If C is the part of the circle (x2)2+(y2)2=1(x2)2+(y2)2=1 in the
first quadrant, find the following line integral with respect to
arc length.
∫C(8x−6y)ds
Consider the unit sphere x2 +y2 +z2 = 1 and the cone (z+√2)2 =
x2 +y2. Show that these surfaces are tangent where they intersect,
that is, for a point on the intersection, these surfaces have the
same tangent plane
Solve the following differential equations.
i) y'''-6y''+10y'=0
ii) dy/dx= x2/(1+y2) with y(1)=3
iii) (x2-2y)y'+2x+2xy=0
iv) Use substitution to solve t2y'+2ty=y5
for t>0
use newton raphson method to solve the following system of
nonlinear equations
x1^2+x2^2=50 ; x1*x2=25
stop after three iterations. initial guess : (x1,x2) = (2,1)
Using Matlab
1. Solve the following equations set
f1 (x1,x2) = sin (sin (x1)) +x2
f2 (x1,x2) = x1+ e^(x2)
a) Can this equation set be solved by the fixed - point
method with the following expressions? And why? Show your analysis
with a 2D graph.
g1 (x1,x2) = -e^(x2)
g2 (x1,x2) = -sin(x1)
b) Use Newton Raphson Method with initial values x1 =
-2, x2 = 1.5. (8 significant figures. Please submit the code and
results.)
Show that the set ℝ2R2, equipped with operations
(?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1)
? ⋅˜ (?,?)=(??+?−1,??−?+1)
(1)defines a vector space over ℝR.
(2)Show that the vector space ?V defined in question 1 is
isomorphic to ℝ2R2 equipped with its usual vector space operations.
This means you need to define an invertible linear map
?:?→ℝ2T:V→R2.
Evaluate or solve the following
A) dy/dx=
-(2x2+y2)/(2xy+3y2)
B)dy/dx=(1+y2)/(1+x2)xy
C) (x2+1)dy/dx+2xy=4x2 given that when
x=3,y=4
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