Mat 7 Calculus III Updated on /0/07 Dr. Firoz Review for Final Eam on Saturda December 8, 007 Chapter Vectors and Geometr of Space Section. Three-Dimensional Coordinate Sstem. Find the equation of a sphere center at (,, ) and radius. Solution: + + z + = ( ) ( ) ( ). Determine whether the points lie on a straight line a) A(5,,), B(7,9 ), C(, 5,) Solution: Check that AB =, BC = 6, AC = 4 and AB + AC = BC, The points are on a line. b) K(0,, 4), L(,, ), C(,0,). Like in a) ou can show the points are not on the same line.. Find the center and radius of the sphere given b Solution: Complete the square as ( ) ( ) ( z ) 5 (,,) and radius 5. + + 6 + 4 = z z + + + = and then center is at 4. Draw the solid rectangular bo in the first octant bounded b the plans =, =, z =. Find also the length of one of its diagonal. 5. Find the volume of the solid that lies inside both the spheres + + z + 4 + 4z + 5 = 0; + + z = 4 It is known that the volume of a cup of height h, radius r is h = π h ( r h) 6. Describe in words the region of R represented b the equations or inequalities a) = 5 b) = 5 c) > 4 d) 0 e) 0 z 6 f) = z g) i) z z + + < j) Further practice problems: + + z > h) z = 0 + = k). Given a =<, >, b =< 5, >. Find a + b, a b, a b and a + b, a b, a b + z 9
Mat 7 Calculus III Updated on /0/07 Dr. Firoz. Find a vector that has same direction as the vector <,4,5 > and magnitude 6.. Find the vector represented b the directed line segment a. initial point A(,, 4) and terminal point B(,,) b. initial point A(, ) and terminal point B(, 4) c. initial point A(4,0, ) and terminal point B (4,,) d. initial point A (5,0) and terminal point B (0, 5) 4. Find a unit vector that has the same direction as the vector a) < 9, 5 > b) i j c) 8i j + 4k 5. Given a = i k, b = i j + k, find a + b Section. The Dot Product of Vectors. Find the dot product between two given vectors. a) u =, 4 and v =< 8, > b) u = 4i k and v = i + j 4k. Given that u = 4, and v = 0, θ = 0 0 find u v. Find the angle between the vectors u =, 4 and v =< 8, > 4. Show that u =<, 6, 4 > and v =<, 9, 6 > are parallel 5. Find a unit vector that is orthogonal to both i + j and i + k Solution: Suppose a =< a, b, c > is the unit vector. Now a + b + c = and < a, b, c > i <,,0 >= 0 and < a, b, c > i <, 0, >= 0. Solve for a, b, and c for the unit vector a =< a, b, c >= ± <,, >. 6. Find direction cosines and direction angles of the vector a =<,, 6 > 7. Solution: cosα = = α = arccos( / 7), 4 + 9 + 6 7 cos β = = β = arccos(/ 7) and 4 + 9 + 6 7 6 6 cosγ = = γ = arccos( 6 / 7) 4 + 9 + 6 7 Further practice problems:. a and b are vectors such that a = 4, b = 0 and the angle between them is 0 o, find a b
Mat 7 Calculus III Updated on /0/07 Dr. Firoz. a = 4 j k and b = i + 4 j + 6k are given vectors, find a b. Show that for the basis vectors i, j, k, i i =, j j =, i k = 0, i j = 0 4. Find two unit vectors that make an angle of 60 o with the vector <, 4 > < 6, b, >, < b, b, b > orthogonal? 5. For what value of b are the vectors 6. Determine the scalar and vector projection of b =< 4, > onto a =<, > 7. If two direction angles are given α = π / 4, β = π /, find γ Section.4 The Cross Product of Vectors. Show that a =<, 4, 7 >, b =<,, 4 > Solution: One needs to verif that a ( b c) = 0 and c =< 0, 9,8 >. Find a vector perpendicular to the plane that that passes thru P(,4,6), Q(,5, ) and R(,,) Solution: Find PQ PR =< 40, 5,5 > are coplanar. Find the area of a triangle with vertices P(,4,6), Q(,5, ) and R(,,) 5 8 Solution: Area A = / PQ PR = 4. Find a b for a =<, 4, 7 >, b =<,, 4 > and show that the cross product is orthogonal to both a and b 5. Find two unit vectors orthogonal to both <,, > and <,0, > 6. Show that a ( b a) = 0 a c b c 7. Show that ( a b) ( c d ) = a d b d Solution: ( a b) ( c d ) = ( a b) v = a ( b v) = a ( b c d ) = a [( b d ) c ( b c) d ] = ( a c)( b d) ( b c)( a d ) Further practice problems:. a = 4 j k and b = i + 4 j + 6k are given vectors, find a b. Show that for an vector a, a a = 0 a zero vector. For an two vectors a,b find the angle between a and a b. 4. What can ou sa about the vectors a,b if a b = 0? 5. What can ou sa about the vectors a,b and c if a ( c b) = 0?
