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Calc midterm 2
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Gravity
Terms in this set (43)
For curve r(t) = ti+tj+(4-t^2)^(1/2) find the unit tangent vector T(t) and parametric equation of the line tangent to the curve at point (1,1,(3)^(1/2))
1) find derivitive of r(t) ( r prime)
2) find the magnitute of r' (t)
3) use unit tangent vector =
T(t)=r'(t)/ ||r'(t)||
4) plug point into r'(t)
5) tangent line =
L(t) = (r'( point) ) •t) + original point
6) solve fie x , y, z
Find T(t), N(t) , at, and an for curve r(t)=4t^2i+ 6tj
1) find r prime
2) find magnitude of r'(t)
3) use unit tangent vector =
T(t)=r'(t)/||r'(t)||
4) take derivative of T(t) = T'(t)
5) find magnitude of T'(t) = ||T'(t)||
6) use normal vector = N(t)= T'(t)/||T'(t) ||
7 ) use v(t) = r'(t) and
And a(t)= v'(t)
8) tangental componen of acceleration =
at=(v•a)/||v||
9) normal component of acceleration = (||a||^2+ at^2)^(1/2)
If T'(t)=0
Principle normal vector N(t) does not exist because you need T'(t) to calculate.
Tangent to the line is the line itself
A particle moves along a path given by r(t) =<t-Sin t, 1-cos t> , t> = 0
Let C be the curve represented by r(t) find all t > = 0 for which c is not smooth
1) find v(t)= r'(t)
2) c is not smooth if x(t)= 0 and y(t) = 0
Or in other words
||v(t)||= 0
3) find magnitude v(t) and solve for equal to 0 these are not smooth points
A particle moves along a path given by r(t) =<t-Sin t, 1-cos t> , t> 0
Determine whether speed is increasing or decreasing at each mentioned value of t
1) find at = (v•a)/||v||
2) solve at at each point of mentioned t
3 ) if at> 0 is increasing
If at = 0 is extrema
If at < 0 is decreasing
A particle moves along a path given by r(t) =<t-Sin t, 1-cos t> , t> 0
Find the distance ( length of the path) traveled by the particle from t=0 to t=2pi
1) find r'(t)= v(t)
2) L = length
=integral ||v(t)|| dt
3) solve integral
Find curvature of
Y=3x^2-1
At x=2
1) find y'
2) find y"
3) curvature =k=
||T'(t)||/||r'(t)|| or
= ||r'(t) x r"(t)||/
||r'(t)||^3
4) find curvature then solve at point
Find the arc length parametrization for then curve
R(t) = 4ti+3cost j + 3 Sint k
1) find r'(t)
2) find magnitude of r'(t)
3) find s(t)= integral from 0 to t ||r'(t)||dt
And solve
4) plug s from 3 into t of r(t)
Consider y= ln X
Find the point on curve at which the curvature k is at maximum
1) find Y' and y"
2) find curvature = k(X)
3) find magnitude of k(X) by making g(X) = k(X) and finding g'(X) then solving
Consider y= ln X
Find limit as K(X) approaches infinity
Solve for K(X) and then take limit
Find all points of zero curvature on graph y= sinx
- find y' and y''
- find k(X) and solve for all points of X where K(X) = 0 and use n as a scalar if needed
A circle of radius 2016 is dropped into the parabola y=x^2
What is the radius of the largest circle that will touch the vertex
1) find y' and y''
2) find curvature= k=
||r' X r'' ||/ ||r'||
3) find maximum using y ' = 0
4) solve for k at maximum
5 ) radius of largest circle using k=1/r
A circle of radius 2016 is dropped into the parabola y=x^2
How close will the circle come to the vertex of the parabola
1) calculate radius as in previous example
2) if r> 1/2 the circle the circle will be stuck between the 2 sides of the parabola
3) find distance between vertex and edge of circle
Determine whether z Is a function of X and y
If z can be eliminated it is a function
Z= 1 definite equation
If z cannot be eliminated it is not a function
Z= +- equation , mult equations
Find domain
F= (9-x^2-y^2 ) ^(1/2)
