12.2 VECTORS. Click here for answers. Click here for solutions. (a) AB l BC l (b) CD l DA l (c) BC l DC l (d) BC l CD l DA l.

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1 ECTION. VECTOR. VECTOR A Click here for answers.. Express w in terms of the vectors u and v in the figure. v u 6 9 Find the sum of the given vectors and illustrate geometrically. 6.,,, 4 w 7.,, 5,. Write each combination of vectors as a single vector. (a) AB l BC l (b) CD l DA l (c) BC l DC l (d) BC l CD l DA l D 8., 0,, 9. 0,,, 0, 0,, 0, 0 5 Find a unit vector that has the same direction as the given vector. 0.,., 5 A., 4,. 4. i j 5., 4, 8 i 4j 7k B 5 Find a vector with representation given by the directed line segment AB l a. Draw AB l and the equivalent representation starting at the origin.. A,, B 4, 4 4. A 4,, C B, 6. A quadrilateral has one pair of opposite sides parallel and of equal length. Use vectors to prove that the other pair of opposite sides is parallel and of equal length. 5. A,, 0, B,,

2 ECTION. VECTOR. ANWER E Click here for exercises.. w = v u. (a) AC (b) CA (c) BD (d) BA 9.,,., 4., 5. 0, 0, 6. 5, 0. 5, 5. 4, , 4 9,. 9, 4 9, i + j 69 i 4 69 j k 7. 4, 5 8., 0,

3 ECTION. THE DOT PRODUCT. THE DOT PRODUCT A Click here for answers. 7 Find a b.. a, 5,. a, 8,. a 4, 7,, 4. a,,, 5. a i j 4 k, 6. a i k, a 7.,, the angle between a and b is 8 Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) 8. a,,, 9. a 6, 0,, 0. a,,. a,, b. a 6i j k,. a i j k, b, b i j b, 5 b, Find, correct to the nearest degree, the three angles of the triangle with the given vertices. 4. A,,, B 6,, 5, b 6, 4 b,, 4 b, 8, 6 b i j k b, 4, 0 b 5,, b i j k b j k C,, 0 5. P 0,, 6, Q,,, R 5, 4, 6 Determine whether the given vectors are orthogonal, parallel, or neither. 6. a, 4, b, 7. a, 4, 8. a, 8,, 9. a, 5,, 0. a i j k,. a i j 5k,. Find the values of x such that the vectors x, x and 4, x are orthogonal.. For what values of c is the angle between the vectors,, and, 0, c equal to 60? 4 8 Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 4.,, 5. 4,, 6. 8i j k 7. i 5j 4k 8.,., 0.8 b 4, b,, 5 b 4,, 9 0 Find the scalar and vector projections of b onto a. 9. a,, b 4, 0. a,, b, b i j k b i 4j k

4 ECTION. THE DOT PRODUCT. ANWER E Click here for exercises cos ( ) 5 4 ( ) 9. cos cos ( 95 5 ) 86 ( ). cos =45. cos ( 7 ) 85. cos ( 4 78 ) ,, 5. 4, 58, Parallel 7. Orthogonal 8. Neither 9. Orthogonal 0. Orthogonal. Orthogonal. 0, 6. ± 4.,, ; 7, 48, ,, ; 5, 0, , , 77 ; 56, 70, 77 5,, 4 5 ; 65, 45, 4 5 8, 8,,, 0, 9 0, 0 8 ; 6, 6, 7

5 ECTION.4 THE CRO PRODUCT.4 THE CRO PRODUCT A Click here for answers. 9 Find the cross product a b. 4. Find the area of the parallelogram with vertices P 0, 0, 0, Q 5, 0, 0, R, 6, 6, and 7, 6, 6.. a, 0,, b 0,, 0. a, 4, 0, b,, (a) Find a vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.. a,, 4, b, 0, 5. P, 0,, Q, 4, 5, R,, 7 4. a,,, b 5,, 6. P 0, 0, 0, Q,,, R 4,, 7 5. a i j k, b i j k 7. P 4, 4, 4, Q 0, 5,, R,, 6. a i j k, b i j 7 k 7. a i k, b i j 8. a,, 0, b,, 9. a,,, b 6,, 0. If a 0,, and b,, 0, find a b and b a.. If a 4, 0,, b,, 0, and c 0,, 5, show that a b c a b c.. Find two unit vectors orthogonal to both i j and. 8 9 Find the volume of the parallelepiped determined by the vectors a, b, and c. 8. a, 0, 6, b,, 8, c 8, 5, 6 9. a i j k, b i j, c i k 0. Given the points P,,, Q, 0,, R 4,, 7, and,,, find the volume of the parallelepiped with adjacent edges PQ, PR, and P.. Find the area of the parallelogram with vertices A 0,, B, 0, C 5,, and D,.

6 ECTION.4 THE CRO PRODUCT.4 ANWER E Click here for exercises.. i + k. 4i j +4k. i +4j 9k 4. 7i j k 5. i +j 6. i 0j 7k 7. i j +4k 8. i j +5k 9. 4i +5j k 0. i +6j k, i 6j +k. ± 6,, (a) 6, 4, 7 (b) (a) 0,, 7 (b) (a) 9,, 4 (b)

7 ECTION.5 EQUATION OF LINE AND PLANE.5 EQUATION OF LINE AND PLANE A Click here for answers. 4 Find a vector equation and parametric equations for the line passing through the given point and parallel to the vector a..,, 8,., 4, 5,. 0,,, 4.,,, a,, 5 a,, 6 a 6 i j k a i 7 k 5 0 Find parametric equations and symmetric equations for the line through the given points. 5.,, 8, 6, 0, 6., 0, 5, 4,, 7.,,,,, 6 8. (,, ),, 4, 9. (,, ), 0, 5, 8 0., 7, 5, 4,, 5 9 Find an equation of the plane passing through the given point and parallel to the specified plane. 9. 6, 5,, 0., 0, 8,.,, 8,., 4, 5, 6 Find an equation of the plane passing through the three given points.. 0, 0, 0,,,, x y z 0 x 5y 8z 7 4.,,,,,, 5., 0,, 0,, 4, 6.,,, 5,, 4, x 4y 6z 9 z x y,, 4, 0, 4,, 6,, 4. how that the line through the points 0,, and,, 6 is perpendicular to the line through the points 4,, and, 6,. 4 Determine whether the lines L and L are parallel, skew, or intersecting. If they intersect, find the point of intersection. x 4. L : y 5 z, 4 L : x. : y z L, 4 L : 4. L : x t, y t, z t : x s, y s, z 4 s L x 5 8 Find an equation of the plane passing through the given point and with normal vector n. 5., 4, 5, 6. 5,,, 7.,,, y z x y z n 7,, 4 n, 5, n 5 i 9 j k 8., 6, 4, n 5 i j k 7 0 Find an equation of the plane that passes through the given point and contains the specified line. 7., 6, 4 ; x t, y t, z t 8.,, ; x t, y 4t, z t 9. 0,, ; 0., 0, ; x 5t, y t, z t 4 Find the point at which the line intersects the given plane.. x t, y t, z t;. x 5, y 4 t, z t;. x t, y, z t; 4. x t, y t, z t; 5 40 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. 5. x z, x y z 6. 8x 6y z, 7. x 4y z, y z 8. x y z 4, 9. x 4y z, x y z 5 z 4x y x y z x 6y 7z 0 6x y z 5 x 6y z x 5y z, 4x y z x y z 5 0 z x y

8 ECTION.5 EQUATION OF LINE AND PLANE.5 ANWER E Click here for exercises.. r =(+t) i +( +t) j +(8+5t) k; x =+t, y = +t, z =8+5t. r =( +t) i +(4 t) j +(5+6t) k; x = +t, y =4 t, z =5+6t. r =(6t) i +(+t) j +(+t) k; x =6t, y =+t, z =+t 4. r =(+t) i j +( 7t) k; x =+t, y =, z = ( + 7t) 5. x =+4t, y = t, z =8 5t; x = y 4 = z x = +5t, y = t, z =5 t; x + 5 = y = z 5 7. x =, y =+t, z = 5t; x =, y = z x = 4t, y =4+t, z =+ t; x + 4 = y 4 = z / 9. x = + t, y =+4t, z = 9t; x +/ / = y 4 = z 9 0. x = 6t, y = 7+9t, z =5; x 6 = y +7 9, z =5. kew. Intersecting, (, 0, ) 4. kew 5. 7x + y +4z = 6. x 5y +z = x +y 4z = 8. 5x +y z = 9. x + y z = 0. x +5y +8z =70. x 4y 6z =. x +y z =. x y + z =0 4. 5x +7y +8z =4 5. 7x +6y +5z = 6. 7x y 9z =0 7. 5x +4y +8z =77 8. x +z = 9. x y + z =0 0. y + z =. (, 0, 0) (. 5,, ). (,, ) 4. (0,, ) 5. Neither, Parallel 7. Perpendicular 8. Neither, Parallel 40. Perpendicular

