1. Calculate the second degree Taylor polynomial Tgfcr] about a for the following func tions. {a} x] = sinl:3:} where a = a. {h} HI} = {\"E where a =1. {e} HI) = e2\": around 3 =1. 2. {a} Approximate sinfo} using the second degree Taylor polynomial T2 [at] of at} 2 sink} [at a = 11'} found in Problem 1(a). (h) Approximate {5% using the seoond degree Taylor polynomial Tate} of r} = {WE (at a = 1} found in Problem 1(h}. 3. Find the rst three nonzero terms of the Maclaurin series of {a} HI) = eh sink}, and ('3) HI} = CDEKT} 111\" + It using the Maclaurin series listed below. in .- 8\" =3 e' 2' H=oo rme{1}m_1+?_+m I" :In-L :l I. T \":1 -1 _ I\" _ _*_ 1-1 m. :"Ei 1] 2u+1 3 3 + s 'r + _ I .._L:"_ 3* 2" e" 3:: In[1+:r,'l EH} 7 e?+?? 4. {a} Find a oonstant s such that y = e\" is a solution of y" 41:\". + 31: = U. (h) Find a constant e such that y = me\" is a solution of y" n19" + 41: = I]. [c] Find constants o and I? such that y = or + b is a solution of the differential equation 3;" = y2 - Eye + $2. 5. Draw direction elds for the following differential equations. (a) 9" = 1; {h} y'=yz-5y+; {c} y' = y\" - 4; {d} y\" = I - y- 5. Use Euler's Method with step size [ll to estimate y[.2} where 3; satises: {a} y' = 3:: 3:03} = 1- 03} r\" = my: em} = 3