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21 Cards in this Set

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What is a vector-valued function (or vector function)?
A vector valued function is simply a linear function whose domain is a set of real numbers and whose range is a set of vectors. Vector valued functions are like normal functions except each component is it's own function.
How do you take the limit of a vector valued function?
The limit of a vector valued function is taken by taking the limits of the component functions as follows:
If r(t) = <f(t), g(t), h(t)>

lim r(t) as t->a is defined as,

lim r(t) as t->a = <lim f(t) as t->a, lim g(t) as t-a, lim h(t) as t-a>

provided the limit of each component exist.
What is a parametric equation of a vector valued function?
A parametric equation of a vector valued function is like such:

given r(t) = <f(t),g(t),h(t)>

parametrically defined,
x = f(t)
y = g(t)
z = h(t)
How do you take the derivative of a vector valued function?
Given
r(t) = <f(t), g(t), h(t)>

The derivative is defined as,
r'(t) = <f'(t), g'(t), h'(t)> = f'(t)i + g'(t)j + h'(t)k
How do you integrate a vector valued function?
given r(t)= f(t)i + g(t)j + h(t)k

Integrate[r(t)dt] = Integrate[f(t)dt]i + Integrate[g(t)dt]j + Integrate[h(t)dt]k
What is the domain of a vector valued function?
The domain of a vector valued function is the domain of all the component functions anded together.
What is the general formula for a circular helix vector valued function?
r(t) = <cos(t), sin(t), t>
How do you define whether a function is continuous at pont a?
vector valued function r(t) is continous at t=a iff lim t->a of r(t) = r(a)
What is the Unit Tangent (sorta a capitol looking T) of r(t)?
T(t) = r'(t) / |r'(t)|
In general what does the derivative of any function at a specific point give?
The tangent line at that point
What is the Unit Normal vector of r(t) (Defined as a capital N)? What does the normal vector indicate conceptually?
The unit normal vector for r(t) is defined as

N(t) = T'(t) / |T'(t)|

The unit normal vector indicates conceptually the direction that the curve is turning towards.
What is the Arc Length from point a to point b of a three dimentional vector valued function r(t)?
given r(t) = <x(t), y(t), z(t)>

L = Integrate[ Sqrt[ (x'(t))^2 + (y'(t))^2 + (z'(t)^2) ], t, from a, to b ]
What is curvature of a vector valued function? How is it calculated?
given the line generated by vector valued function r(t), the curvature defines the curviness of the function at a specific point.

Defined as,
Curvature = K(t) = |T'(t)| / |r'(t)| =
|Cross[r'(t), r''(t)]
/
(|r'(t)|)^3
Given a position vector of a particle r(t), how do you define the velocity vector of the particle, how do you define the acceleration of the acceleration vector of the particle, how do you define the speed of the particle?
Given a position vector r(t)
velocity is: r'(t)
speed is : |r'(t)|
acceleration is: r''(t)
What should you do given a "Projectile is fired with initial velocity $v at an angle of of $a, where does it land?" type problem.
Fuck it and use this formula, the thing you screwed up on the quiz is that you didn't know how to do the initial velocity.

x = ($v*cos[$a])*t
y = ($v*sin[$a])*t -(.5)*g*t^2
What is the definition of a function of two variables?
A function of two variables is a rule that assigns to each ordered pair of real numbers (x,y) in a set D a unique real number denoted by f(x,y). The set D is the domain of f and its range is the set of values that f takes on, that is {f(x,y)|(x,y) is a subset of D}.
How do you determine the domain of a function of severable variables?
The domain exist wherever the function is properly defined, use all real numbers minus any numbers that make the funtion invalid (such as negative square roots, and devides by zero).
What is a level or contour curve of a function?
The level curve of a function is a contour map on which points of constant elevation are joined.
What is a partial derivative? What is the partial derivative of Fxxyz if f(x, y, z) = sin (3x + yz)?
A partial derivative is a derivative where you differentiate according to one independent variable, while treating the other variables as fixed values. According to the notation you derive from left to right, so the first derivate is on x, the second is on x, the third is on y, and the fourth is on z. You should end up with
Fxxxyz(x,y,z) = -9*cos(3x + yz) + 9*y*z*sin(3x+yz)
Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z = f(x,y) at the point P($x,$y,$z) is
z - $z = Fx($x,$y)(x-$x) + Fy($x,$y)(y-$y)
cos(t)^2 + sin(t)^2 = ?
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