a.Express y in terms of x given that dy/dx = (y + 2)(2x + 1)
given that y = 2 at x = 0.
b. Solve (x^2 + 1)dy/dx + 3xy = 6x.
c) Obtain a general solution of dy/dx + y/x = sin x.
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}
Let (X, dX) and (Y, dY) be metric
spaces.Define the function
d : (X × Y ) × (X × Y ) → R
by
d ((x1, y1),
(x2,y2)) =
dx(x1,x2)+dy(y1,y2)
Prove that d is a metric on X × Y .
Suppose (X, dX) and (Y, dY ) are metric spaces. Define d : (X ×Y
)×(X × Y ) → R by d((x, y),(a, b)) = dX(x, a) + dY (y, b). Prove d
is a metric on X × Y .
dx dt =ax+by dy dt =−x − y,
2. As the values of a and b are changed so that the point (a,b)
moves from one region to another, the type of the linear system
changes, that is, a bifurcation occurs. Which of these bifurcations
is important for the long-term behavior of solutions? Which of
these bifurcations corresponds to a dramatic change in the phase
plane or the x(t)and y(t)-graphs?
Let ∬[a,b]×[c,d]f(x,y)dA denote the integral of f(x,y)over the
region with a≤x≤b and c≤y≤d. Find ∬[0,1]×[0,1]f(x,y)dA given the
following: ∬[0,1]×[1,5]f(x,y)dA=2, ∬[1,2]×[0,1]f(x,y)dA=−1,
∬[1,2]×[1,5]f(x,y)dA=4, and ∬[0,2]×[0,5]f(x,y)dA=3.
Group of answer choices
2
-2
8
0
None of the above.