Name Math 125 Project 4 (10.1-10.3) (rwo pages) 15pts For problem do the followi

Question

Name Math 125 Project 4 (10.1-10.3) (rwo pages) 15pts For problem do the following: (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Graph the Cartesian equation first, then graph the actual curve for the parametric equations with arrows showing direction. Also label IP and TP. (c) Set up the integral to find the length of the curve. Do not evaluate. 2, 1st 3 nt y -t 2. a) Find dy x 1+ lnt, y t +2 for the parametric curve b) Find an equation of the tangent line when t-1.

Explanation / Answer

1) (a) x=ln t^2 =2lnt

x/2 = lnt

t=e^(x/2) now we will plug this in 2nd equation y= t^2-2 we will get

y= (e^x/2)^2-2

y= e^x-2 answer

c) x= 2ln2

y= e^x -2

dy/dx=e^x

if t= 1 then x= 2ln(1) =0

if t=3 then x= 2ln3

so we will have length of curve

L= integral ( from 0 to 2ln3) sqrt( 1+(dy/dx)^2)

= integral ( 0 to 2ln3) sqrt( 1+e^(2x))

2) a) x= 1+lnt

dx/dt= 1/t

y= t^2+2

dy/dt= 2t

dy/dx= dy/dt/dx/dt=2t/1/t=2t^2

dy/dx= 2t^2

b) dy/dx = 2t^2  

if t= 1

then dy/dx = 2

if t=1   then x= 1+ln(1) =1=0 =1

x=1

and y= t^2+2

y= 1+2 = 3 so w e got

dx/dt= 1/t

y= t^2+2

dy/dt= 2t

dy/dx /dx/dt = dy/dx= 2t/1/t= 2t^2

dy/dx = 2t^2 Answer

b) dy/dx = 2t^2  

if t= 1

then dy/dx = 2

if t=1   then x= 1+ln(1) =1=0 =1

x=1

and y= t^2+2

y= 1+2 = 3 so w e got

y-y1=m(x-x1)    m= 2 , x1= 1 and y1=3

we got

y-3=2(x-1)

y-3=2x-2

y=2x+1 answer

3) (x, y) = (-1, sqrt3)

x= r cost

y= rsint

-1 = rcost

sqrt3=r sin t

squaring and adding both sides

1+3= r^2(cos^2t+sin^2t)

4= r^2

r= 2

also divide both we got sqrt3/-1 = tant

tan t= - sqrt3

t= 2pi/3

so we got polar coordinates   ( 2, 2pi/3)

4) r^2 =-3sec theta

we know that sectheta= 1/costheta

and x= r costheta

costheta = x/r   so w e will get

r^2 = -3/x/r

r^2*1/r= -3/x

r= -3/x

but r= sqrt(x^2+y^2)

so w e got sqrt(x^2+y^2) = -3/x

squaring both sides

x^2+y^2 =9/x^2

x^2(x^2+y^2)=9 answers

5) a) x^2+y^2 =9

we know that x^2+y^2 = r^2

so wewill have r^2 =9

so r= 3 and r= -3   answers

b) x+y = 9

x=rcostheta   and y= r sintheta

rcostheta+r sintheta =9

r(costheta+sintheta) =9

r= 9/(costheta+sintheta)   answer

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