Mat 7 Calculus III Updated on /0/07 Dr. Firoz 6. Find two unit vectors orthogonal to both the vectors <,,> and <,0,>. 7. Show that the following vectors are orthogonal a) a = ti + t j + t k and b = i + tj + t k t t t t b) a = i + e j + e k and b = i + e j e k Section.5 Equations of lines and planes A. Lines. Find a vector equation and parametric equations for the line that passes thru (5,, ) and is parallel to v =<, 4, > Vector equation <,, z >=< 5,, > + t <, 4, > =< 5 + t,+ 4 t, t >= (5 + t) i + ( + 4 t) j + ( t) k Parametric equation = 5 + t, = + 4 t, z = t. Find a smmetric equation and parametric equations for the line that passes thru (, 4, ) and (,, ) The smmetric equation 0 0 z z0 = = z z 0 0 0 4 z = = The parametric form is = + t, = 4 t, z = t. In eample, find intersection of the line with -plane. On the -plane z = 0. Then = + t, = 4 t, z = 0 = t t = / We have the point (7/, -/, 0) B. Planes 4. Find an equation of a plane through (, 4, -) with a normal vector n =<,,4 > The plane has equation n ( r r 0) = 0 <,, 4 > <, 4, z + >= 0 ( ) + ( 4) + 4( z + ) = 0 5. Find the equation of a plane thru P(,, ), Q(, -, 6) and R(5,, 0) Derive vectors PQ =<, 4, 4 >, PR =< 4,, >, and n = PQ PR. Now ou can consider the point P(,, ) and the normal vector to find our plane ( ) + 0( ) + 4( z ) = 0
Mat 7 Calculus III Updated on /0/07 Dr. Firoz 6. Find the angle between two given planes + + z = and + z =. Notice that we have n =<,, > n n and n =<,, >. Now find cosθ = n n 7. Find the smmetric equations of the line of intersection L of two planes + + z = and + z =. Suppose n and n are the normal vectors to the given planes. Then n =<,, > and n =<,, >. The line L has direction vector v = n n =< 5,, >. Let us find a point common to both the planes letting z = 0, which could be (, 0, 0). Thus we have the z equation of L in smmetric form, = = 5 Further practice problems:. Find vector equation, parametric equations and smmetric equation for the line that passes through (5,,) and parallel to the vector b = i + 4 j k. Find parametric and smmetric equations of the line through the points (,4,-) and (,-,). At what point does the line intersect the plane?. Given two vectors r 0 =< 0,, > and r =<,,5 >. Write the line segment from r0 to r. 4. Find the equation of the plane through (,,), (,-,6) and (5,,0). 5. Find the angle between the planes + + z = and + z =. Find also the smmetric equations for the line of intersection L of these two planes. 6. Show that the planes0 + z = 5 and 5 + z = are parallel. Find the distance between them. 7. Find parametric and smmetric equations of the line through (,,0) and perpendicular to both the vectors i+j and j+k. 8. Find the cosine of angles between the planes + + z = 0 and + + z = Section.6 Clinder and quadric surfaces Homework problems: 8. Since z is missing in =, we consider = with z = k, is a hperbola on the z = k plane. The surface is hperbolic clinder.. Find the traces of 4 = + z in the planes = k, = k, and z = k. When = k: 4 = k + z is a parabola, = k: 4k = + z is a circle and z = k: 4 = + k is also a parabola Thus the surface is a circular paraboloid with ais in the ais and verte (0, 0, 0)
