Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
78AYU79AYU80AYU81AYU82AYU83AYU84AYU85AYU86AYU87AYU88AYU89AYU90AYU91AYU92AYU93AYU94AYU95AYU96AYU1AYU2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYU32AYU33AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYU41AYU42AYU43AYU44AYUThe distance d from P 1 =( x 1 , y 1 ) to P 1 =( x 1 , y 1 ) is d= _______. (p. 4)In space, points of the form ( x,y,0 ) lie in a plane called the _______.If v=ai+bj+ck is a vector in space, the scalars a , b , c are called the _________ of v .The squares of the direction cosines of a vector in space add up to _____.True or False In space, the dot product of two vectors is a positive number.6AYUIn Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). y=0In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). x=0In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). z=2In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). y=3In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). x=4In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). z=3In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). x=1 and y=2In Problems 7-14, describe the set of points ( x,y,z ) defined by the equation(s). x=3 and z=1In Problems 15-20, find the distance from P 1 to P 2 . P 1 =( 0,0,0 ) and P 2 =( 4,1,2 )In Problems 15-20, find the distance from P 1 to P 2 . P 1 =( 0,0,0 ) and P 2 =( 1,2,3 )In Problems 15-20, find the distance from P 1 to P 2 . P 1 =( 1,2,3 ) and P 2 =( 0,2,1 )In Problems 15-20, find the distance from P 1 to P 2 . P 1 =( 2,2,3 ) and P 2 =( 4,0,3 )In Problems 15-20, find the distance from P 1 to P 2 . P 1 =( 4,2,2 ) and P 2 =( 3,2,1 )In Problems 15-20, find the distance from P 1 to P 2 . P 1 =( 2,3,3 ) and P 2 =( 4,1,1 )In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. ( 0,0,0 );( 2,1,3 )In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. ( 0,0,0 );( 4,2,2 )In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. ( 1,2,3 );( 3,4,5 )In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. ( 5,6,1 );( 3,8,2 )In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. ( 1,0,2 );( 4,2,5 )In Problems 21-26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. ( 2,3,0 );( 6,7,1 )In Problems 27-32, the vector v has initial point P and terminal point Q . Write v in the form ai+bj+ck ; that is, find its position vector. P=( 0,0,0 ) ; Q=( 3,4,1 )In Problems 27-32, the vector v s has initial point P and terminal point Q . Write v in the form ai+bj+ck ; that is, find its position vector. P=( 0,0,0 ) ; Q=( 3,5,4 )In Problems 27-32, the vector v has initial point P and terminal point Q . Write v in the form ai+bj+ck ; that is, find its position vector. P=( 3,2,1 ) ; Q=( 5,6,0 )In Problems 27-32, the vector v has initial point P and terminal point Q . Write v in the form ai+bj+ck ; that is, find its position vector. P=( 3,2,0 ) ; Q=( 6,5,1 )In Problems 27-32, the vector v has initial point P and terminal point Q . Write v in the form ai+bj+ck ; that is, find its position vector. P=( 2,1,4 ) ; Q=( 6,2,4 )In Problems 27-32, the vector v has initial point P and terminal point Q . Write v in the form ai+bj+ck ; that is, find its position vector. P=( 1,4,2 ) ; Q=( 6,2,2 )In Problems 33-38, find v . v=3i6j2kIn Problems 33-38, find v . v=6i+12j+4kIn Problems 33-38, find v . v=ij+kIn Problems 33-38, find v . v=ij+kIn Problems 33-38, find v . v=2i+3j3kIn Problems 33-38, find v . v=6i+2j2kIn Problems 39-44, find each quantity if v=3i5j+2k and w=2i+3j2k . 2v+3wIn Problems 39-44, find each quantity if v=3i5j+2k and w=2i+3j2k . 3v2wIn Problems 39-44, find each quantity if v=3i5j+2k and w=2i+3j2k . vwIn Problems 39-44, find each quantity if v=3i5j+2k and w=2i+3j2k . v+wIn Problems 39-44, find each quantity if v=3i5j+2k and w=2i+3j2k . v wIn Problems 39-44, find each quantity if v=3i5j+2k and w=2i+3j2k . v + w45AYUIn Problems 45-50, find the unit vector in the same direction as v . v=3jIn Problems 45-50, find the unit vector in the same direction as v . v=3i6j2kIn Problems 45-50, find the unit vector in the same direction as v . v=6i+12j+4kIn Problems 45-50, find the unit vector in the same direction as v . v=i+j+kIn Problems 45-50, find the unit vector in the same direction as v . v=2ij+k51AYU52AYU53AYU54AYU55AYU56AYU57AYU58AYU59AYU60AYUIn Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). v=i+j+kIn Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). v=ijkIn Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). v=i+jIn Problems 59-66, find the direction angles of each vector. Write each vector in the form of equation (7). v=j+k65AYU66AYU67AYUThe Sphere In space, the