1/ show the curvature of r(t)= is equal 1/2 2/ Find equa

1/ show the curvature of r(t)= <cos t^2, sin t^2, t^2> is equal 1/2 2/ Find equa

ID: 3372575 • Letter: 1

1/ show the curvature of r(t)= <cos t^2, sin t^2, t^2> is equal 1/2

2/ Find equation of tangent line to z(x,y)= (y-x)^3*e^2y above (3,0) along y-axis

3/ k(x,y)= 4xy^4-5x^2

a) find the contour of k at (1,2)

b) find the rate of greatest decrease in g at (1,2)

c) find the derivative of g at (1,2) that toward to (4,-3) in xy-plane

d) find the equation of tangent line to surface above (1,2) that towards to (4,-3) in xy-plane

4. if t: time, r(t): position of object. Which types of acceleration that this object experience? (Explanation+ calculation)

Please explain for me all the questions for specific details. Thanks ! I'll rate 5 too

magnitude of r(t)=sqrt(cos^2 t^2+sin ^2 t^2+t^4)=sqrt(t^4+1)

so,unit vector of r(t)=(1/sqrt(t^4+1))*<cos t^2 , sin t^2 , t^2>

the tangent vector=r'(t)

then the curvature=magnitude of r'(t)/magnitude of r(t)=2

2)along y-axis means z=0

so (y-x)^3*e^(2*y)=0

differenting w.r.t x,

3*(y-x)^2*exp(2*y)*(y'-1)+(y-x)^3*exp(2*y)*2y'=0

putting x=3,y=0

we get

y'=27/21=9/7

so equation of line:

(y-0)/(x-3)=9/7

7*y=9*x-27

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