$ $\kappa \nu^2 \vec{N} + \nu' \vec{T} = \vec{a}(t)$ • We have written acceleration as a sum of $$\vec{T}$$ and $$\vec{N}$$. • $a_\vec{T} = \nu', a_\vec{N} = kappa \nu^2$ ## Week 5 Monday (14.1) • Multivariable functions • A real function $$f$$ of two variables assigns each pair • Domain: the set of all values that are allowed. • Range: the set of all values the output takes on. • Look for domain restrictions - square roots, logarithms, rationals, absolute values, etc. • Intercepts are helpful (esp. when planes are involved). • Use traces to generate level curves - contour map/elevation map as a projection of level curves. ## Week 5 Wednesday (14.7) • Critical numbers - places where the critical number is zero or not defined in the domain. • Secpmd derivative test - check concavity • A local maximum occurs at $$9a, b)$$ if $$f(a, b) \ge f(x, y)4 for all points$$(x, y)$$near$$(a, b)$$. This is the local minimum value. • Some extrema can happen at special locations. A critcial point is a point at which $$f_x(a, b) = 0$$ and $$f_y(a, b)=0$$, or $$f_x(a, b)44 or$$f_y(a, b)$$ do not exist. • Saddle point - point which is not a minimum or maximum • Second derivative test - directional derivatives. • Compute $$D = f_{xx} (a, b) f_{yy}(a, b) - f_{xy}^2(a, b)$$. • If $$D > 044, concavity is the same in all directions. If$$f_{xx} > 0$$, it is a local minimum (concave up in all directions). If$$f_{xx} < 0$\$, it is concave down in all directions, a local maximum.
• If $$D < 0$$, saddle point.
• If $$D = 0$$, inconclusive.

## Week 6 Wednesday (15.1)

• Double integral - signed volume between a function and the $$xy$$ plane.
• Integrate across both dimensions.

## Week 6 Friday (15.2)

• Given $$x$$ in the range $$a \le x \le b$$, we have $$g_1(x) \le y \le g_2(x)$$. Top/bottom. Compute integral as $$\int_a^b \left( \int_{g_1(x)}^{g_2(x)} f(x, y) dy \right)$$

## Week 7 Monday (10.3)

• Polar coordinates - stand at the origin facing the positive $$x$$-axis.
• Rotate by $$\theta$$
• Conversion: $$x = r \cos \theta$$, $$y = r \sin theta$$, $$\tan \theta = \frac{y}{x}$$, $$x^2 + y^2 = r^2$$

## Week 7 Wednesday (15.3)

• In polar coordinates, $$x = r \cos \theta, y = r \sin \theta$$
• To set up a double integral, describe the region with polar coordinates, replace $$x$$ and $$y$$, and replace $$dA$$ with $$r dr d \theta$$.
• Why replace $$dA$$ with $$r dr d \theta$$? The arc length is $$r d \theta$$, the ‘length’ is $$dr$$.

## Week 7 Friday (15.4)

• Mass and center of mass
• The density $$\rho(x, y)$$ of some lamina varies with $$x, y$$; $$\rho(x, y)$$ is measured in units of mass p unit area.
• Mass of whole lamina: $$\int \int_D \rho (x, y) dA$$
• Inverse/proportionality: adapt a proportionality constant $$k$$
• Moment of the lamina about the $$x$$-axis: $$M_x = \int \int_D y \rho(x, y) dA$$
• The center of mass can lie outside of the lamina.

## Week 8 Wednesday (Taylor Notes 1)

• Taylor polynomial of degree 1: $$T_1(x) = f(b) + f'(b)(x - b)$$
• Approximation error
• Bounds - given a function $$f(x)$$ - a number is an upper bound if all values are less than or equal to that value.
• You don’t need a ‘best’ bound, just a bound that works.
• Error boudns: given an interval $$I$$ aroudn $$b$$, $$b - \epsilon \le x \le b + \epsilon$$: if $$\| f''(x) \| \le M$$ for all $$x$$ in $$I$$, then
$\| f(x) - T_1(x) \| \le \frac{M}{2} \| x - b \|^2$

## Week 8 Friday (Taylor Notes 2)

• 1st Taylor Polynomial: the tangent line. The error is bounded by $$\frac{M}{2} \| x - b \|^2$$ for $$\| f''(x) \| M$$. You can make this independent of $$x$$ by finding the maximum possible RHS expression on the domain.
• The second Taylor polynomial of a function $$f(x)$$ based at $$b$$ is
$T_2(x) = T_1(x) + \frac{1}{2} f''(b)(x-b)^2$
• This error will be less than or equal to $$\frac{M}{6} \| x - b \|^3$$ for $$\| f'''(x) \| le M$$.
$T_3(x) = T_2(x) + \frac{1}{6} f'''(b)(x-b)^3$ $T_n(x) = T_{n-1}(x) + \frac{1}{n!} f^n(b)(x-b)^n$
• The error for the $$n$$th Taylor polynomial is $$\frac{M}{(n+1)!} \| x-b \|^{n+1}$$

• We are matching the derivative on multiple different levels.

## Week 9 Monday (Taylor Notes 3)

• Sigma notation - allows us to express complex patterns, like exponentials of $$-1$$ to change sign
• $$n$$th Taylor polynomial - as $$n$$ increases, the error bound decreases.
• We let $$n \to \infty$$

Taylor Series for $$f(x)$$ at base $$b$$:

$\lim_{n\to\infty} T_n(x) = \sum_{k=0}^\infty \frac{1}{k!} f^k (b)(x-b)^k$
• If the limit exists for an $$x$$, it converges at $$x$$. Otherwise, it diverges at $$x$$.
• Interval of convergence - largest open interval over which the series converges.
• Convergence only happens within an interval.
• Important series:
• $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...$$, converges for all $$x$$
• $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \pm ...$$, converges for all $$x$$
• $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \pm ...$$, converges for all $$x$$

## Week 9 Wednesday (Taylor Notes 4)

• $$n$$th Taylor polynomial of $$f$$ on $$I$$: $$T_n(x) = \sum_{k=0}^n \frac{1}{k!} f^k (b) (x-b)^k$$
• Error bound: error $$\lke \frac{M}{(n+1)!} \|x-b\|^{n+1}$$
• Taylor series - take $$n\to\infty$$, we see error $$\to 0$$.
• Word of caution - sometimes a function is differentiable and $$T_n$$ doesn’t diverge, but it does not work out.
• Interval of convergence: when do we see convergence? Which $$x$$ leads to convergence when $$n\to\infty$$?
• Consider $$(1 - x) T_n(x) = ...$$, slve for $$T_n (x)$$.

## Week 9 Friday (Taylor Series 5)

• Developing additional ability to write Taylor series
• On the interval of convergence, we can pull in constants and merge Taylor series.
• Whenever we have an integral/derivative and a sum, we can distribute across it term-wise.
$\frac{d}{dx} A(x) = \sum_{k=0}^\infty \frac{d}{dx} \left( \frac{f^k (b)}{k!} (x - b)^k \right) = \sum_{k = 0}^\infty \frac{f^k (b)}{(k-1)!} (x - b)^{k-1}$
• Same logic can be applied to the integral
• The differentiation/integration will have the same interval of convergence as the original

## Week 10 Wednesday

• Take derivatives repeatedly and find patterns in evaluation at certain points.
• For $$f(x) = \sum_{k=0}^\infty _k (x - b)^k$$, $$f^n b = c_n \cdot n!$$, $$c_n = \frac{f^n (b)}{n!}$$