collection of all points that arc the same distance from some fixed point is called a sphere. See the illustration. The constant distance is called the radius, and the fixed point is the center of the sphere. Show that an equation of a sphere with center at ( x 0 , y 0 , z 0 ) and radius r is ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 = r 2 [Hint: Use the Distance Formula (1).]In Problems 69 and 70, find an equation of a sphere with radius r and center P 0 . r=1 ; P 0 =( 3,1,1 )In Problems 69 and 70, find an equation of a sphere with radius r and center P 0 . r=2 ; P 0 =( 1,2,2 )In Problems 71-76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] x 2 + y 2 + z 2 +2x2y=2In Problems 71-76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] x 2 + y 2 + z 2 +2x2z=1In Problems 71-76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] x 2 + y 2 + z 2 4x+4y+2z=0In Problems 71-76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] x 2 + y 2 + z 2 4x=0In Problems 71-76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] 2 x 2 +2 y 2 +2 z 2 8x+4z=1In Problems 71-76, find the radius and center of each sphere. [Hint: Complete the square in each variable.] 3 x 2 +3 y 2 +3 z 2 +6x6y=3Work Find the work done by a force of 3 newtons acting in the direction 2i+j+2k in moving an object 2 meters from ( 0,0,0 ) to ( 0,2,0 ) .Work Find the work done by a force of 1 newton acting in the direction 2i+2j+k in moving an object 3 meters from ( 0,0,0 ) to ( 1,2,2 ) .79AYU1AYUTrue or False For any vector v,vv=0 .3AYUTrue or False uv is a vector that is parallel to both uandv .5AYUTrue or False The area of the parallelogram having uandv as adjacent sides is the magnitude of the cross product of uandv .In Problems 7-14, find the value of each determinant. [ 3 4 1 2 ]In Problems 7-14, find the value of each determinant. [ 2 5 2 3 ]In Problems 7-14, find the value of each determinant. [ 6 5 2 1 ]In Problems 7-14, find the value of each determinant. [ 4 0 5 3 ]In Problems 7-14, find the value of each determinant. [ A B C 2 1 4 1 3 1 ]In Problems 7-14, find the value of each determinant. [ A B C 0 2 4 3 1 3 ]In Problems 7-14, find the value of each determinant. [ A B C 1 3 5 5 0 2 ]In Problems 7-14, find the value of each determinant. [ A B C 1 2 3 0 2 2 ]In Problems 15-22, find (a) vw , (b) wv , (c) ww , and (d) vv . v=2i3j+k w=3i2jkIn Problems 15-22, find (a) vw , (b) wv , (c) ww , and (d) vv . v=i+3j+2k w=3i2jkIn Problems 15-22, find (a) vw , (b) wv , (c) ww , and (d) vv . v=i+j w=2i+j+kIn Problems 15-22, find (a) vw , (b) wv , (c) ww , and (d) vv . v=i4j+2k w=3i+2j+kIn Problems 15-22, find (a) vw , (b) wv , (c) ww , and (d) vv . v=2ij+2k w=jkIn Problems 15-22, find (a) vw , (b) wv , (c) ww , and (d) vv . v=3i+j+3k w=ik21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYU32AYUIn Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k u( uv )In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k v( vw )In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k u( vw )In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k ( uv )wIn Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k v( uw )In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k ( vu )wIn Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k u( vv )In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k ( ww )vIn Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k Find a vector orthogonal to both uandv .In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k Find a vector orthogonal to both uandw .In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k Find a vector orthogonal to both uandi+j .In Problems 23-44, use the given vectors u,v,andw to find each expression. u=2i3j+kv=3i+3j+2kw=i+j+3k Find a vector orthogonal to both uandj+k .In Problems 45-48, find the area of the parallelogram with one corner at P 1 and adjacent sides P 1 P 2 and P 1 P 3 . P 1 =( 0,0,0 ), P 2 =( 1,2,3 ), P 3 =( 2,3,0 )In Problems 45-48, find the area of the parallelogram with one corner at P 1 and adjacent sides P 1 P 2 and P 1 P 3 . P 1 =( 0,0,0 ), P 2 =( 2,3,1 ), P 3 =( 2,4,1 )In Problems 45-48, find the area of the parallelogram with one corner at P 1 and adjacent sides P 1 P 2 and P 1 P 3 . P 1 =( 1,2,0 ), P 2 =( 2,3,4 ), P 3 =( 0,2,3 )In Problems 45-48, find the area of the parallelogram with one corner at P 1 and adjacent sides P 1 P 2 and P 1 P 3 . P 1 =( 2,0,2 ), P 2 =( 2,1,1 ), P 3 =( 2,1,2 )In Problems 49-52, find the area of the parallelogram with vertices P 1 , P 2 , P 3 and P 4 . P 1 =( 1,1,2 ), P 2 =( 1,2,3 ), P 3 =( 2,3,0 ), P 4 =( 2,4,1 )50AYUIn Problems 49-52, find the area of the parallelogram