9 ECTION.6 CYLINDER AND QUADRIC URFACE.6 CYLINDER AND QUADRIC URFACE A Click here for answers. 5 Find the traces of the given surface in the planes x k, y k, z k. Then identify the surface and sketch it.. x y z. x z z x y x 4y z x 0. x y z 4. 4z x y 0. x y 4z 4x 6y 8z 5. 9x y z 9. 4x y z. x y 4y z 4 6 Reduce the equation to one of the standard forms, classify the surface, and sketch it.. 9x y z y z 0 6. z x 4y 7. 4x 9y z 6 0

10 ECTION.6 CYLINDER AND QUADRIC URFACE.6 ANWER E Click here for exercises.. x = k, y + z = k,circle(k>0); y = k, x k = z, parabola; z = k, x k = y,parabola Circular paraboloid with axis the x-axis 5. x = k, y + z =9 ( k ),circle( k > ); y = k, 9x z =9+k,hyperbola; z = k, 9x y =9+k,hyperbola Hyperboloid of two sheets with axis the x-axis. x = k, z = ± 4 k, two parallel lines ( k < ); y = k, x + z =4, ellipse; z = k, x = ± (k /), two parallel lines ( k < ) Elliptic cylinder with axis the y-axis x y z = Hyperboloid of one sheet with axis the z-axis 7. x + y z 6 = Hyperboloid of two sheets with axis the y-axis. x = k, z y = k, hyperbola; y = k, x + z =+k, circle; z = k, x y = k, hyperbola Hyperboloid of one sheet with axis the y-axis 4. x = k, 4z y =+k, hyperbola; y = k, 4z x =+k, hyperbola; z = k, x + y =4k,circle( k > ) Hyperboloid of two sheets with axis the z-axis 8. z =x + y Circular paraboloid with axis the z-axis 9. (x ) +4y + z = Ellipsoid

11 ECTION.6 CYLINDER AND QUADRIC URFACE 0. (x +) +(y ) 4(z +) = Hyperboloid of one sheet with center (,, ) and axis parallel to the z-axis. z =(y ) x Hyperbolic paraboloid with center (0,, 0). x = y z ( ) Hyperbolic paraboloid with saddle point (0, 0, 0). (z ) = x +(y ) (/) Elliptic cone with axis parallel to the z-axis

12 ECTION. VECTOR FUNCTION AND PACE CURVE. VECTOR FUNCTION AND PACE CURVE A Click here for answers.. Find the domain of the vector function r t ln t i t t j e t k 5 Find the limit... lim t, cos t, t l 0 lim t l 0 cos t, t t, e t 4. lim i t tan t j t l st t t 5. lim t l e t i t t j tan t k k 6 8 ketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases r t t, t, r t t, t, t r t sin t, t, cos t ; 9 0 Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve. 9. r t t, t t, t 0. r t st, t, t

13 ECTION. VECTOR FUNCTION AND PACE CURVE. ANWER E Click here for exercises.. (0, ) (, ). 0,,. 0, 0, 0 4.,, tan 5. 0,, π

14 ECTION. DERIVATIVE AND INTEGRAL OF VECTOR FUNCTION. DERIVATIVE AND INTEGRAL OF VECTOR FUNCTION A Click here for answers. (a) ketch the plane curve with the given vector equation. (b) Find r t. (c) ketch the position vector r t and the tangent vector r t for the given value of t.. r t t, t, t. r t e t i e t j,. r t sec t i tan t j, 4 7 Find the domain and derivative of the vector function. 4. r t t, t, t r t i tan t j sec t k Find the derivative of the vector function. 8. r t ln 4 t i s t j 4e t k 9. t 0 t 4 r t t 4, st 4, s6 t r t te t i t t j tan t k r t e t cos t i e t sin t j ln t k 0 4 Find the unit tangent vector T t at the point with the given value of the parameter t. 0. r t st, t t, tan t, t. r t t i sin t j cos t k, t 6. r t e t cos t i e t sin t j e t k,. r t t, t, 4t, 4. r t e t, e t, te t, 5 0 Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. 5. x t, y t, z t ;,, 6. x t, y t t, z t t t ;,, 7. x t cos t, y t sin t, z 4t; 8. x sin t, y st, z cos t; 9. x t, y s cos t, z s sin t; 0. x cos t, y e t, z e t ; Evaluate the integral.... y y t i t j t k dt 0 y t i 4t 4 j t k dt 0 4 t t 0 cos t i sin t j t sin t k dt 0,, 4,,,, t (0, 4, )

15 ECTION. DERIVATIVE AND INTEGRAL OF VECTOR FUNCTION. ANWER E Click here for exercises.. (a), (c) (b) t, t. (a), (c) (b) e t i e t j. (a), (c) (b) sec t tan t i +sec t j 4. R, r (t) =, t, t 5. {t 4 t 6}, r (t) = t, t 4, 6 t 6. { t t (n +) π,nan integer}, r (t) = ( sec t ) j + (sec t tan t) k 7. {t t }, r (t) =(+t) e t i + (t +) j + +t k 8. r (t) = t 4 t i + +t j et k 9. r (t) = e t (cos t + sin t) i + e t (cos t sin t) j + t k 0. 6,, 6. i + j k i + j + k. 46, 46, 4.,, x =+t, y =+t, z =+t 6. x =+t, y =+t, z = t 7. x = π t, y = + t, z =+4t 4 8. x = πt, y =+ t, z = 9. x = π + t, y = t, z =+t 4 0. x =, y =+6t, z = 6t... i + j + 4 k 0 i 4 j 4 k 5 i + j + 4 π 4 k

16 ECTION. ARC LENGTH AND CURVATURE. ARC LENGTH AND CURVATURE A Click here for answers. Find the length of the given curve.. r t t, sin t, cot t, a t b 5 9 Use Theorem 0 to find the curvature. 5. r t i t j t k. r t e t, e t sin t, e t cos t, 0 t 6. r t t i t j t k. r t 6t i st j t k, 0 t 7. r t t i t j 6t k 4. x t, y t, z t, 0 t 8. r t t i t 4t j t k 5. x t cos t, y t sin t, z t, 0 t 9. r t sin t i cos t j sin t k 6 8 Reparametrize the curve with respect to arc length mea-sured from the point where t 0 in the direction of increasing t. 6. r t e t sin t i e t cos t j 7. r t t i t j 5t k 8. r t cos t i sin t j cos t k, 0 t 0. Find the curvature of r t st, e t, e t at the point (0,, ). Use Formula to find the curvature.. y sx. y sin x. y ln x 9 4 (a) Find the unit tangent and unit normal vectors T t and N t. (b) Use Formula 9 to find the curvature. 9. r t sin 4t, t, cos 4t \ 0. r t 6t, st, t. r t s cos t, sin t, sin t. r t t, t, t. r t st, e t, e t 4 5 Use the formula where the dots indicate derivatives with respect to t (see Exercise 6 in the text) to find the curvature of the parametric curve. 4. x t, y t 5. x t sin t, y t cos t x y y x x y 4. r t t, t, t