Mat 7 Calculus III Updated on /0/07 Dr. Firoz z. 9 + 4 + z = + + = is an ellipsoid with intercepts / 9 / 4 ( ± /,0,0), (0, ± /,0), (0,0, ± ) 4. Reduce the equation 4 + z 6 4z + 0 = 0 to one of the standard forms a classif the surface and make a rough sketch. ( ) ( z ) Solution: We find the form = + is an elliptic paraboloid verte at 4 4 (0,, ) and ais is the horizontal line =, z =. Section.6 Clinder and quadric surfaces. Plot the point with clindrical coordinates (, π /,) and find its Cartesian form.. Find the clindrical form of the Cartesian coordinate (,-,-7). Given = ρ sinφ cos θ, = ρ sinφ sin θ, z = ρ cosφ, show that + + z = ρ Section. Vectors Functions and Space Curves Eamples:. Find the domain of r t,ln, t =< > Solution: For, t, t is in I, set of real numbers. For ln( t), t < and for t, t 0. Thus the domain is [0, ). Determine lim r where t 0 sin t r = + t,, t t Solution: lim r =<,, > t 0. Describe the curve defined b r =< + t, + 5 t, + 6t > Solution: The parametric form of the curve is = + t, = + 5 t, z = + 6t, which represents a straight line thru (,, -) in the direction of the vector <,5,6 > 4. Sketch the curve r =< cos t,sin t, t > Solution: The parametric form of the curve is = cos t, = sin t, z = t, which represents a curve that spirals around a circular clinder with level curves + =. The curve is known as circular heli. See the figure at page # 85 in ou tet. 5. Find the vector function that represents the curve of intersection of the clinder + = and the plane + z =
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Solution: Consider = cos t, = sin t, then z = sin t, 0 t π. Now r =< cos t,sin t, sint >= icost + jsin t + k( sin t) 6. Find the vector equation of a line thru the points P(, 0, ) and Q(,, ) Solution: Use the formula thru two given points as r = ( t) r0 + tr, 0 t We have r = ( t) <, 0, > + t <,, >, 0 t t t t 7. Identif the curve r =< e cos0 t, e sin0 t, e > The curve is a spiral around a cone whose level curves are circles Observe that + = z Further practice problems:. Find the value of + + z in terms of t, when t + = e. t t t = e cos0 t, = e sin0 t, z = e. Find the vector equation and parametric equations of the line segment that joins a) P(,-,) to Q(4,,7) b) Q(4,,7) to P(,-,). Show that the curve with parametric equations = t cos0 t, = t sin0 t, z = t, lies on z = + 4. Given t r t, te,(sin t) / t =< + > find lim r 5. Sketch the graph of the parametric equations = cos t, = sin t, z = t. Section. Derivatives and Integrals of Vector Functions. Given Solution: r = + t,,sin t t / = t,,cost and ( t ) r t dt = t + t + c t + c t + c 4 4/ ( ),ln( ), cos t 0, find and r dt t. Find a unit tangent vector at (, 0, 0) to the vector r =< + t, te,sin t > Solution: We have ( ), t t t =< t e te,cos t > and (0) =< 0,, >, (0) =. 5 (0) The unit tangent vector at t = 0 is T (0) = = < 0,, >= j + k (0) 5 5 5