with vertices P 1 , P 2 , P 3 and P 4 . P 1 =( 1,2,1 ), P 2 =( 4,2,3 ), P 3 =( 6,5,2 ), P 4 =( 9,5,0 )In Problems 49-52, find the area of the parallelogram with vertices P 1 , P 2 , P 3 and P 4 . P 1 =( 1,1,1 ), P 2 =( 1,2,2 ), P 3 =( 3,4,5 ), P 4 =( 3,5,4 )53AYU54AYU55AYU56AYUProve for vectors uandv that uv 2 = u 2 v 2 ( uv ) 2 [Hint: Proceed as in the proof of property (4), computing first the left side and then the right side.]58AYUShow that if uandv are orthogonal unit vectors, then uv is also a unit vector.Prove property (3).Prove property (5).Prove property (9). [Hint: Use the result of Problem 57 and the fact that if the angle between uandv , then uv= u v cos .]63AYUIn Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. y 2 =16xIn Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. x 2 25 y 2 =1In Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. y 2 25 + x 2 16 =14REIn Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. 4 x 2 y 2 =8In Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. x 2 4x=2yIn Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. y 2 4y4 x 2 +8x=4In Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. 4 x 2 +9 y 2 16x18y=119REIn Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes. 9 x 2 +4 y 2 18x+8y=23In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Parabola; focus at ( 2,0 ) ; directrix the line x=2In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Hyperbola; center at ( 0,0 ) ; focus at ( 0,4 ) ;vertex at ( 0,2 )In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Ellipse; foci at ( 3,0 ) and ( 3,0 ) ; vertex at ( 4,0 )In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Parabola; vertex at ( 2,3 ) ; focus at ( 2,4 )In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Hyperbola; center at ( 2,3 ) ; focus at ( 4,3 ) ; vertex at ( 3,3 )In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Ellipse; foci at ( 4,2 ) and ( 4,8 ) ; vertex at ( 4,10 )In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Center at ( 1,2 ) ; a=3 ; c=4 ; transverse axis parallel to the x-axis .In Problems 11-18, find an equation of the conic described. Graph the equation by hand. Vertices at ( 0,1 ) and ( 6,1 ) ; asymptote the line 3y+2x=9In Problems 19-23, identify each conic without completing the squares and without applying a rotation of axes. y 2 +4x+3y8=020RE21RE22RE23REIn Problems 24-26, rotate the axes so that the new equation contains no xy-term . Analyze and graph the new equation. 2 x 2 +5xy+2 y 2 9 2 =0In Problems 24-26, rotate the axes so that the new equation contains no xy-term . Analyze and graph the new equation. 6 x 2 +4xy+9 y 2 20=0In Problems 24-26, rotate the axes so that the new equation contains no xy-term . Analyze and graph the new equation. 4 x 2 12xy+9 y 2 +12x+8y=027RE28RE29RE30RE31REIn Problems 32-34, graph the curve whose parametric equations are given by hand and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=4t2 , y=1t ; t33REIn Problems 32-34, graph the curve whose parametric equations are given by hand and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= sec 2 t , y= tan 2 t ; 0t 435RE36RE37RE38RE39RE40RE41RE42RE43RE44RE1CT2CT3CT4CT5CT6CT7CT8CT9CT10CT11CT12CT13CT1CR2CR3CR4CR5CR6CR7CR8CR9CR10CR11CR12CRThe formula for the distance d from P 1 =( x 1 , y 1 ) to P 2 =( x 2 , y 2 ) is d= _______.(p.4)To complete the square of x 2 4x , add_______ .(pp. A28-A29)Use the Square Root Method to find the real solutions of ( x+4 ) 2 =9 .(p.A48)The point that is symmetric with respect to the x-axis to the point is_______. (pp.19-21)To graph y= ( x3 ) 2 +1 , shift the graph of y= x 2 to the right_____units and then ______1 unit.(pp. 106-114)6AYU7AYU8AYU9AYU10AYUIn Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 13-20, the graph of a parabola is given. Match each graph to its equation. (A) y 2 =4x (C) y 2 =4x (E) ( y1 ) 2 =4( x1 ) (G) ( y1 ) 2 =4( x1 ) (B) x 2 =4y (D) x 2 =4y (F) ( x+1 ) 2 =4( y+1 ) (H) ( x+1 ) 2 =4( y+1 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 4,0 ) ; vertex at ( 0,0 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 0,2 ) ; vertex at ( 0,0 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 0,3 ) ; vertex at ( 0,0 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 4,0 ) ; vertex at ( 0,0 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 2,0 ) ; directrix the line x=2In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 0,1 ) ; directrix the line y=1In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Directrix the line y= 1 2 ; vertex at ( 0,0 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Directrix the line x= 1 2 ; vertex at ( 0,0 )