Evaluate the line integral
∫CF⋅dr∫CF⋅dr,
where
F(x,y,z)=sin(x)i+cos(y)j+4xzkF(x,y,z)=sin(x)i+cos(y)j+4xzk
and C is given by the vector function
r(t)=t3i−t2j+tkr(t)=t3i−t2j+tk
, 0≤t≤10≤t≤1.
Consider the vector fields F = ze^xz cos y i − e^xz sin y j +
xe^xz cos y k G = x^2 z i + xy^2 j + yz^3 k (a) Compute div F. (b)
Use curl to determine which of F and G is conservative. (c) Find a
function f such that your answer to part (b) is equal to ∇f. (d)
Find ∇^2f.
Find dy/dx for a & b
a) sin x+cos y=1
b) cos x^2 = xe^y
c)Let f(x) = 5 /2 x^2 − e^x . Find the value of x for which the
second derivative f'' (x) equals zero.
d) For what value of the constant c is the function f continuous
on (−∞,∞)?
f(x) = {cx^2 + 2x, x < 2 ,
2x + 4, x ≥ 2}
a. 1 1 cos(x)cos(y) = -cos(x-y) + -cos(x + y) 1 l
sin(x)sin(y) = -cos(x-y)--cos(x+ y) 1 l sin(x)cos(y) =—sin(x-y)
+-sin(x + y) A DSB-FC (double sideband-full carrier) signal s(t) is
given by, s(t) = n cos(2rr/cf)+ cos(2«-/mt)cos(2«-fct) What is the
numeric value for the AM index of modulation, m, fors(f) ?
Let f(x, y) = − cos(x + y2
) and let a be the point a = ( π/2,
0).
(a) Find the direction in which f
increases most quickly at the point
a.
(b) Find the directional derivative
Duf(a) of f at
a in the direction u = (−5/13 , 12/13)
.
(c) Use Taylor’s formula to calculate a quadratic approximation
to f at a.
Let f ( x , y ) = x^ 2 + y ^3 + sin ( x ^2 + y ^3 ). Determine
the line integral of f ( x , y ) with respect to arc length over
the unit circle centered at the origin (0, 0).
Consider the function given as example in lecture: f(x, y) = (e
x cos(y), ex sin(y)) (6.2) Denote a = (0, π/3) and b = f(a). Let f
−1 be a continuous inverse of f defined in a neighborhood of b.
Find an explicit formula for f −1 and compute Df−1 (b). Compare
this with the derivative formula given by the Inverse Function
Theorem.
evaluate
C
(y + 4 sin x)
dx + (z2 + 8 cos
y) dy +
x3dz
where C is the curve
r(t) =
sin t, cos t, sin
2t
, 0 ≤ t ≤ 2π.
(Hint: Observe that C lies on the surface
z = 2xy.)