17 ECTION. ARC LENGTH AND CURVATURE. ANWER E Click here for exercises.. (b a) (. e π ) ln ( + ) ( ) π 5. ln ++ π + π π + ln ( )[ ( ( )) 6. r (t (s)) = s + sin ln s + i ( ( )) ] +cos ln s + j 7. r (t (s)) = ( ) ( ) + 0 s i s j 5 0 s k 8. r (t (s)) = cos [ cos ( 4 5 s)] i + sin [ cos ( 4 5 s)] j + [ 4 5 s] k 9. (a) 4cos4t,, 4 sin 4t, sin 4t, 0, cos 4t 5 (b) (a), t, t, +t ( ) t, t, t (+t ) (b). (a) (b) (+t ) sin t, cos t, cos t, cost, sin t, sin t. (a) t, t,, t, t, t t + t + (b) (t +). (a) e t,e t,, e t + e t, e t, e t e t + e t (b) (e t +) 4. (a) t, t,, t, t, t t + t + (b) (t +) (4t +) / ( + 8t ) / +4t + t (t 4 + t +) / 6 8. (t 4t +5) / ( + cos t) / 4 (4x +) / sin x ( + cos x) / x (x +) / 6 t (9t +4) / +t 5. ( + t ) /

18 ECTION.4 MOTION IN PACE:VELOCITY AND ACCELERATION.4 MOTION IN PACE: VELOCITY AND ACCELERATION A Click here for answers.. Find the velocity, acceleration, and speed of a particle with position function r t st, t r t e t i t j e t k r t cosh t i sinh t j t k ketch the path of the particle and draw the velocity and acceleration vectors for t. 7 Find the velocity, acceleration, and speed of a particle with the given position function.. r t t, t, t 8. A gun has muzzle speed 0 m s. What angle of elevation should be used to hit an object 500 m away? 9 Find the tangential and normal components of the acceleration vector. 9. r t t 4 i t j. 4. r t t, t, t r t st, t, tst 0.. r t t sin t i cos t j r t t i 4 sin t j 4 cos t k 5. r t t i j t k. r t t i t j t k

19 ECTION.4 MOTION IN PACE:VELOCITY AND ACCELERATION.4 ANWER. E Click here for exercises. t /,, 4 t /, 0, 4 t + 5. t, 0, t, t, 0,, 4t6 + t 6. e t,, e t, e t, 0,e t, e t +4+e t 7. sinh t, cosh t,, cosh t, sinh t, 0, cosh t t t +, t +., t, t, 0,, 6t, +4t +9t 4. t, t, t, 6t,, 6t, t 8t t /,,, t/ +4t +9t t 4 t /, 0, 4 t /, 0. sin t cos t, ( cos t). 0, 4. 8t +4t 9t4 +4t +, 9t 4 +9t + 9t4 +4t +

20 ECTION 4. FUNCTION OF EVERAL VARIABLE 4. FUNCTION OF EVERAL VARIABLE A Click here for answers.. If f x, y x y 4xy 7x 0, find (a) f, (b) f, 5 (c) f x h, y (d) f x, y k (e) f x, x. If t x, y ln xy y, find (a) t, (b) t e, (c) t x, (d) t x h, y (e) t x, y k. If F x, y xy x y, find (a) F, (b) F, (c) F t, (d) F, y (e) F x, x 4. If G x, y, z x sin y cos z, find (a) G, 6, (b) G 4, 4, 0 (c) G t, t, t (d) G u, v, 0 (e) G x, x y, x 5 0 Find the domain and range of the function. 5. f x, y x y 5 6. f x, y sx y 7. f x, y x y 8. f x, y tan y x f x, y, z x yz f x, y, z x sin y z. Let f x, y e x y. (a) Evaluate f, 4. (b) Find the domain of f. (c) Find the range of f.. Let t x, y s6 9x 4y. (a) Evaluate t,. (b) Find and sketch the domain of t. (c) Find the range of t.. Let f x, y, z x ln x y z. (a) Evaluate f, 6, 4. (b) Find the domain of f. (c) Find the range of f. 4. Let f x, y, z sx y z. (a) Evaluate f,, 4. (b) Find the domain of f. (c) Find the range of f. 5 5 Find and sketch the domain of the function ketch the graph of the function. 6. f x, y x f x, y xysx y f x, y s9 x y x y 7. f x, y x y 8. f x, y tan x y x y 9. f x, y ln xy 0. f x, y ln x y. f x, y x sec y. f x, y sin x y f x, y s4 x y f x, y ln x ln sin y f x, y sy x ln y x f x, y sin y f x, y x 9y f x, y y f x, y s6 x 6y f x, y y x f x, y x f x, y x y 4x y Draw a contour map of the function showing several level curves. 4. f x, y x 5. f x, y x y y x y f x, y e (x y f x, y y cos x ) 8. f x, y x 9y 9. f x, y e xy

21 ECTION 4. FUNCTION OF EVERAL VARIABLE 4. ANWER E. (a) 7 Click here for exercises. (b) 45 (c) x +xh + h y +4xy +4hy 7x 7h +0 (d) x y ky k +4xy +4xk 7x +0 (e) 4x 7x +0. (a) 0 (b) ln e = (c) ln x (d) ln (xy + hy + y ) (e) ln (xy + kx + y + k ). (a) (b) t (c) t + (d) y +y (e) x +x { } 5. (x, y) y x 6. { (x, y) y x and x + y 9 } 7. {(x, y) y x and y x} 4. (a) (b) (c) t sin t cos t (d) u sin v (e) x cos x [sin x cos y +siny cos x] 5. R, R 6. {(x, y) x y}, {z z 0} 7. {(x, y) x + y 0}, {z z 0} 8. {(x, y) x 0}, { z π <z< π 9. {(x, y, z) yz 0}, R 0. R, R. (a) (b) R (c) {z z>0}. (a) (b) { (x, y) 4 x + 9 y } (c) {z 0 z 6}. (a) 0 (b) {(x, y, z) x + z>y} (c) R 4. (a) 5 (b) { (x, y, z) x + y + z > } (c) (0, ) } 8. { (x, y) x y π + nπ, n an integer} 9. {(x, y) xy > } 0. {(x, y) y < x }

22 ECTION 4. FUNCTION OF EVERAL VARIABLE. { (x, y) y π + nπ, n an integer} 0... {(x, y) x y and y x}.. {. (x, y) x + y 4 } {(x, y) x>0 and nπ < y < (n +)π, n an integer} {(x, y) y<x y, y>0}

23 ECTION 4. PARTIAL DERIVATIVE 4. PARTIAL DERIVATIVE A Click here for answers Find the indicated partial derivatives. u xy z ln x y z u x y z. f x, y x y 5 ; f x, 40. f x, y, z, t x y 4.. f x, y sx y; f y, 4 z t f x, y, z, t xy z t 4. f x, y xe y y; f, 0 y 4 45 Use implicit differentiation to find z x and z y. 4. f x, y sin y x ; f 4. xy yz xz 4. xyz cos x y z, y 44. x y z x y z 5. z z z x y 45. xy z x y z x y z ;, x y x y 6. z z z xsy y ;, sx x y 46. Find z x and z y if z f ax by. 7. z z x ; y y 47 5 Find all the second partial derivatives. x x 47. f x, y x y xsy 8. z xy x 4 4 ; z z 48. f x, y sin x y cos x y, x y 49. z x y 50. z cos 5x y 9. u xy sec xy ; u x 5. z t sin sx 5. z x ln t u u 0. u x ;, 5 56 Verify that the conclusion of Clairaut s Theorem holds, x t x t that is, u xy u yx.. f x, y, z xyz; f y 0,, 5. u x 5 y 4 x y x 54. u sin x cos y. f x, y, z sx y z ; f z 0,, u sin xy 56. u x y z 4. u xy yz zx; u x, u y, u z 4. u x y t 4 ; u x, u y, u t 57 6 Find the indicated partial derivative. 57. f x, y x y x 4 y; f xxx 5 4 Find the first partial derivatives of the function. 58. f x, y e xy ; f xxy 5. f x, y x y 5 x y x 59. f x, y, z x 5 x 4 y 4 z yz ; f xyz 6. f x, y x y x 4 y f x, y, z e xyz ; f yzy 7. f x, y x 4 x y y 4 8. f x, y ln x y z 6. z x sin y; 9. f x, y e x tan x y 0. f s, t s ss t y x. t x, y y tan x y. t x, y ln x ln y 6. z ln sin x y ; z y x. f x, y e xy cos x sin y 4. f s, t s s 5t u 6. u ln x y ; 5. z sinhsx 4y 6. z log x y z x y z 7. f u, v tan u v 8. f x, t e sin t x 9. z ln(x sx y ) 0. z x xy 64. If f and t are twice differentiable functions of a single variable, show that the function.. f x, y y x e t f x, y y y e t dt dt x y t u x, y xf x y yt x y f x, y, z x yz xy z f x, y, z xsyz f x, y, z x yz f x, y, z xe y ye z ze x satisfies the equation u xx u xy u yy how that the function f x,..., x n x xn n satisfies the equation y f 7. u z sin x z x f xn 0