Solve for function under square root = 0
X^2+ y^2 = 3^2
Equation of a disk with center (0,0) and radius 3
Find domain
F= ln (1-xy)
Solve for 1-xy > 0
Describe level curve of 2x^2 + 4y^2 for c=2
Solve and get
(X^2)/1^2+ y^2/ (1/(1/(2)^(1/2))^ 2 =1
Equation of ellipse with semis is 1 and 1/ (2^(1/2))
Evaluate parametric equation at s(0)
Solve x(0) and y(0)
Indicate dorection of curve
Make a table plugging in at least 2 values
Compute dy/dx
1) take derivative of y ( dy/dt)
2 ) take derivative X ( dx/dt)
3) place derivative of y over derivative of X = dy/dx and simplify
Find an equation of the tangent line to the curve at t=0
Plug 0 into dy/dx
Plug 0 into X and y = x0 and y0
Equation of - line
Y-y0= slope (X-x0)
Find velocity of scrambler
V(t) = s't= < X ' , y'>
Determine times at which speed is zero
Take magnitude v(t) and square
Set equal to zero and solve
Find horizontal tangent points
Dy= 0
Dx can't equal 0
Find vertical tangent points
Dx = 0
Dy can't equal 0
Find area enclosed by scrambler
Area = integral a/ b y dx
Find arc length of scrambler
Arc length = || v(t) ||dt
Find curvature of scrambler at t=0
Find s"
Solve a' and s" at find curvature
Evaluate fx and fy ( partials ) at given point
Take derivative with respect to X
Take derivative with respect to y
Plug in points
Find all values X and y such that fx = 0 and fy = 0
Take derivative with respect to X
Take derivative with respect to y
Set equal to 0 and solve
May have to plug fy into fx to solve this will give specific sets
Find all second order derivatives
Fxx
fyy
Fxy= fyx
Show that the mixed partial derivatives are equal
Solve along both paths until equal
Show function satisfies heat equation
Take partial derivatives fx and ft and fxx
Plug in to make equations equal
Compute partials fx and fy using limit def
Fx = lim as X goes to zero (f( X +delta X ) - f (X))/ delta X
Repeat with y
Determine greatest negative factor at point
Find partials vi and vr
Plug in point
Lesser number equals greates negative factor
Find total differential
Fx = partial f/ partial X
Fy = partial f / partial y
Find derivative with respect to X and y
Df= total differential = <fx, fy> •<dx , dy>
Use total differential to approximate a square root
Make f (X,y) equal to base equation
Take partials fx and fy
Set xo and yo equal to two close easy numbers
And set delta X and delta y equal to original X - new x0 and same for Y
Solve f (x0, y0 )
Solve partial X and partial y at point after plugging in new x0 and yo
Delta f = < fx (xo, yo), fy (xo, yo > • < delta X, delta y >
Solve and add delta f to f(x0 , y0)
Find dz over dt using chain rule
Z= function of X,y
And X and y are functions of t
Calculate zx and zy
Gradient of z = < zx, zy >
Plug in fx and fy into z
Set r(t) = to <X,y>
Set r' t = <fx,fy>
Gradient of z • r' solve
Find dz over dt by fingering z into function first
Z= function of X,y
And X and y are functions of t
Plug in X y into z drift entrusts
Find partial z/ partial X
Set equation equal to f(X,y,z)
Find fx fy and fz
Partial z/ partial X = - fx/ fx
Find directional derivative at point p in directio n of point [
Find pq ( q-p)
Find pq / magnitude pq
Find fx and fy
Directional derivative = gradient p • unit vector
Find maximum directional derivative at point find gradient <fx fy fz >
Solve gradient at point = point Take magnitude= value
Always in direction of gradient
...
Find directions X Reno of highest increase
Find gradient <fx fy fz >
Solve gradient at point
Gradient = direction greatest increase
Normalize to get unit vector
Take magnitude
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