Mat 7 Calculus III Updated on /0/07 Dr. Firoz. Find parametric equation for the tangent line to the heli with parametric equations = cos t, = sin t, and z = t at P(0,, π / ) Solution: We have r =< cos t, sin t, t >, =< sin t,cos t, > ( π / ) =<,0, >. The tangent line passes thru P(0,, π / ) in the direction of ( π / ) =<,0, > having parametric equation = t, =, z = π / + t 4. Find the tangent to the heli with parametric equations = cos t, = sin t, and z = t at P(0,, π / ) Solution: Tangent at the given point is ( π/) T( π/) = = <,0, > ( π/) 5 We have r =< cos t,sin t, t >, =< sin t,cos t, > ( π / ) =<,0, > and ( π / ) = 5 5. Determine whether the semicubical parabola r =< + t, t > is smooth. Solution: ( ) =<, >=< 0,0 > for t = 0. Thus it is not smooth. r t t t 6. Given r =< cos t,sin t,t >, find Solution: π / 0 π / 0 r dt π / r dt = i sin t j cost + kt = i + j + π / 4k 7. Find the point of intersection of the tangent lines to the curve r =< sin πt,sin πt,cosπt > at the points where t = 0, and t = 0.5 Solution: =< π cos πt, π cos πt, π sinπt >, r (0) =< 0, 0, >, r (0.5) =<,,0 > and (0) =< π, π,0 >, (0.5) =< 0,0, π > The equation of the tangent line at t = 0 is <,, z >=< 0,0, > + u < π, π,0 >=< πu, uπ, > and Tangent at t = 0.5 is <,, z >=<,,0 > + v < 0,0, π >=<,, πv >. We have at the point of intersection πu =, πv = u = / π, v = / π. The point is (,, ) Further practice problems:. Given r =< cos t, sin t >, find ( π / 4). Given r =< cos t, t,sin t >, find ( π / 4). Given r e, e, te 4. Evaluate the integral t t t =< >, find r (0), r (0) r (0), T (0) 0 4 t ( + + / ) t i te j t k dt
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Section. The Arc Length and Curvature. Find the length of the arc of the circular heli with vector equation r = i cost + j sin t + kt from the point (, 0, 0) to (,0, π ) Solution: π π L = dt = dt = π 0 0 Where = < sin t,cos t, > = + =. Reparametrize the heli r = i cost + j sin t + kt with respect to arc length s measured from (, 0, 0) in the direction of increasing t. ds Solution: We have = = from eample. Now dt t s s = dt = t t = and r( t( s)) = i cos( s / ) + j sin( s / ) + k( s / ) 0. Show that the curvature of a circle of radius r is /r. T Solution: Consider u =< r cos t, r sin t > a circle of radius r. Now κ = and u u < r sin t, r cost > T = = =< sin t,cost >, T =< cos t, sin t >, T = u r T The curvature of the circle of radius r is κ = = u r 4. Find the unit normal vector and binormal vector for the circular heli r =< cos t,sin t, t > Solution: =< sin t,cos t, >, T = = < sin t,cos t, > T And N = =< cos t, sin t,0 >, T i j k B = T N = sin t cost sin t, cos t, = < > cost sin t 0 5. Find the equation of a normal plane and a osculating plane for the circular heli r =< cos t,sin t, t > at the point P(0,, π / ) Solution: The normal plane has normal vector ( π / ) =<,0, >.
