No calculators, books, notebooks or any other written materials are allowed. Question Points Score Total: 40

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1 Be sure this exam has 6 ages including the cover The University of British Columbia MATH 317, Section 11, Instructor Tai-Peng Tsai Midterm Exam 1 October 16 Family Name Student Number Given Name Signature No calculators, books, notebooks or any other written materials are allowed Question Points Score Total: 4 For a given curve r(t), Z t 1. s = r ( ) d, ds dt = r, ds = r (t) dt. T = r r, N = T T, B = T N 3. ale = dt ds = T r = r r, alen = dt r 3 ds 4. For y = f(x), ale(x) = f (x) [1 + (f (x)) ] 3/

2 October 16 Math 317 Midterm Exam I Page of 6 (5 oints) 1. (a) Find a vector function r(t) thatreresentsthecurveofintersectionofthe semiellisoid x + y +4z =4,y, and the cylinder x + z =1. Secifythe interval of the arameter t so that no two values of t corresond to the same oint. This is 13. #46. From x + z =1wemayset x =cost, z =sint, ( ale t< ). Substituting z =1 x into the first equation we get y =4 x 4z =3x, y = 3 x = 3 cos t Thus r(t) =(cost, 3 cos t, sin t), ( ale t< ). Remark. Many of you set x = t and z = 1 z<. t. Then you miss the art where (5 oints) (b) Find A = d dt u(t) v(t) for u(t) =(t, t=1 t,t 3 )andv(t) satisfying v(1) = (, 1, 1) and v (1) = (1, 1, 3). u(1) = (1, 1, 1), u (t) =(1, t, 3t ), u (1) = (1,, 3). By roduct rule, A = u(1) v (1) + u (1) v(1) =(1, 1, 1) (1, 1, 3) + (1,, 3) (, 1, 1) =(,, ) + ( 1, 1, 1) = (1, 3, 1)

3 October 16 Math 317 Midterm Exam I Page 3 of 6 (5 oints). (a) Find the length of the curve r(t) = t î + e tˆ + e t ˆk, ale t ale 1. r (t) =(,e t, e t ), r (t) = +e t + e t 1/ = e t + e t. The length is L = Z 1 r (t) dt = Z 1 e t + e t dt = e t e t 1 = e e 1. (5 oints) (b) Find the curvature of y = f(x) =x at (, 4). Thus ale(x) = f (x) =x, f (x) =, f (x) (1 + f (x) ) = 3/ (1 + 4x ) 3/ ale() = 17 3/.

4 October 16 Math 317 Midterm Exam I Page 4 of 6 3. Consider the curve r(t) =(cost, sin t, ln(cos t)) near P (1,, ) = r(). r (t) =( sin t, cos t, tan t), r (t) =sect = 1 cos t, T(t) = r r =( sin t cos t, cos t, sin t), r (t) =( cos t, sin t, sec t). (4 oints) (a) Find the unit normal vector N of r(t) attheointp. T (t) =(sin t cos t, sint cos t, cos t), T () = ( 1,, 1). Thus T () = and N() = T () T () = 1 ( 1,, 1). Remark. Many comuted T () by showing T (t) = 1+cos t. It takes more time. (3 oints) (b) Find the curvature of r(t) attheointp. and, by the curvature formula, r () =1, T () = ale() = T () r () =. Alternative solution. Note r () = (, 1, ), r () = ( 1,, 1), r () r () = ( 1,, 1). Thus ale() = r () r () = r () 3 1 =. 3 (3 oints) (c) Find the radius and the center of the osculating circle of r(t) attheointp. The center is The radius is P + RN =(1,, ) + R = 1 ale =. 1 ( 1,, 1) = ( 1,, 1 ). Remark. (i) Eight of you confused the circle with the shere (x 1/) + y +(z +1/) =1/. The circle is a curve. The shere is a surface. They are di erent! (ii) This roblem was quite easy for most of you.

5 October 16 Math 317 Midterm Exam I Page 5 of 6 (5 oints) 4. Suose rofessor Hinton hits a baseball 1 m above the ground toward the center field fence, which is 1 m higher than the home late and 8 3mfromthehomelate,and the ball leaves the bat at seed V (m/s) and angle 3 above the horizontal. What is the minimal seed V so that it is a home run? For simlicity, assume the gravity constant is g =1m/s, and there is no air resistance. Recall sin 3 =1/, cos 3 = 3/. Let r(t) denotetheositionoftheball,v = r,anda = v.wehave r() = (, 1), v() = V (cos 3, sin 3 )=( 3V, V ), a =(, 1). Z t 3V v(t) =v() + a( )d =(, V 1t), Z t 3Vt r(t) =r() + v( )d =(, 1+ Vt 5t ). Suose the ball flies over the fence at time t 1 >. x(t 1 )=8 3Vt1 3=, t 1 = 16 V. At time t 1 the height of the ball is at least 1, that is We get 1 ale y(t 1 )=1+ Vt 1 5t 1. V 1t 1 = 16, thus V 4. V Alternative solution. After deriving r(t) formula, with minimal seed V = V min we have 3Vt1 r(t 1 )=(, 1+ Vt 1 5t 1)=(8 3, 1). The equation for y(t 1 )givesv =1t 1. Substituting this in the x(t 1 )equation,weget t 1 =4andV =4. Remarks. (i) The air resistance is imortant: In the vacuum, the otimal angle is 45. Due to air resistance, the average homerun is launched at an angle of 9 ± 5, at seed around 45 m/s. (ii) Math rofessor C. Hinton invented the itching machine for baseball, see htts://en.wikiedia.org/wiki/pitching_machine (iii) Some of you remembered the formula of r(t) withoutderivation,andlost1oint.

2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4).

2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4). Exam 3 Study Guide Math 223 Section 12 Fall 2015 Instructor: Dr. Gilbert 1. Which of the following vector fields are conservative? If you determine that a vector field is conservative, find a valid potential