math

Math263Exam2Revi.ee# Section 13.213.3

, 14.1-1406

④ Exam problems may look like problems on this Review sheet, homework , quizzes, or class problems.. ↳ study everything !!

1) Find the unit tangent vectors Ttt) at the point w/given value of the parameter t Ahs: a) Flt)=L 4te

't

, 2 tan

'Ct) , get> , t --o a) LEist, E)

b) Flt) -- 4VII +2Ejt4tk , t--I b) LI,E.E>

2)Evaluate the following integral 14¥I+7¥ Ans: Zayn + (in I

3) Find parametric eqns for the tangent line to thecurve w/ Ahs: C l-444T, I-4T> the given parametric equations at the specified point. *e-

'" cos 14T)

, y - - e-45in(4T)

, z=E4t ; (1,0, I)

4) Find Fct) if F'A)=3ET t 4t3jtVI Te and FCD= it j Ans : test + t4j + (2(t})k

5) Find the length of the curve Flt)= cos19T) itsin (at)j -19 In ( cosGDI Ans: 9 In(VIti) I f-OR oETETH

G) Consider the vector function Ahs: Fct)= 29VIE, eat, e-

at ) a)e¥, LVTeat, e'set, -D a) Find Flt) b) e¥, C l - e'

St , vIe9t,Viet)

b) Find Nlt) c)VIe C) Find curvature KH) (else+ 1)

2

7) Find F. IT, B- at the given point Ans: F-Co, 1,0> Flt)=L6cosIt), 6sink), 6In(costs)) D= f-Tz , o,

-Tz) @ (6,0, O) B-=L

-Fei 01¥)

8) Reparametrize the curve with Respect to the arc length Ahs ! with Ezo Fest FaZe + G-E⇒j+(7tZ⇒k Fct) = 4to t (5-at)j t (7-13T)K

9) Find eggs of the normal plane & osculatingplane Ahs :

at the given point Normal : y - 15x - T

X = 5sinCst) , y = t , 2-=505134 Osculating : Xt 152=154

@ Co, it, -s)

#Y ¥ 10) Find & Sketch the domain of the function

f-(x.g)=In (4- XZ-4yd

a)Sketch the graph of the function f-Cxy) = 11-4×-52

#B/ (2) Match contour curves with surfaces

B) Draw a contour map of the function f-(x.g)= XZ- y

'

14) find the limit , if it exists

a) Iim X4-145 Ans : (x.g)→co, a) DNE

b) lim x2tyZ b) 8 (x.g)→ too) V×ztyz#-4

15) find hcxid-gcflx.us)) & the set which his continuous Ahs:

got)= telnet) & fix.D=z-xy_hcx.yj-3.IE#-ln(3IET)2tX2y2 Ahs:

b) Find the first partials a) 2-× = sin (x.y) txycoslxy)

a) fcx,y)=xsincxg) Zy = X'coscxy)

b) w --I b) wu= -I ut V8 Kutv)

'

Wv = evlutvt-sp) -

cut ✓8)2

(7) Find equation of the tangent plane @ (1,3, -10) Ahs: 2- =4x-y -H

z=2×2t YZ-Ty

I 8)Find the differential dz Ans: dz=3x2y8dx t 8x3y >dy Z =×3y8

19) find d¥ where z=×y9-Ey Ans: 2T C-x2t9xyk2xyty9) X=t2tl , y=t2-I

↳Find EE and IF Ahs: 2-= tan Cua) tE= (zu-su) secy%) U= Is-15T

✓=5s -2E LE=#(2ut5v)seEC4v)

a) Find dd# ; 9g coscx) =XZ-15 Ahs: Gysin (x) t2x- 9 cos (x) - 2g

⇒ Find the directional derivative off at the given Ans: 31-124VI

point & direction [email protected] , D= -%

237 Find the directional derivative at Coals) w/ Ahs: 12-962 direction vector J=L-6,83

10

f-Cx.g)=3 inly)

24) Find the maximum Rate of change of f at (16,3) Ahs: max= 27 and the direction in

'

which it occurs J=L 1,0)

fcx,g) = SryVI

257 Find the gradient of f-(x.g) =sinC2xt3y) Ahs: Tf- L2 (2×+22,3cos(2x-12g))

267Find the equation of the tangentplane Ans: Lox-y -182=19 y=×2 -ZZ @ (10119,9)