Mat 7 Calculus III Updated on /0/07 Dr. Firoz The equation of the plane thru P is ( 0) + 0( ) + ( z π / ) = 0 z = + π /. The osculating plane at P contains the vectors T and N. So from eample 4, its normal vector is B = T N = < sin t, cos t, >, B( π / ) = <,0, >. The equation of the osculating plane is ( 0) + ( z π / ) = 0 z = + π / 6. Find the curvature of the parabola = + at the point (, ) Solution: The curvature is κ( ) = =. / [ + ( ) ] ( + 4 ) At =, κ () = 0.8 / ( + 4) 7. Find the osculating circle of = at (0, 0) Solution: κ( ) = = =, = 0. The radius of the osculating / [ + ( ) ] ( + 4 ) circle is ρ = /, center at (0, ½). The osculating circle has the equation + ( / ) = / 4 t t 8. Find curvature of r =< e cos t, e sin t, t > at t = 0. Solution: κ = =< e t cost e t sin t, e t sin t + e t cos t, >, (0) =<,, > t t =< e sin t,e cos t,0 >, (0) =< 0,,0 > Check that (0) (0) =<, 0, > (0) (0) Then κ(0) = = (0) t t t 9. Given r =< e, e sin t, e cost >. Find N, T, B at (, 0, ) T Solution: T =, N = and B = T N T = e t <,sin t + cos t,cost sin t > t When e = t = 0, (0) =<,, >, (0) = and
Mat 7 Calculus III Updated on /0/07 Dr. Firoz t T = = e <,sin t + cos t,cost sin t > T =< 0, cos t sin t, sin t cost > (0) T (0) T (0) = = <,, >, N(0) = = < 0,, > (0) T (0) B(0) = T (0) N(0) = <,, > 6 Further practice problems:. Find curvatures of = at the following points separatel a) (0,0), b) (,) and c) (,4). Find the arc length L for curve represented b r =< t,sint t cos t,cost + t sin t >,0 t π. Find unit tangent and unit normal vector to r =< sin t,5 t, cos t > t t 4. Find curvature of r =< e cos t, e sin t, t > at the point (0,0,0) t t t 5. Find the tangent(t) normal (N) and binormal(b) of r =< e, e sin t, e cost > at the point (,0,) Section.4 Motion in Space: Velocit and Acceleration. Find the velocit, acceleration and speed of a particle with position function r =< t +, t, t > Solution: Velocit v = =< t, t, t > Acceleration a = v = =<,6 t, > Speed v = = + ( t ) +. Find the velocit, acceleration and speed of a particle with position function r = t i + t j+ t k Solution: Velocit v = = ti + t j+ tk Acceleration a = v = = i + 6tj+ k Speed v = = + ( t ) +. Given the acceleration vector a = i + j+ tk, find velocit vector and position vector when v(0) = 0, r(0) = i + k dv Solution: a = v = = i + j+ tk. Now integrating we have velocit dt v = ti + t j+ t k+c, using condition v (0) = 0, c = 0.
Mat 7 Calculus III Updated on /0/07 Dr. Firoz dr Again, v = = = ti + t j+ t k, integrating we have position dt r = t i + t j+ t k+c, using initial condition r (0) = i + k we find c = i + k 4. Find the tangential and normal components of the acceleration vector r = ti + t j+ tk Solution: tangential component a N at = v = and the normal component is = κv =. You can find them now. See also eample 7 in our tet at page number 875. Further practice problems:. Find velocit, acceleration and speed where r =< cos t, t, sin t >. Find velocit, acceleration and speed where r =< t,4 t > at t =. Find the tangent and normal component where Chapter 4 Partial Derivatives r t, t t =< + > Section 4. Functions of Several Variables + +. Find the domain of a) f (, ) = and b) f (, ) = + + Solution: a) The domain of f is D = {(, ) } b) The domain of f is D = {(, ) + + 0, }. Find the domain and range of f (, ) = 6 Solution: The domain of f is D the circle of radius 6. = {(, ) + 6} that is all points inside and on And the range of the function of f is { z z = 6,(, ) D}. Find the domain and range of f is z = h(, ) = 4 + Solution: We have seen in chapter that the function h(, ) is an elliptic paraboloid with verte at (0, 0, 0), and opens upward. Horizontal traces are ellipses and vertical traces are parabolas. The domain is all the ordered pairs (, ) in R, that is the plane. The range is the set [0, ) of all nonnegative real numbers.