24 ECTION 4. PARTIAL DERIVATIVE 4. ANWER E Click here for exercises x 4 +x y xy (x + y ), x y + y 4 yx (x + y ) 6. y + 7. y y x y x /, x y x 8. 4 ( xy x 4 + ) ( y 4x ), 4xy ( xy x 4 + ) 9. y sec (xy)[+xy tan (xy)] t (x + t), x (x + t) 4 5. y + z, x + z, y + x 4. xy t 4, x y t 4, 4x y t 5. f x (x, y) =x y 5 4xy +, f y (x, y) =5x y 4 x 6. f x (x, y) =6x 5 y +xy 6, f y (x, y) =6y 5 x +yx 6 7. f x (x, y) =4x +xy, f y (x, y) =x y +4y 8. f x (x, y) = x, x fy (x, y) = y + y x + y 9. f x (x, y) =e x [ tan (x y) + sec (x y) ], f y (x, y) = e x sec (x y) 0. f s (s, t) = t (s + t ) /, ft (s, t) = st (s + t ) /. g x (x, y) =xy 4 sec ( x y ), g y (x, y) =tan ( x y ) +x y sec ( x y ). g x (x, y) = x +lny, g y (x, y) = y (x +lny). f x (x, y) =e xy sin y (y cos x sin x), f y (x, y) =e xy cos x (x sin y +cosy) s 4. f s (s, t) = s 5t, t f t (s, t) = s 5t z x = cosh x +4y, z x +4y y = cosh x +4y x +4y z x = ln y x (ln x), z y = y ln x v 7. f u (u, v) = u + v, f v (u, v) = u u + v ( ) t e sin(t/x) 8. f x (x, t) = t cos, x x ( ) f t (x, t) = esin(t/x) t cos x x z x = x + y, z y = y x x + y + x + y z x = xy x x y z ( + y ln x), y = y xx +y (ln x). f x (x, y) = e x, f y (x, y) =e y. f x (x, y) = ex x, f y (x, y) = ey y. f x (x, y, z) =xyz + y, f y (x, y, z) =x z + x, f z (x, y, z) =x yz 4. f x (x, y, z) = yz, f y (x, y, z) = xz yz, f z (x, y, z) = xy yz 5. f x (x, y, z) =yzx yz, f y (x, y, z) =zx yz ln x, f z (x, y, z) =yx yz ln x 6. f x (x, y, z) =e y + ze x, f y (x, y, z) =xe y + e z, f z (x, y, z) =ye z + e x 7. u x = yz ( ) y (x + z) cos, u y = z x + z x + z cos u z = sin ( y x + z ) yz (x + z) cos 8. u x = y z [ln (x +y +z)+ u y =xyz [ln (x +y +z)+ u z =xy z [ln (x +y +z)+ ( y x + z ) x x +y +z ], y x +y +z z x +y +z ( ) y, x + z 9. u x = y z x yz, u y = x yz y z z ln x, u z = x yz y z ln x ln y 40. f x (x, y, z, t) =, fy (x, y, z, t) = z t z t, f z (x, y, z, t) = y x x y, ft (x, y, z, t) = (z t) (z t) 4. f x (x, y, z, t) =y z t 4, f y (x, y, z, t) =xyz t 4, 4. f z (x, y, z, t) =xy z t 4, f t (x, y, z, t) =4xy z t z y y x, x + z x y yz +sin(x + y + z) xz + sin (x + y + z) 4., xy +sin(x + y + z) xy + sin (x + y + z) ], ]

25 ECTION 4. PARTIAL DERIVATIVE x y z, y x x + z x + z y z x y z xy z + x y, xyz x yz xy z + x y 46. af (ax + by), bf (ax + by) 47. f xx =y, f xy =x +, fyx =x + y y, f yy = x 4y / 48. f xx = sin (x + y) cos (x y), f xy = sin (x + y)+cos(x y), f yx = sin (x + y)+cos(x y), f yy = sin (x + y) cos (x y) 49. z xx = ( x + y ) x + y, zxy = xy x + y, zyx = xy x + y, z yy = ( x +y ) x + y t (x ) 5. z xx = 4(x x ), / zxt = x x, ztx = x x, z tt =0 5. z xx =(lnt)[(lnt) ] x (ln t), z xt = x (ln t) +lntln x, z tx = x (ln t) +lntln x, t t z tt = x ln t (ln x) ln x t xy 58. y e xy ( +xy ) 59. f xyz =48x y z 60. x z ( + xyz) e xyz 6. sin y 6. csc (x y)cot(x y) 6. 7yz (x +y +z ) 50. z xx =50 [ sin (5x +y) cos (5x +y) ], z xy =0 [ sin (5x +y) cos (5x +y) ], z yx =0 [ sin (5x +y) cos (5x +y) ], z yy =8 [ sin (5x +y) cos (5x +y) ]

26 ECTION 4.4 TANGENT PLANE AND LINEAR APPROXIMATION 4.4 TANGENT PLANE AND LINEAR APPROXIMATION A Click here for answers. 9 Find an equation of the tangent plane to the given surface at the specified point.. z x 4y,. z x y,. z 5 x y, 4. z xy, 5. z sx y, 6. z y x, 7. z sin x y, 8. z ln x y, 9. z e x ln y, ; 0 Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable. 0. z xy,. z sx y,,,,, Explain why the function is differentiable at the given point. Then find the linearization L x, y of the function at that point.. f x, y y ln x,,, 8,, 5 5,, 4, 5, 9,, 0,, 0,, 0 5,,,. f x, y s x y, 0,, 0, 0 4 Find the differential of the function. 4. z x y 5. v ln x y 6. w x sin yz 7. z x 4 5x y 6xy 0 8. z x y 9. z ye xy 0. u e x cos xy. w x y y z. 6 Use differentials to approximate the value of f at the given point.. f x, y s0 x 7y,.95, f x, y ln x y, 6.9, f x, y, z x y z 4,.05, 0.9,.0 6. f x, y, z xy sin z,.99, 4.98, Use differentials to approximate the number s w x y y z (s99 s 4) 4 s0.99 e s

27 ECTION 4.4 TANGENT PLANE AND LINEAR APPROXIMATION 4.4 ANWER E Click here for exercises.. 4x +8y z =8. 6x +4y z =5. x +4y z = 6 4. x y z = 5. x y 4z = 4 6. z =8x +0y 9 7. z = x + y 8. z =x + y 9. z = e y e x +(ln)y 4. xy dx +x y dy 5. ( dx dy) x y 6. (sin yz) dx +(xz cos yz) dy +(xy cos yz) dz ( 7. 4x 0xy +6y ) dx + ( 5x +8xy ) dy 8. (x + y (xdx+ ydy) ) 9. y e xy dx + e xy ( + xy) dy 0. e x (cos xy y sin xy) dx (xe x sin xy) dy. xy dx + ( x +yz ) dy + y dz. (y + z) dx +(z x) dy (x + y) dz (y + z) π ,

28 ECTION 4.5 THE CHAIN RULE 4.5 THE CHAIN RULE A Click here for answers. 8 Use the Chain Rule to find dz dt or dw dt. 8. z x, x re st, y rse t ; y. z x y, x t, y t z z z,, when r, s, t 0. z x y, x st, y st r s t. z ln x y, x s t, 4. z xe x y, x cos t, 5. z 6x xy y, x e t, 6. z xs y, x te t, t y e t y e 7. w xy z, x sin t, y cos t, 8. w x, x st, y cos t, y y z y st y cos t z e t z e t 9. u x y, y z x p r t, y p r t, z p r t; u u u,, p r t t t 0. t z sec xy, x uv, y vw, z wu;,, u v. w cos x y, x rs t sin, y r st cos ; w w w w,,, r s t t w 9 4 Use the Chain Rule to find z s and z t. 9. z x sin y, x s t, 0. z sin x cos y, x s t,. z x x y, x se t,. z x tan xy, x t,. z x y, x s t, 4. z xe y ye x, x e t, y st y st y se t t y se y st y s t 5 Use the Chain Rule to find the indicated partial derivatives. 5. w x y z, x st, y s cos t, z s sin t; w w, when s, t 0 s t 6. u xy yz zx, x st, y e st, z t ; u u, when s 0, t s t 7. z y tan x, x t uv, y u tv ; z z z,, when t, u, v 0 t u v x. u pq p r s, p x y, q x y, r, y u u 4 s xy ;, x y 6 Use Equation 6 to find dy dx.. x xy y 8 4. y 5 x y 5x 4 5. x cos y y cos x 6. y s xy x 7 7 Use Equations 7 to find z x and z y. 7. xy yz xz x y z x y z xy z x y z x y z 0. y ze x y sin xyz 0. xy yz zx. xe y yz ze x 0. ln x yz xy z 4. The radius of a right cylinder is decreasing at a rate of. cm s while its height is increasing at a rate of cm s. At what rate is the volume of the cylinder changing when the radius is 80 cm and the height is 50 cm?

