Web Assign Review

Review of the web assign practice for the mid term review

Vectors Stuff

We are given:

$$x^2+y^2=64$$ $$z=x^{-}{y}^2$$

Find the curve of intersection

standard parameterization of a circle of radius 8

$$x=8\cos t$$ $$y=8\sin t$$

once we have the first 2, the third comes for free

$$z=64{\cos }^{2}t-64{\sin }^{2}t$$

alternatively

$$\vec{r}(t)=⟨8\cos t,8\sin t,64\cos 2t⟩$$

we use the cosine double angle here

$$\cos (2t)={\cos }^{2}t-{\sin }^{2}t$$ $$z=64\cos (2t)$$

Another Problem Example

$$x^2+y^2=36$$ $$z=xy$$

vector

$$\vec{r}(t)=⟨6\cos t,6\sin t,18\sin (2t)⟩$$

Curvature Practice

e/trig problem

$$\vec{r}(t)=⟨e^t\cos t,e^t\sin t,3t⟩$$

find the curvature at point (1,0,0)

we need to find the value of t that corresponds to this point

t=0 is a good example

curvature equations that we need:

$$K=\frac{|{\vec{r}}^{\prime }\times {\vec{r}}^{\prime \prime }|}{|{\vec{r}}^{\prime }{|}^3}$$ $$K=|{\vec{r}}^{\prime \prime }(\partial )|$$ $$K=|\frac{dT}{d\partial }|$$ $$K=\frac{|{\vec{T}}^{\prime }|}{|{\vec{r}}^{\prime }|}$$

plugging in for ${\vec{r}}^{\prime }$

$${\vec{r}}^{\prime }=⟨e^t\cos t-e^t\sin t,e^t\sin t+e^t\cos t,3⟩$$ $$=⟨e^t(\cos t-\sin t),e^t(\sin t+\cos t),3$$ $${\vec{r}}^{\prime }=⟨e^t(\cos t-\sin t)+e^t(-\sin t-\cos t),e^t(\sin t+\cos t)+e^t(\cos t-\sin t),0⟩$$ $$=⟨e^t(\cos t-\sin t-\sin t-\cos t),e^t(\sin t+\cos t+\cos t+\sin t),0$$ $$=⟨-2e^t\sin t,2e^t\cos tt,0⟩$$ $$r^{\prime }(0)=⟨1,1,3⟩$$ $$r^{\prime \prime }(0)=⟨0,2,0⟩$$ $$|{\vec{r}}^{\prime }(0)|=\sqrt{1+1+9}=\sqrt{11}$$ $${\vec{r}}^{\prime }\times {\vec{r}}^{\prime \prime }=\left[\begin{array}{ccc}i& j& k\\ 1& 1& 3\\ 0& 2& 0\end{array}\right]=⟨-6,0,2⟩$$

Polar Coordinates Problem

Find the area in the top right quadrant and outside of the smaller circle

on desmos:

Desmos

we can get r from the point of intersection

settin up the double integral:

Tangent Planes / Normal Lines Problem

Find the tangent plane and normal line at the point (0,0,3)

write as a function

do chain rule and make into a vector

$$\nabla f={⟨1-3yz{e}^{xyz},1-3xz{e}^{xyz},1-3xy{e}^{xyz}⟩|}_{(0,0,3)}$$ $$\nabla f=⟨1,1,1⟩$$ $$⟨1,1,1⟩\cdot ⟨x-0,y-0,z-3⟩=0$$ $$x+y+z-3=0$$

Equation of the Tangent Plane of This

$$\nabla f\cdot \nabla r=0$$

this is usefull on test

$$\frac{x-0}{1}=\frac{y-0}{1}=\frac{z-0}{1}$$

Random Practice I Forgot

$$r_1=1.2$$ $$h_1=-2.8$$ $$v_1=?whenr=135,h=186$$ $$V=\frac{\pi r^{2}h}{3}$$

chain rule problem

flowchart TD
	V --> r
	V --> h
	r --> t1
	h --> t2

$$V_t=V_r{r}_t+V_h{h}_t$$