Mat 7 Calculus III Updated on /0/07 Dr. Firoz 4. Sketch all the level curves of the function 5. Find the level surfaces of the function f (, ) 6 = for k = 0,,, f (,, z) = + + z Solution: Choose different numerical values of f (,, z) and observe that k z our tet. = + + represents spheres as level surfaces. See eample 5, page # 897 at Further practice problems. Sketch the level curve of g(, ) 9. f(, ) = ln( + ), find (,), (,) = for k = f f e and sketch the domain of f. P L. Given P = bl α K α show that ln = ln b + α ln K K Section 4. Limits and Continuit. Given f (, ) =. Find the limits when (, ) (0,0) along + a) the ais b) the ais c) the line = d) the line = - e) the parabola = (0) Solution: a) Along ais = 0: lim f (,0) = = 0 (, ) (0,0) + 0 (0) b) Along ais = 0: lim f (0, ) = = 0 (, ) (0,0) 0 + c) Along the line = : lim f (, ) = = / (, ) (0,0) + d) Along the line =- : lim f (, ) = = / (, ) (0,0) + e) Along the parabola = lim f (, ) = = 0 (, ) (0,0) 4 +. Given f (, ) =. Find the limit if eists. + In Eample, we have seen different values along different lines/curves, thus the limit does not eist.. Evaluate lim (, ) (,) + 4. Evaluate lim ( + ) ln( + ) (, ) (0,0) Answer: DNE
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Solution: Where 5. Evaluate ln r / r / r lim ( + ) ln( + ) = lim r ln r = lim = lim = 0 (, ) (0,0) + + + r 0 r 0 r 0 + = r + sin lim + (, ) (0,0) Solution: DNE. + sin 0 lim = / + 0 (, ) (0,0) along ais and 0 + sin lim =. Limit (0) + (, ) (0,0) 6. Evaluate 7. Evaluate lim (, ) (0,0) 6 + 5 lim (, ) (0,0) + Answer: DNE Answer: 0 8. Homework problems: 0. 6. 6 (, ) (0,0) 4 4 + lim lim (, ) (0,0) 8 + 4 Answer: DNE Answer: DNE 8.. lim (,, z) (0,0,0) lim (, ) (0,0) 6 + + + z + + z Answer: DNE Answer: DNE 6. (, ), (, ) f = + + (0,0) 0, (, ) = (0,0) The limit lim does not eist, therefore it is discontinuous. (, ) (0,0) + + Section 4. Partial Derivatives. Find f (,), f (,), f (,), f (,), f (,), f (,) for f (, ) = + + 4
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Solution: f find the rest f (,) (,) = = 54 + 4 = 58, f (,) f (,) = = 4 + = 4. You can.. f (, ) 5 4 = + +, find f, f (, ) = e, find f, f 4. f (, t) = arctan( t ), find f, f 5. f (, ) = ln( + 5 ), find f, f, f, f, f Further practice problems t. f (, ) = sin, find f, f + 5 4. f (, ) = + +, find f, f. u(, t) = sin( at), show that u t f u = a Section 4.4 Tangent Planes and Linear Approimations + 4 + 4. Given f (, ) =, find equation of the tangent plane at (,,) 4 4 + + Solution: The equation is z = f (,)( ) + f (,)( ) Where f (,) = f (,) = 8 / 9. w z = e, find = + + z Further practice problems. Find the linearization of dw w d w d w dz f (, ) = + at the point (,) Section 4.5 The Chain Rule Forms:. Given = f ( ), = g, then. Given z = f (, ), = g, = h then d d d = dt d dt dz f d f d = + dt dt dt
Mat 7 Calculus III Updated on /0/07 Dr. Firoz z f f. Given z = f (, ), = g( s, t), = h( s, t) then = + s s s and z f f = + t t t d F 4. Implicit differentiation: =, F(, ) = 0, = f ( ) d F Eamples:. Given. Given z = + 4, = cos t, = sin t, find dz dt z z z = tan ( + ), = s t, = sln t, find,and s t z z z 4st ln t Solution: = + = + s s s + ( + ) + ( + ) z z z s s / t = + = + t t t + ( + ) + ( + ) and. Given z = f (, ), = g, = h, g() =, h() = 7, g () = 5, h () = 4 dz f (, 7) = 6, f (, 7) = 8, find, t = dt dz f d f d Solution: and are functions of one variable onl. We have = +. dt dt dt dz f d f d When t =, = + = f (,7) g () + f (,7) h () = 6 dt dt dt z M M 4. Given M = e, = uv, = u v, z = u + v find, when u =, v = u v M M M M z M M M M z Solution: use = + + and = + + u u u z u v v v z v Further practice problems z z. z = e sin, = st, = s t, find, s t g g. g( s, t) = f ( s t, t s ) and f is differentiable show that t + s = 0 s t z z. z = sinα tan β, α = s + t, β = s t find, s t