29 ECTION 4.5 THE CHAIN RULE 4.5 ANWER E Click here for exercises.. 6t 5 +4t +4t ( t) ( t ) ( t). t t (. +t ++ t +t + + ) t t [( 4. e cos t/et + cos t ) sin t et cos ] t e t e 4t ( 5. 8e t cost ) e t + ( e t 4cost ) sin t 6. e t +e t ( + t) t +e t 7. y z (cos t)+xyz ( sin t)+xy z ( e t) ( x 8. y + (sin t) t y ) + y z z e t 9. 4sx sin y +tx cos y, 4xt sin y +sx cos y 0. (s t)cosxcos y s sin x sin y, (t s)cosxcos y +tsin x sin y ( ). x 6xy e t 9x y e t, ( x 6xy ) se t +9x y se t x e t [ ]. +x y, tan xy x (xy)+ (t)+ +x y +x y set (. x y ln )( st t ), ( x y ln )( s 6st ) ( 4. xe y + e x) t, ( e y ye x) e t + ( xe y + e x) st 5., 0 6., 7. 0, 0, , 4, 9. t/ ( p ), 0, /p 0. sec (xy)[w + vzy tan (xy)], z sec (xy)tan(xy)[yu + xw], sec (xy)[u + vzx tan (xy)]. st sin (x y) [ r cos θ st sin θ ], [rt sin (x y)] ( r cos θ st sin θ ), [sr sin (x y)] ( r cos θ st sin θ ), [ rst sin (x y)] ( st cos θ + r sin θ ). x 8x (x +y)(x + y) y /, y + (x +y) x y / (5x +y) y 8 y x y x y sin x cos y cos x x sin y z y y x, x + z x y 9. y z +x y z xy z + x y, xyz +x yz xy z + x y z cos (xyz) yze x+y 0. ye x+y x cos (xyz), xz cos (xyz) ( ) ex+y yz + y z y e x+y xy cos (xyz). y +zx xy + z, yz + x yz + x. ey + ze x y + e, xey + z x y + e x. y z (x + yz) y xy z (x + yz), xyz (x + yz) z y xy z (x + yz) π cm /s 4. 6xy +0x 5y 4 +6x y x x / y / y + x / y / x y z z + x, y x z + x

30 ECTION 4.6 DIRECTIONAL DERIVATIVE AND THE GRADIENT VECTOR 4.6 DIRECTIONAL DERIVATIVE AND THE GRADIENT VECTOR A Click here for answers. 5 Find the directional derivative of f at the given point in the direction indicated by the angle. 4. f x, y, z sxyz,, 4,, v 4,, 4. f x, y x y x 4 y,,, 5. t x, y, z xe yz xye z,,,, v i j k. f x, y sin x y, 4,, 6. t x, y, z x tan y z,,,, v i j k. f x, y xe y, 5, 0, 7. t x, y, z z x y,, 6,, v i 4j k 4 4. f x, y x y,,, 5. f x, y y x,,, 6 9 (a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u. 6. f x, y x 4x y y, P 0,, 7. f x, y e x sin y, P, 4, 8. f x, y, z xy z, P,,, 9. f x, y, z xy yz xz, P, 0,, u,, 0 7 Find the directional derivative of the function at the given point in the direction of the vector v. 0. f x, y x y, 6,,. f x, y sx y, 5,,. t x, y xe xy,, 0, 4 u s5, v, v, 5 v i j. t x, y e x cos y,, 6, v i j u 5, 4 5 s5 u s, s, s 8 Find the maximum rate of change of f at the given point and the direction in which it occurs. 8. f x, y sx y, 9. f x, y cos x y, 0. f x, y xe y y,. f x, y ln x y,. f x, y, z x y z,. f x, y, z x, y y z 4 0 Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 4. xy yz zx, 5. xyz 6, 6. x y z xy 4xz 4, 7. x y z xyz 4, 8. xe yz,,,, 0, x y z 4, 4, 0, 0, 4,, 4,,,, 6, 8,, 0. x y z,,,, 0,,,

31 ECTION 4.6 DIRECTIONAL DERIVATIVE AND THE GRADIENT VECTOR 4.6 ANWER E Click here for exercises (a) ( x 8xy ) i + ( y 4x ) j (b) j (c) (a) e x sin y i + e x cos y j (b) e (i + j) (c) 0 e 8. (a) y z, xyz, xy z (b) 4, 4, (c) 0 9. (a) y + z,x+ z, yz +xz (b) 7,, 54 (c) e 6 6. π e , 4, 6 9.,, 0.. 5,,. 5,, 5,,,. 7,, 0, 4 4. (a) x + y + z = (b) x = y = z 5. (a) 6x +y +z =8 (b) (x ) = (y ) = (z ) 6 6. (a) x y + z =4 (b) x = y = z 7. (a) 8x +5y =4 (b) x = y + 8 5, z = 8. (a) x +5y = (b) x = y 5, z =5 9. (a) 4x + y + z = (b) x = y =z 4 0. (a) x +y +z +=0 (b) x += y = z +

32 ECTION 4.7 MAXIMUM AND MINUMUM VALUE 4.7 MAXIMUM AND MINUMUM VALUE A Click here for answers. 9 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function f x, y x y 4x 6y f x, y 4x y 4x y f x, y x y xy x y f x, y x xy y f x, y x y x y 4 f x, y xy x y f x, y ysx y x 6y f x, y x y 8x y xy f x, y x y x y 0 4 Find the absolute maximum and minimum values of f on the set D. 0. f x, y 5 x 4y, D is the closed triangular region with vertices 0, 0, 4, 0, and 4, 5. f x, y x xy y, D is the closed triangular region with vertices,,,, and,. f x, y ysx y x 6y, D x, y 0 x 9, 0 y 5. f x, y xy x y, D is the region bounded by the parabola y x and the line y 4 4. f x, y x x y, D x, y x y 4 5. Find the shortest distance from the point,, to the plane 6x 4y z. 6. Find the point on the plane x y z that is closest to the point 4,,. 7. Find the point on the plane x y z 4 that is closest to the origin. 8. Find the shortest distance from the point x 0, y 0, z 0 to the plane Ax By Cz D 0.

33 ECTION 4.7 MAXIMUM AND MINUMUM VALUE 4.7 ANWER E Click here for exercises.. Minimum f (, ) =. Minimum f (, ) =. Minimum f (0, ) = 4. Minimum f (, ) =, saddle point (0, 0) 5. Minimum f (0, 0) = 4,saddlepoints ( ±, ) 6. addle point (, ) 7. Maximum f (4, 4) = 8. Maximum f (, 4) = 6 9. Minima f ( ( + y),y)=0, maximum f (, ) = 0. Maximum f (4, 5) =,minimumf (4, 0) = 7. Maximum f (, ) = 7, minimum f (0, 0) = 0. Maximum f ( 5, 5) = f ( ) 9, 9 4 = 45, 4 minimum f (9, 0) = 9. Maximum f (, 4) =, minimum f (, 4) = 9 4. Maximum f (, 0) = 8, minimum f ( 4, 0) = ( 6. 5, 6, ) 5 6 ( 7., 4, ) Ax 0 + By 0 + Cz 0 + D A + B + C

34 ECTION 4.8 LAGRANGE MULTIPLIER 4.8 Lagrange Multipliers A Click here for answers. 4 Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).. f x, y x y;. f x, y xy; 9x y 4. f x, y, z x y 5z; x y z 4. f x, y, z x y z; x y 4z 4 x 4y 5 8 Use Lagrange multipliers to give an alternate solution to the indicated archived problem from ection [Archived problem 4.7.5] Find the shortest distance from the point,, to the plane 6x 4y z. 6. [Archived problem 4.7.6] Find the point on the plane x y z that is closest to the point 4,,. 7. [Archived problem 4.7.7] Find the point on the plane x y z 4 that is closest to the origin. 8. [Archived problem 4.7.8] Find the shortest distance from the point x 0, y 0, z 0 to the plane Ax By Cz D 0.

