Find c. Show that. is an equation of a sphere, and find its center and radius. This n That. 3D Space is like, far out

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1 D Space is like, far out Introspective Intersections Inverses Gone wild Transcendental Computations This n That c <,,6 > Find c. c + ( ) Show that y + z is an equation of a sphere, and find its center and radius. + y + z y+ z + y + z ( ) + ( y + y) + ( z z) ( ) ( y y ) ( z z ) ( ) + ( y+ ) + ( z ) 8 center (,,) radius 8 Bonus (Low-00) What is the distance from the center to the z-plane? Solution: Bonus (All-00) What is the equation of the intersection of this sphere with the the z-plane? Solution: z-plane contains points of the form (,0,z) ( ) + (0 + ) + ( ) 8 ( ) + ( ), z z circle

2 c <,, 6 > Find a vector with the same direction as c but has length. Unit Vector: (length ) Vector: (length ) c <,, 6 > u c 6 <,, > 6 u <,, > 8 <,, > Find the equation of the plane that contains the points (,,-),(,0,) and(0,-,). a <, 0, ( ) ><,, > b < 0,0 ( ), ><,, > i j k a b i j + k Normal i[ 6] j[ 6] + k[ + ] 7i+ j+ k Vector Plane: 7( 0) + ( y + ) + ( z ) y + z Find the area of the parallelogram with vertices at P(0,,0), Q(,0,0) and R(,-,0) PQ < 0,0,0 0><,,0> PR < 0,,0 0><,,0> i j k 0 0 PQ PR 0 i j + k i 0j k PQ PR ( )

3 Two vectors are given a,0,, b, 7, What is the angle between aand b?. a,0,, b, 7, a b + 0+ cosθ 0 a b 69 θ cos (0) π θ 90 rad The vectors are orthogonal. Find an equation for the line through the point (,0,-) and perpendicular to the plane + y + z n <,, > direction vector for the line Point (,0,-) an equation for the line is + t, y t, z + t y 0 z + or or <, y, z >< + t,t, + t > (There are actually an infinite number of correct solutions.) L: +t y 6-t z +t and the plane +y-z-0 Determine the intersection of the line L: +t y 6-t z +t and the plane +y-z-0 (+t)+(6-t) -(+t)-0 9+6t+-0t-8-t-0-6t0 t/6/ +(/)9/ y6-(/)9/ z+(/)7/ (,y,z)(9/,9/,7/)

4 Write the equation of a solid cylinder with a diameter of whose intersection with the y-plane is a circle. + y ( ) + y 9 Diameter radius / Also, recall that any cylinder has only two of the three possible variable present. Determine whether the lines are parallel, skew, or intersecting. If they intersect, find the point of intersection. L : + t, y 6 6t, z t L : 7 + s, y + s, z 7 + s v <, 6, > v and v aren't parallel v <,, > L, L not parallel Intersection? - + t 7 + s s t- 6-6t + s 6-6t + (t-) 6-6t + 8t- 8 t t s ()- - Check z equation: t 7+? s -() 7 + (-)? - - L, L intersect ( yz,, ) ( + (),6 6(), ()) (7 + ( ),+ ( ),7+ ( )) (, 6, ) (0,~.8) (,) (,) Thefunction ( ) + is increasingand-for. f Thus, theinverse function, ( )isdefinedfor. f Remember, the graph of a function is related to a graph of its inverse by a reflection over the line y. Sketch a graphof theinversefunction, ( ). Labelthecoordinatescorrespondingtothereddotson thegraphof ( ). f f

5 Find the inverse of y State the domain and range of the inverse. e. y e y+ e ln( y+ ) ln( e ) ln( y+ ) ln( e) ln( y+ ) ln( y + ) ln( + ) f ( ) Domain : need + > 0 so, > Range : ( ) Given f ( ) e +, find f (). f ( ) e + f ( ) e Note : ( f ) f 0 (0) e (0) + () 0 () f f f 0 ( ()) (0) e f sin() d sin( ) d sin( u) du sin udu ( cos( u)) + C ln() ln() ln() u du ln() d cos() + C ln() du d ln()

6 Suppose that the differentiable function yg() has an inverse and that the graph of g passes through the origin with slope. Find the slope of the graph of g - at the origin. Given g 0 and g(0) 0. ( g ) ( ) ( ) ( ) ( g) ( ) g(0) bythrm7: ( g ) ( ) g g ( ( )) 0. g ( g (0)) g (0) Thus, ( g ) ( 0 ). dy Compute given y d, ln( + ) ln( ) y + dy + + d + dy + ln( + ) d + ln( ) (ln ) d d du ln u C ln ln C ln + + u u ln( ) du d 6

7 Find the equation of the line tangent to the curve f ( ) ln( + ) + e at 0. f( ) ln( + ) + e m f e 0 + f '( ) e e 0 '(0) 0 tan at 0 y f + e 0 (0) ln(0 ) y + b (0) + b b y + dy Compute, given y sin( ) d ( ) cos( ) Thisderivativemayonlybecomputed usinglogarithmicdifferentiation. cos( ) y ( sin( ) ) cos( ) ln y ln( sin( ) ) ln y cos( ) ln( sin( ) ) dy cos() cos () + lnsin() ( ) sin() yd sin( ) dy cos( ) cos( ) ln sin( d + ( ) ) sin( ) sin( ) y dy cos( ) cos( ) ln ( sin( )) sin( ) ( sin( )) cos( + ) d sin( ) ( e ) d ( e ) d + + ln ln + + e C 7

8 Find a unit vector parallel to the line: y+ z+ L : y+ z+ L : directionvector : v <,, > + + ( ) v v unit vector : u <,, >,, v Determine the traces and identify the surface. y + z y + z Traces parallel to the yz plane ( k): + k y z z y k hyperbolas Traces parallel to the z plane ( y k): k + z + z + k circles r fork ( 0) Traces parallel to the y plane ( z k): y + k y k hyperbolas Hyperboloid of One Sheet Compute : a)arctan(sec(0)) b π )cos (csc( )) π a)arctan(sec(0)) arctan() 0.78 sec(0) cos(0) π b)cos (csc( )) cos (.) undefined π csc( ). π sin( ) π π y cos ( ) y tan ( ), Note Thedomainof and. >. : cos ( ) is[,] 8

9 d d Given ( e ) e, show that (ln ). d d Hint: This will require differentiating implicitly. d (ln ) d Proof: y let y ln e ( Def. pg0) y dy Differentiating implicitly withrespect to : e d dy dy Solve for : d d e y y dy Since e, d. or d (ln ) d Proof: e Def pg ln let (. 0) Differentiating implicitly withrespect to e d d d d Solve for (ln ) : (ln ) ln d d e Since e d, (ln ) d. ln ln : (ln ) To close a sliding door, a person pulls on a rope with a constant force of 0 pounds at a constant angle of 60 o. Find the work done in moving the door feet to its closed position. (see pretty figure) o W cos(60 ) F d W (0)() 00 foot pounds 60 o 9