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Section 4.6 Directional Derivatives. Find the directional derivatives: a) f (, ) =, u = i + j at the point (, ) b) f (, ) = e, u = cosθi + sin θ j, θ = π / at the point (-, 0) c) f (,, z) = z + z, a = i + j k at the point (, -, 0) d) f (,, z) =, + z at the point P(,, -) in the direction from P to Q(-,, 0) 4 4. Suppose that Du f (,) = 5, Dv (,) = 0, u = i j, v = i + j. find 5 5 5 5 f (,), f (,),and Du f (, ) in the direction of origin.. 4. f e D f (, ) =,find ma u (,0) (, ) =,find min u (,0) f e D f 5. Find the equation of the tangent plane to + 4 + z = 8 at P(,, ) and determine the acute angle that the plane makes with the plane. Solution: F(,, z) =<,8,z >. Now F(,,) =<,6, > The plane thru P(,, ) has the equation ( ) + 6( ) + ( z ) = 0 The angle between two planes is the angle between the normal to the planes. Let us call the normals F(,,) =<,6, >= n and on plane < 0, 0, >= n. The angle n n θ = = n n cos cos (/ 66) Further practice problems. Find directional derivative of a) direction of origin and b) vector v =<, > f (, ) z = + + at the point (,,) in the f (, ) = ln( + ) at (,) in the direction of the p q. Find maimum rate of change of f ( p, q) = qe + pe at the point (0,0) Section 4.7 Maimum and Minimum Values. The surface. The surface. The surface 4. The surface z f = (, ) = + has relative min (absolute min) at (0, 0) z f = (, ) = ( + ) has relative ma (absolute ma) at (0, 0) z f = (, ) = + has relative min (absolute min) at (0, 0) z f (, ) 8 = = + has relative min at (, 6)
Mat 7 Calculus III Updated on /0/07 Dr. Firoz 4 4 5. The surface z = f (, ) = + 4 + has relative min at (, ) and at (-, -) and a saddle point at (0, 0) 6. Find all absolute ma and min of z = f (, ) = 6 + 7 on the closed triangular region R with vertices P(0, 0), Q(, 0) and M(0, 5) Section 4.8 The Lagrange Multiplier Method. At what point or points on the circle + = does f (, ) = have an absolute maimum, and what is that maimum? Answer: (/,/ ), ( /, / ), ½. Use Lagrange multiplier method to prove that the triangle with maimum area that has a given perimeter P is equilateral. Solution: p = + + z, s = p /, where s is the half of perimeter = constant. We need to maimize area A = s( s )( s )( s z). Let us consider f (,, z) = A = s( s )( s )( s z), + + z = p, where g(, ) = + + z. Now f = λ g s( s )( s z) = λ, s( s )( s z) = λ = and s( s )( s z) = λ, s( s )( s ) = λ = z. Use Lagrange multiplier method to find the point on the plane + z = 4 that is closest to the point (,,) Solution: Let us consider a point (,, z) on the given plane. To minimize d z = ( ) + ( ) + ( ) with z 4 + =. We consider f (,, z) = ( ) + ( ) + ( z ), g(,, z) = + z Now set f = λ g ( ) = λ, ( ) = λ,( z ) = λ, + z = 4. Solving we get λ = 4 /, = 5/, = 4 /, z = /, which is the point on the plane that has minimum distance from the given point. 4. The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass. Use Lagrange multiplier method to find the dimensions of the aquarium that minimize the cost of the materials. Solution: Hint volume V = z, C(,, z) = 5 + ( z + z) = cost to minimize. Answer: = = / 5 V, z = 5/ 4V Further practice problems. Find etreme values of multiplier method.. Find etreme values of f (, ) = + on the circle + = using Lagrange f (, ) = + 4 5inside + 6.