35 ECTION 4.8 LAGRANGE MULTIPLIER Answers E Click here for exercises. (. Maximum f minimum f 4 7, (. Maximum f ) 7 ( 4 7, 7 ), = f ( ) minimum f, = f (. Maximum f 5, 5, minimum f = 7, ) = 7 ( ), =, ) = (, 5 5 ) = 5, ( 5, 5, 5 5 ) = 5 ( 4 4. Maximum f 7, 4 ( minimum f 4 7, ( 6. 5,, ) ( 7., 4, ) , Ax 0 + By 0 + Cz 0 + D A + B + C 7 ) = 7, ) 4 7, 7

36 ECTION 5. DOUBLE INTEGRAL OVER RECTANGLE 5. DOUBLE INTEGRAL OVER RECTANGLE A Click here for answers.. Find approximations to xx R x y da using the same subrectangles as in Example but choosing the sample point to be the (a) upper left corner, (b) upper right corner, (c) lower left corner, (d) lower right corner of each subrectangle.. Find the approximation to the volume in Example if the Midpoint Rule is used.. (a) Estimate the volume of the solid that lies below the surface z x 4y and above the rectangle R x, y 0 x, 0 y. Use a Riemann sum with m, n, and take the sample point to be the upper right corner of each subrectangle. (b) Use the Midpoint Rule to estimate the volume of the solid in part (a). 4. If R,,, use a Riemann sum with m n 4 to estimate the value of xx R x x y da. Take the sample points to be the lower left corners of the subrectangles.

37 ECTION 5. DOUBLE INTEGRAL OVER RECTANGLE 5. ANWER E Click here for exercises.. (a) 7.75 (b) 5.75 (c) 8.75 (d) (a) 6 (b)

38 ECTION 5. ITERATED INTEGRAL 5. ITERATED INTEGRAL A Click here for answers. 4 Find and x f x, y dx x sin y da,. f x, y x y. f x, y xy x. f x, y xe x y 4. f x, y x x 6. yy, y da y R 5 Calculate the iterated integral. y 4 0 y 0 5. xsy dx dy 6. y y 0 7. x y xy dy dx 8. y 0 y 0 y 0 y e x y dy dx x 4 y dx dy yy R yy R 7. xye y da, yy R 8. xe xy da, 9., x y da yy R R x, y x 4, 0 y 6 R x, y x, 0 y R x, y 0 x, 0 y R 0, 0, R, 0, y 4 y sin x dy dx 0. y 0 y 0. sx y dx dy. 9 Calculate the double integral. yy R. y xy da, yy R y y 0 0 y y 0 0 sin x cos y dy dx sin x y dy dx R x, y x, 0 y 4. xy y da, R x, y x, y 0 x 0. Find the volume of the solid lying under the plane z x 5y and above the rectangle x, y x 0, y 4.. Find the volume of the solid lying under the circular paraboloid z x y and above the rectangle R,,.. Find the volume of the solid lying under the hyperbolic paraboloid z y x and above the square R,,.. Find the average value of f x, y x sin xy over the rectangle R 0, 0,.

39 ECTION 5. ITERATED INTEGRAL 5. ANWER E Click here for exercises.. 4x, y. 4x 6x, y. xe x ( e ), e y 4. x tan, 5. (y +) 6. e e +e ) 9. ( ( 9 ) ln ( ) 4 9 ln 8. e 9. ln π

40 ECTION 5. DOUBLE INTEGRAL OVER GENERAL REGION 5. DOUBLE INTEGRAL OVER GENERAL REGION A Click here for answers. 6 Evaluate the iterated integral.. x dx dy.. x y dy dx sin x dy dx Evaluate the double integral y 0 yy 0 y 0 y sx y 0 yx 0 yy D yy D D x, y x, x y x yy D D x, y 0 x, sx y x yy D D x, y 0 y, 0 x cos y yy D yy D D x, y 6 x 4, sin x y cos x yy D D x, y 0 y, y x y yy D 4. x y da, D is bounded by y x, y x yy D xy da, x y da, x xy da, x sin y da, x da, x y da, y xy da, y 0 yy 0 y y x 0 x y 0 y0 x y dx dy y dy dx x D x, y 0 x, x y sx D x, y y e, y x y 4 x y dy dx 5. xy da, D is bounded by y x, y x 4x 4 yy D 6. e x y da, D is bounded by y 0, y x, x yy D 7. xy da, D is the first-quadrant part of the disk with center 0, 0 and radius yy D 8. y x da, D is bounded by x y, x y yy D 9. ye x da, D is the triangular region with vertices 0, 0,, 4, and 6, Find the volume of the given solid. 0. Under the paraboloid z x y and above the region bounded by y x and x y. Under the paraboloid z x y and above the region bounded by y x and x y y. Bounded by the paraboloid z x y 4 and the planes x 0, y 0, z 0, x y. Bounded by the cylinder x z 9 and the planes x 0, y 0, z 0, x y in the first octant 4. Bounded by the planes y 0, z 0, y x, and 6x y z 6 5. Under the surface z xy and above the triangle with vertices,, 4,, and, 6. Bounded by the cylinder y z 9 and the planes y x, y 0, z 0 in the first octant 7 0 ketch the region of integration and change the order of integration. y y x f x, y dy dx 8. y y y 0 y 9. f x, y dx dy 0. y 4 y y sin x y y f x, y dy dx f x, y dx dy

41 ECTION 5. DOUBLE INTEGRAL OVER GENERAL REGION 5. ANWER E Click here for exercises.. 6 (. 6 ) π/ 8. f (x, y) dx dy 0 sin y ( cos ) ln π e 6 9e ( ) ( 5 7 ) + 9 sin ( e e + ) x f (x, y) dy dx + x f (x, y) dy dx 0 0 x 0 f (x, y) dy dx 7. f (x, y) dx dy 0 y

42 ECTION 5.4 DOUBLE INTEGRAL IN POLAR COORDINATE 5.4 DOUBLE INTEGRAL IN POLAR COORDINATE A Click here for answers. 9 Evaluate the given integral by changing to polar coordinates.. xx R x da, where R is the disk with center the origin and radius 5. xx R y da, where R is the region in the first quadrant bounded by the circle x y 9 and the lines y x and y 0. xx R xy da, where R is the region in the first quadrant that lies between the circles x y 4 and x y 5 4. xx R sx y da, where R x, y x y 9, y 0 5. xx R sin x y da, where R is the annular region x y 6 6. xx D sx y da, where D is the region that lies inside the cardioid r sin and outside the circle r 7. xx D sx y da, where D is the region bounded by the cardioid r cos 0 Use a double integral to find the area of the region. 0. The region enclosed by the cardioid r sin. The region enclosed by the lemniscate r 4 cos. The region inside the circle r cos and outside the cardioid r cos r. The smaller region bounded by the spiral, the circles r and r, and the polar axis 4 7 Use polar coordinates to find the volume of the given solid. 4. Under the cone z sx y and above the ring 4 x y 5 5. Under the plane 6x 4y z and above the disk with boundary circle x y y 6. Inside the sphere x y z 4a and outside the cylinder x y ax 7. A sphere of radius a 8. yy, where D is the region in the first x y da D quadrant enclosed by the circle x y 6 9. xx D x y da, where D is the region bounded by the spirals r and r for 0 8. Evaluate the iterated integral y y s9 x 0 0 arctan y x by converting to polar coordinates. dy dx

43 ECTION 5.4 DOUBLE INTEGRAL IN POLAR COORDINATE 5.4 ANWER E Click here for exercises.. 0 ). 9 ( π 5. π (cos cos 6) π π 9. 4π 5 ( ) π. π π 5. 5π 6. 6a ( π π a 9 4 π )

44 ECTION 5.5 APPLICATION OF DOUBLE INTEGRAL 5.5 APPLICATION OF DOUBLE INTEGRAL A Click here for answers. x, y. Electric charge is distributed over the rectangle 0 x, 5. D is the region in the first quadrant bounded by the parabola y so that the charge density at is y x and the line y ; x, y xy x, y x y (measured in coulombs per square meter). Find the total charge on the rectangle. CA 6. D is bounded by the parabola y 9 x and the x-axis; x, y y. Electric charge is distributed over the unit disk x y so that the charge density at x, y is+ x, y x y 7. D is bounded by the cardioid r sin ; x, y (measured in coulombs per square meter). Find the total charge on the disk. 8. D x, y 0 y sin x, 0 x ; x, y y 9 Find the mass and center of mass of the lamina that occupies the region D and has the given density function.. D x, y x, 0 y ; x, y x 4. D x, y 0 x, 0 y ; x, y y 9. D x, y 0 y cos x, 0 x ; x, y x 0. Find the moments of inertia I x, I y, I 0 for the lamina of Problem 5.

