Calc 3 Exam 2 flashs

Equation for Arc Length

When is a function r(t) continuous?

if and only if, f(t), g(t), and h(t) are continuous on [a,b]

if r(t) is differentiable, then……

r’(t) = <f’(t), …>

If R(t) is a Vec-Val-Func such that R’(t) = r(t), then its the ………

antiderv of r(t)

How do you find the der of a Vec-Val_fun * a scalar ( cu(t) )

Scalar Der of Func (c u’(t) )

How do you find the der of a sclar function and Vec-Val-Func?

(f(t) * u(t) )

Product rule of both
f’(t) u(t) + f(t) u’(t)

How do you find the der of 2 separate Vec-Val-Fun being added together?

u(t) + v(t)

Product Rule

v’(t) u(t) + v(t) u’(t)

How do you find the der of 2 separate Vec-Val-Fun multiplied?
u(t) * v(t)

Product Rule

v’(t) u(t) + v(t) u’(t)

The der of a Vec-Val-Func can be seen as……

The der of each func individually

< /f(t) dt, /g(t) dt, /h(t) dt >

Tang vector equals…….

T(t) = r’(t) / ||r’(t)||

If y= f(x) then the curvaturecan be found as……

k = |f’’’(x)| / (1+(f ‘ (x))²)^{3/2 /}

If r(t) is a Vec-Val-Func, the curvature can be found as…….

k = ||T ‘ (t) || / ||r’(t)||

If r(t) is a 3-dimensional Vec-Val-Func, the curvature can be found as…….

or on a 2D VVF if given a 3rd component of 0

k = ||r’(t) x r”(t)|| / ||r’(t)||³

The curve of a xy-plane (2d) is the radius of a curvature or……

1 / k

The center of a curvature is defined as……..

r(t) + (1/k)*N(t)

The principal unit normal vector at t is defined as…….

N(t) = T’(t) / ||T’(t)||

The binormal vector at t is defined as

T(t) x N(t) = ( r’(t) x r’’(t) ) / ||r’(t) x r’’(t)||

If r(t) represents the position of an object at time t, then what is Velocity?

r’(t) of the func

If r(t) represents the position of an object at time t, then what is acceleration?

r’’(t) of the func

If r(t) represents the position of an object at time t, then what is magnitude of the velocity vector in speed??

ds/dt = ||r’(t)||

If given only one function of the object, and initial of the rest, how would you find the others?

Either get integral or derivate of given function, then add initial that is given for said function

In a word project, what would

Mass = ?
Height = ?

Velocity = ?

angle = ?

Spin at mag = ?

Used Mass*Acc=Force

A = <0, 0, grav> + <0, -mag/mass, 0>
Velocity = a’(t) = <….> + V(0)

V(0) = <cosO, 0, sinO>

height = r(0)

How would you find the speed and landing position of an object?

Landing position = 0

r(t) → <z> = 0, => t, input into r(t)

||r(t)|| = m/s

What is the domain of a variable?

When the domain is not in-valid or non-existent

ie. 1/0, //0, ……

What is the method of proving that a limit DNE?

set x(t) and y(t) to values of t to equal each other
to have the Num y/x equal the denom y.

For 3D, do 3 cases with z added

Do so in 2 variations (if not, set (-), if they do not equal the same value, the limit DNE

What is the method of proving that a limit does exist?

use the func |f(x) - L| with L = g(x,y) or limit
then reduce such to prove it goes to limit
Ie. |3xy /x+y| < |3xy/ x| < |3y| = g(x,y)

Suppose f(x,y) is continuous at (a,b) and g(x) is continuous at f(a,b), what function would this equal?

h(x,y) = g(f(x,y))

For a partDer, Fx, if the limit h→0, what would the function equal?

Fx = Lim h→ ( f(x+h, y, z) - (x, y, z) ) / h

replace y or z for …

If z = f(x, y) is differentiable at (a, b), (or a,b,c..) what is the linear approximation?

L(x, y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

Suppose f(x,y) and Fx and Fy exist at (a,b), then what 2 vectors are true?

u = <1, 0, Fx(a,b)> and v = <0, 1, Fy(a,b)>
n = u x v

What would x, y, and z equal on the normal line (orthogonal to the tang plane) in parametric form?

x = a + Fx(a,b)t

y = b + Fy(x,y)

z = c + Fz(a,b)

If a surface is described by the equitation F(x,y,z) = C, where C is a constant, what is the tang plane surface?

Fx(a,b,c)(x-a) + Fy(a,b,c)(y-b) + Fz(a,b,c)(z-c) = 0

If z = f(x,y) is differentiable at (a,b) then the increment and differential of z are what?

Δz = f(a+Δx, b+Δy) - f(z,b)
dz = Fx(a,b)Δx + Fy(a,b)Δy

what will x, y and z equal if the normal line to the surface z= f(x,y) when (x,y) = (a,b)

x = a + Fx(a,b)t

y = a + Fy(a,b)t

z = f(a,b) - t

If z = f(x(t), y(t)) then……

derivate of z = Fx (der of X) + Fy (der of Y)

If z = f(x(r, t), y(r, t)) then……

derivate of Zs = Fx Xs + Fy Ys
and Zt = Fx Xt + Fy Yt