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Chapter 5 and Chapter 6 Evaluate the double integrals (-6). (5 ) da, R = {(, ) 0,0 5}. R R, R = {(, ) 0,0 } + +. cos( + ) da, R = {(, ) 0 π,0 π / } R 4. e + e da, R = [0, 4] [0,] R 5. ( + ) da, R is the region bounded b the curves 6. R R π da R cos( ) cos ( ), = {(, ) 0 0.5,0 } = and π = + 7. Sketch the region of integration and change the order of integration of 4 / a) b) 0 0 4 0 4 f (, ) dd f (, ) dd 8. Find the volume of the sphere b) triple integration. 9. Use polar coordinate to evaluate 0. Evaluate E + + z = a using a) double integration and 0 ( + ) / dd dv where E is the region bounded b the planes + + z = 4, = 0, = 0, z = 0.. Find the volume of the solid in the first octant bounded b the clinder z = 9 and the plane =. Evaluate zdv where E is the region in the first octant that is bounded b E clindrical solid. Evaluate the limit 4. Evaluate e d + z = 9 b the planes =, = 0, z = 0 5. Use the change of variables lim( e ( + ) + 5) = u v, = uv to evaluate the integral where R is the region bounded b the -ais and the parabola = 4 4, = 4 + 4 R da,
Mat 7 Calculus III Updated on /0/07 Dr. Firoz 6. Evaluate e da, where R is the region bounded b R = /, =, = /, = / 7. Find the volume of the region enclosed b the ellipsoid + + z = a b c 8. Evaluate sin(9 + 4 ) da, where R is the region bounded b the ellipse R 9 + 4 =. Use u =, v =, J ( u, v) = / 6 9. Evaluate ( + z ) ds, from (,0,0) to (,0, π ) along the heli C that is C represented b = cos t, = sin t, z = t, 0 t π 0. Evaluate. Evaluate π sin mt( sin mt) dt, m is a constant C 0 zd + d + dz C = t = t z = t t, :,,,0. Evaluate F dr ; F(,, z) = zi + j k, r = ti + sintj + cos tk,0 t π C. Is F(, ) =< + + ln, > conservative? If conservative then there eist a function f such that f = F, find f. 4. Use Green s theorem to show that d d = π, where C is the circle C + = 5. Show that for a scalar function f, ( f ) = 0, a zero vector 6. For a vector field F, if curl F = 0 a zero vector, what can ou sa about F? 7. Show that F =< z, z, > is conservative. Find a potential function f for which f = F 8. Is there a vector field G on R such that curl G = i + z j + z k?, where 9. Use the surface integral formula f (,, z) ds = f (,, g(, )) ds dg dg ds = + + da to evaluate d d z = +, 0, 0 s s D ds, where s is the surface given b 0. Given that r( φ, θ ) = sinφ cosθ i + sinφ sinθ j + cosφ k, show that r r = sin. Evaluate ds, where s is the surface given b s / / z = ( + ), 0, 0 φ θ φ