45 ECTION 5.5 APPLICATION OF DOUBLE INTEGRAL 5.5 ANWER E Click here for exercises C π C, ( 0, 4. 9, (, ) 5. ), ( 4, ) , ( ) 0, π, ( ) 0, 5 6 π 8., ( π, ) 6 4 9π π,,, ( π 8 (π ), π 4 )

46 ECTION 5.6 TRIPLE INTEGRAL 5.6 TRIPLE INTEGRAL A Click here for answers. xxx E x yz dv. xxx E z dv, where E is bounded by the planes x 0, y 0,. Evaluate the integral, where E x, y, z 0 x, y 0, z z 0, y z, and x z using three different orders of integration. 6 Evaluate the iterated integral.. xyz dx dy dz. 4. z sin y dx dz dy y 0 yz 0 yy 0 y ys4 z 0 y 0 0 y y y yx Evaluate the triple integral. xxx E yz dv 7., where E x, y, z 0 z, 0 y z, 0 x z xxx E e x dv 8., where E x, y, z 0 y, 0 x y, 0 z x y xxx E y dv 9., where E lies under the plane z x y and above the region in the xy-plane bounded by the curves y x, y 0, and x xxx E x dv x y z dz dy dx y 0 yx x y ys9 x y x y x y xy dz dy dx 0 yz dy dz dx 0., where E is bounded by the planes x 0, y 0, z 0, and x y z 6 CA 5 Use a triple integral to find the volume of the given solid.. The tetrahedron bounded by the coordinate planes and the plane x y 6z. The solid bounded by the elliptic cylinder 4x z 4 and the planes y 0 and y z 4. The solid bounded by the cylinder x y and the planes z 0 and x z 5. The solid enclosed by the paraboloids z x y and z 8 x y 6. Evaluate the triple integral exactly: y ysin x 0 y z x e x 5y z dy dz dx z x 7. et up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis of the solid bounded by the paraboloid x 4y 4z and the plane x 4 with density function x, y, z x y z. 8. Find the average value of the function f x, y, z x y z over the tetrahedron with vertices 0, 0, 0,, 0, 0, 0,, 0, and 0, 0,.

47 ECTION 5.6 TRIPLE INTEGRAL 5.6 ANWER E Click here for exercises , (7 e) π π 6. e ( cos 4 sin 4) (a) 8. y 4 ( y 4y +4z x + y + z ) dx dz dy (b) (x, y, z) where x = y 4 x ( x m y + y + z ) dx dz dy 4y +4z y = y 4 y ( x m y + y + z ) dx dz dy 4y +4z z = y 4 z ( x m y + y + z ) dx dz dy 4y +4z (c) 4 y 4 ( y 4y +4z x + y )( x + y + z ) dx dz dy

48 ECTION 5.7 TRIPLE INTEGRAL IN CYLINDRICAL COORDINATE 5.7 TRIPLE INTEGRAL IN CYLINDRICAL COORDINATE A Click here for answers. Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point..,,. (s, 4, s) 8 Change from rectangular to cylindrical coordinates.., 0, 0 4.,, 5. (s,, 4) 6. ( s, s, 0) 7. 4, 4, 4 8. (, s, ) 9 Write the equation in cylindrical coordinates. 9. x y z 6 0. x y z 6. x y z 6. x y z 4 ketch the solid whose volume is given by the integral and evaluate the integral. y y y4 r r dz dr d 4. y xxx E x y dv y y 0 r r dz d dr 5. Evaluate, where E is the region bounded by the cylinder x y 4 and the planes z and z. xxx E sx y dv 6. Evaluate, where E is the solid bounded by the paraboloid z 9 x y and the xy-plane. xxx E y dv 7. Evaluate, where E is the solid that lies between the cylinders x y and x y 4, above the xy-plane, and below the plane z x. xxx E xz dv 8. Evaluate, where E is bounded by the planes z 0, z y, and the cylinder x y in the half-space y 0.

49 ECTION 5.7 TRIPLE INTEGRAL IN CYLINDRICAL COORDINATE 5.7 ANWER E Click here for exercises (0,, ). (,π,0) ( 4., π 4, ) ( 5., π, 4) ( 6., π, 0) 6 4 ( 7. 4, π 4, 4) ( 8., π, ) r + z =6 r z =6 r cos θ +r sin θ +z =6 r =z (,, ) 8π 5. 4π 6. 5π 4π

50 ECTION 5.8 TRIPLE INTEGRAL IN PHERICAL COORDINATE 5.8 TRIPLE INTEGRAL IN PHERICAL COORDINATE A Click here for answers. Change from spherical to rectangular coordinates..,, 4. 4, 4, 6., 4, Change from rectangular to spherical coordinates. 4., 0, 0 5. (,, s) 6. (s, 0, ) 7. ( s,, ) 8 Change from cylindrical to spherical coordinates. 8. (s, 4, 0) 9.,, 0. 4,, 4.,, 5 5 Write the equation in spherical coordinates.. x y z 6. x y z ketch the solid whose volume is given by the integral and evaluate the integral y y y y y 0 y sec Use spherical coordinates. 8. Evaluate, where E is the solid that lies between the spheres x y z and x y z 4 in the first octant. 9. Evaluate, where E is bounded below by the cone and above by the sphere. 0. Evaluate, where E lies between the spheres and and above the cone. sin d d d sin d d d xxx E xe x y z dv xxx E sx y z dv 6 xxx E x dv. Find the volume of the solid that lies above the cone and below the sphere. 4 cos 4 4. x y z 6 5. x y z

51 ECTION 5.8 TRIPLE INTEGRAL IN PHERICAL COORDINATE 5.8 ANWER E Click here for exercises. ( ) ( ). 0,,.,, ( ) ( ),,,π, π. 4. (, π, ) ( ) π 4 4, 0, π ( 4, 4π, ) ( π, π 4, ) π (, π, ) ( π 4 4, π, ) π 4 ( ( )),π,cos ρ =4 ρ ( cos φ ) =6 ρ (sin φ cos θ + sin φ sin θ +cosφ) =6 ρ sin φ =cosφ π π π ( e 6 e ) ( ) 6 ( 8 5 π ) 0.. (0, 0,.) 0 π

52 ECTION 5.9 CHANGE OF VARIABLE IN MULTIPLE INTEGRAL 5.9 CHANGE OF VARIABLE IN MULTIPLE INTEGRAL A Click here for answers. 6 Find the Jacobian of the transformation.. x u v,. x u v,. x e u cos v, 4. x se t, y se t 5. x u v w, y u v w, 6. x u, y v, y u v y u v y e u sin v z 4w 7 8 Find the image of the set under the given transformation. 7. u, v 0 u, 0 v ; x u v, y u v 8. u, v 0 u, u v ; x u, y v z u v w 9 0 Use the given transformation to evaluate the integral. 9. xx R x 4y da, where R is the region bounded by the lines y x, y x, y x, and y x; x, y u v v u 0. xx R x y da, where R is the square with vertices 0, 0,,, 5,, and, ; x u v, y u v Evaluate the integral by making an appropriate change of variables.. xx R xy da, where R is the region bounded by the lines x y, x y, x y, and x y x y. yy, where R is the parallelogram bounded cos x y da R by the lines y x, y x, x y 0, and x y

