Motion in space/Domain and Range
Given the position vector r(t), how would you find the velocity vector v(t), acceleration vector a(t), and the speed s(t)?
v(t) is the first derivative of r(t)
a(t) is the second derivative of r(t)
s(t) is the magnitude of velocity v(t)
When taking a partial derivative of a function f(x, y, z) with respect to x, what do we treat the variables y and z as?
constants
What is the equation for a tangent plane to the surface S defined by the function f(x,y) at the point P=(x0, y0).
z = f(x0, y0) + fx(x0, y0)(x - x0) +fy(x0, y0)(y - y0)
What must be true about the direction vector used to find the directional derivative of f(x,y)?
The direction vector must be a unit vector
How can we determine if a point P = (a,b) is a critical point of a function f(x,y)?
If fx = fy = 0 or DNE
Find the domain and range of
f(x,y,z) = (x2) ln( x - y + z)
Domain: (x - y + z) > 0
Range: All real numbers
What is the shortcut for implicit differentiation of a curve?
Set the equation equal to zero then find
dy/dx = - fx/fy
Given a function z = f(x,y), what do fx and fy measure?
fx is the rate of change in of f in the x direction
fy is the rate of change of f in the y direction
How can we find the direction of fastest increase of a function f(x,y) at a point P = (x0,y0)? What about the direction of steepest increase?
For the fastest increase find the gradient of f at the point P. For the steepest increase, it is the negative of gradient of f at point P.
What test must we use to determine if a critical point of a function f(x,y) is local minima, local maxima, or a saddle point?
The second derivative test
Find the equations for two contour lines of
4z + 2y2 - x = 0 and describe what the would look like on the graph.
x = 2y2+ 4c
c = 1 : x = 2y2 + 4 (parabola opening to the right with x - intercept (4,0).
c = 2 : x = 2y2 + 8 (parabola opening to the right with x - intercept (8,0).
Find fx and fy of f(x,y) = exy - x at (-1,1)
fx =(y - 1)exy - x at (-1,1) = 0
fy = xexy - x at (-1,1) = -1
What do fx(x0, y0) and fy(x0, y0) represent in our tangent plane equation?
The slope of the tangent plane in the x and y direction respectively
Let f(x, y, z) = sin(xy + z), and P = (0,0,1), v = <3,0,4>. Find the gradient of f at the point P.
gradient of f
= <ycos(xy + z), xcos(xy + z), cos(xy + z)>
=<0, 0, cos(1)>
What is the formula for the discriminant of a function f(x,y) at a point (a,b)?
D(a, b) = fxx(a, b)fyy(a, b) - (fxy(a, b))2
If the velocity vector v(t) = <2t, 3t2, 1> and
r(0) = <0,0,1>, find the position vector r(t).
r(t) = <t2, t3, t + 1>
Find zt of z = 3x2 - y3 where x = 4t and y = ln(t)
zt = 24(4t) - 3(ln(t))2(1/t)
Approximate the value of f(x, y, z) = xy + yz + xz at f(2.02, 1.4, 3.1).
z = 11 + 4(2.02 - 2) + 5(1.4 - 1) + 3(3.1 - 3)
= 13.38
Find a tangent vector at P = (-1,1) to the level curve of f(x,y) = 3x - 4x2y + 5 - y3
<7,11>
Find all critical points of f(x,y) = (y - 2)x2 - y2.
(0,0), (−2,2), and (2,2)
The position vector of a particle is given by r(t) = <t2, 6t, t2 - 4t>. At what value of t is the speed minimized?
t = 1
Let f(x,y) = ln(xy) and r(s, t) = <4s + 2t, 5s - 3t>. Use the chain rule to find the partial derivatives of f(r(s, t)) with respect to s.
f(r(s, t))s = 4/(4s + 2t) + 5/(5s - 3t)
Find two tangent vectors and a normal vector to the surface S given by
z = 5xy2 - x2y3 at P = (2,1)
v = <1, 0, 5y2 - 2xy3> = <1, 0, 1>
w = <0, 1, 10xy - 3x2y2> = <0, 1, 8>
n = <1, 8, -1>
Given f(x,y) = 3x - xy + 6y2, find the direction in which the directional derivative of f is maximal and what is the maximum value at P = (3, 1)?
Direction of maximal f = <2, 9>
Max value = (85)1/2
The critical points of f(x,y) = 3x2 + 2y3 + 6xy + 12x are (-4, 2) and (-1, -1). Determine whether they are a local maxima, local minima, or saddle point.
(-4,2) is a local minima
(-1, -1) is a saddle point