53 ECTION 5.9 CHANGE OF VARIABLE IN MULTIPLE INTEGRAL 5.9 ANWER E Click here for exercises... 4v. e 4u 4. s vw 7. The parallelogram bounded by the lines y =x, y =x +, x =y, x =y 6 8. The figure bounded by the lines x =0, y =and the 9. parabola x = y ln (sec + tan )

54 ECTION 6. VECTOR FIELD 6. VECTOR FIELD A Click here for answers. 5 ketch the vector field F by drawing a diagram like Figure 5 or Figure F x, y x y, x y F x, y sin x, sin y. F x, y x i y j. F x, y x i y j 8. F x, y ln x y, x. F x, y y i j 4. F x, y x i y j 5. F x, y, z j k 9 4 Find the gradient vector field of f. 9. f x, y x 5 4x y 6 8 Match the vector fields F with the plots labeled I III. Give reasons for your choices. 0. f x, y sin x y I 5 I I 6. f x, y e x cos 4y. f x, y, z xyz _5 5 _ f x, y, z xy yz f x, y, z x ln y z III _5 5 _6 5 6 Find the gradient vector field f of f and sketch it. 5. f x, y x y 6. f x, y ln sx y _5 5 _5

55 ECTION 6. VECTOR FIELD 6. ANWER E Click here for exercises ( 9. 5x 4 8xy ) i ( x y ) j 0. cos(x +y) i +cos(x +y) j. e x cos 4y, 4e x sin 4y. yz, xz, xy. y, xy z, yz x 4. ln (y z), y z, x y z 5. x i y j x i + y j x + y 6. III 7. II 8. I

56 ECTION 6. LINE INTEGRAL 6. LINE INTEGRAL A Click here for answers. 4 Evaluate the line integral, where C is the given curve. 4. x C yz dx xz dy xy dz,. x C x ds,. x C y ds,. x C xy ds, C is the line segment joining, to, 4. x C x y dy, C is the arc of the parabola y x from, 4 to, 5. x C sin x dx, C is the arc of the curve x y 4 from, to, 6. x C xsy dx ysx dy, C consists of the shortest arc of the circle x y from, 0 to 0, and the line segment from 0, to 4, 7. x C xyz ds, C: x t, y sin t, z cos t, 0 t 8. x C x z ds, C: x sin t, y t, z cos t, 0 t 4 9. x C xy z ds, C: x t, y t, 0 t C: x t, y t,0 t C is the line segment from, 0, to 0,, 6 0. x, :, y st, z t C xz ds C x 6t, 0 t. x, :, y t, z t C x y z dz C x t, 0 t. x, :,, z t C yz dy xy dz C x st y t, 0 t. x C z dx z dy y dz, C consists of line segments from 0, 0, 0 to 0,,, from 0,, to,,, and from,, to,, 4 C consists of line segments from 0, 0, 0 to, 0, 0, from, 0, 0 to,,, and from,, to,, Evaluate the line integral x C F dr, where C is given by the vector function r t. 5. F x, y x y i xy j, r t t i t 4 j, 6. F x, y, z y z i x j 4y k, r t t i t j t 4 k, 0 t 7. F x, y, z x i xy j z k, r t sin t i cos t j t k, 0 t 8 9 Use a calculator to find the integral to three decimal places. 8. x x sin y ds, C: x ln t, y e t, t 9. x z ln x y ds, C: x t, y t, z t, 0 t 0. Find the work done by the force field 0 t F x, y, z xz i yx j zy k on a particle that moves along the curve r t t i t j t 4 k, 0 t

57 ECTION 6. LINE INTEGRAL 6. ANWER E... Click here for exercises. ( ) 0 / π [ ] 9 () / π

58 ECTION 6. THE FUNDAMENTAL THEOREM FOR LINE INTEGRAL 6. THE FUNDAMENTAL THEOREM FOR LINE INTEGRAL A Click here for answers. 9 Determine whether or not F is a conservative vector field.. F x, y e y i xe y j, If it is, find a function f such that F f. C: r t te t i t j, 0 t... F x, y x y i y x j F x, y x 4y i 4y x j F x, y x y i x j F x, y, z y i x z j y k, C is the line segment from,, 4 to 8,, F x, y, z xy z 4 i x y z 4 j 4x y z k, C: x t, y t, z t, 0 t 4. F x, y x y i y x j 6. F x, y, z xz sin y i x cos y j x k, 5. F x, y 4x y i x 4 y j C: r t cos t i sin t j t k, 0 t 6. F x, y y cos x cos y i sin x x sin y j 7. F x, y, z 4xe z i cos y j x e z k, 7. F x, y e x x sin y i x cos y j C: r t t i t j t 4 k, 0 t 8. F x, y ye xy 4x y i xe xy x 4 j 9. F x, y x y i xy y j 0 7 (a) Find a function f such that F f and (b) use part (a) to evaluate x C F dr along the given curve C. 0. F x, y x i y j, C is the arc of the parabola y x from, to, 9. F x, y y i x j, C is the arc of the curve y x 4 x from, 0 to, 8. F x, y xy i x y j, C: r t sin t i t j, 0 t 8 9 how that the line integral is independent of path and evaluate the integral. 8. x C x sin y dx x cos y y dy, C is any path from, 0 to 5, 9. x C y x y dx 4xy 9x 4 y dy, C is any path from, to, 0. Find the work done by the force field F x, y x y i x y j in moving an object from P 0, 0 to Q,.

59 ECTION 6. THE FUNDAMENTAL THEOREM FOR LINE INTEGRAL 6. ANWER E Click here for exercises.. f (x, y) =x xy + y + K. Not conservative. Not conservative 4. f (x, y) = x + xy + y + K 5. f (x, y) =x + x 4 y + K 6. f (x, y) =y sin x x cos y + K 7. Not conservative 8. f (x, y) =e xy + x 4 y + K 9. f (x, y) = x + xy + y + K 0. (a) f (x, y) = x + y (b) 44. (a) f (x, y) =xy (b) 6. (a) f (x, y) =x y ( (b) π +4 ) 64. (a) f (x, y) =xe y + y (b) e (a) f (x, y, z) =xy + yz (b) 5 5. (a) f (x, y, z) =x y z 4 (b) 0 6. (a) f (x, y, z) =x z + x sin y (b) π 7. (a) f (x, y, z) =x e z +siny (b) e +sin 8. 5 sin

60 ECTION 6.4 GREEN THEOREM 6.4 GREEN THEOREM A Click here for answers. 4 Evaluate the line integral by two methods: (a) directly and (b) using Green s Theorem.. x C x y dx xy dy, C is the square with vertices (0, 0), (, 0), (, ), and (0, ). x C x dx x y dy, C is the triangle with vertices (0, 0), (, ), and (0, ). x C x y dx x y dy, C consists of the arc of the parabola y x from (0, 0) to (, ) followed by the line segment from (, ) to (0, 0) 4. x C x y dx xy dy, C consists of the arc of the parabola y x from 0, 0 to, 4 and the line segments from, 4 to 0, 4 and from 0, 4 to 0, Use Green s Theorem to evaluate the line integral along the given positively oriented curve. 5. x C xy dx y 5 dy, C is the triangle with vertices 0, 0,, 0, and, 6. x C x y dx xy 5 dy, C is the square with vertices, 7. x C x dx y dy, C is the curve x 6 y 6 8. x C x y dx y dy, C is the circle x y 9. x C xy dx x dy, C is the cardioid r cos. x C y tan x dx x sin y dy, C is the boundary of the region enclosed by the parabola y x and the line y 4. x C xy dx x dy, C consists of the line segment from, 0 to, 0 and the top half of the circle x y 4. x C x y dx x y dy, C is the boundary of the region between the circles x y and x y 9 4. x, where F x, y y x y i xy C F dr j and C consists of the circle x y 4 from, 0 to (s, s) and the line segments from (s, s) to 0, 0 and from 0, 0 to, 0 5. x, where F x, y y 6 i xy 5 C F dr j and C is the ellipse 4x y 6. x, where F x, y x y i x 4 C F dr j and C is the curve x 4 y Find the area of the given region using one of the formulas in Equations The region bounded by the hypocycloid with vector equation r t cos t i sin t j, 0 t 8. The region bounded by the curve with vector equation r t cos t i sin t j, 0 t 0. x C xy e x dx x ln y dy, C consists of the line segment from 0, 0 to, 0 and the curve y sin x, 0 x