MTH 22 Exam Two - Review Problem Set Name Sketch the surface z = f(x,y). ) f(x, y) = - x2 ) 2) f(x, y) = 2 -x2 - y2 2) Find the indicated limit or state that it does not exist. 4x2 + 8xy + 4y2 ) lim (x, y) (, -) x + y ) A) 2 B) 0 C) D) No limit 4) lim (x, y) (0, 0) x2y + xy2 4) A) 2 B) 0 C) D) No limit 5) lim (x, y) (0, 0) x + y x2 + y + y2 5) A) B) 2 C) 0 D) No limit 6) lim (x, y) (0, 0) 8xy 6) A) B) 0 C) π D) - Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). ) f(x, y) = x2y x4 + y2 ) 8) f(x, y) = xy 8) Provide an appropriate response. x2y2 9) Define f(0, 0) in such a way that extends f(x, y) = A) f(0, 0) = B) f(0, 0) = 0 C) f(0, 0) = 2 to be continuous at the origin. 9) D) No definition makes f(x, y) continuous at the origin.
0) Define f(0, 0) in such a way that extends f(x, y) = 8x 2 -x2y + 8y2 to be continuous at the origin. 0) A) f(0, 0) = 8 B) f(0, 0) = 2 C) f(0, 0) = 6 D) f(0, 0) = 0 Find all the first order partial derivatives for the following function. x ) f(x, y) = x + y ) 2) f(x, y) = 2) ) f(x, y) = xye-y ) Find all the second order partial derivatives of the given function. 4) f(x, y) = x2 + y - ex+y 4) Find the value. 5) If F(x, y) = ln (), find Fx(, ) and Fy(, ). 5) A) Fx(, ) = 0, F y(, ) = 0 B) Fx(, ) = 5, F y(, ) = 5 C) Fx(, ) = 5, F y(, ) = 5 D) Fx(, ) = 5, F y(, ) = 2 Find all the second order partial derivatives of the given function. 6) f(x, y) = cos xy2 6) ) Find the slope of the tangent to the curve of the intersection of the cylinder 4z = 8 4 - x2 and the plane y = -9 at the point 5, -9, 4 A) 0 B) - 0 C) 2 D) 0 ) 8) Find the slope of the tangent to the curve of the intersection of the surface 400z = 6x2 + 25y2 and the plane x = -5 at the point -5, -4, 2 8) A) - 2 5 B) - 2 C) - 5 8 D) - 8 25 Give an appropriate answer. 9) If f(x, y, z) = x2y2 + xyz + y2z2 find fxz(x, y, z). 9) A) 2xy B) 2yz C) y D) xz 20) If f(x, y, z) = exyz find fz(x, y, z). A) xyez B) xyzexyz C) xyexyz D) zexyz 20) 2
Find the gradient f. xy2 2) f(x, y) = 2) A) () -2 [2x2yi + y2(y2 - x2)j] B) () -2 [y2(y2 - x2)i + 2x2yj] C) () 2 [y2(y2 - x2)i + 2x2yj] D) ()[2x2yi + y2(y2 - x2)j] 22) f(x, y) = yexy A) exy(xy + )i + y2exyj B) xy2exyi + exy(xy + )j C) y2exyi + yexy(xy + )j D) y2exyi + exy(xy + )j 22) 2) f(x, y, z) = xyex + z A) ex + z[(xy2 + y)i + xy2j + k] B) (xyex + z + yex + z)i + xy2ex + zj + ezk] C) (xyex + y)i + xy2ex + zj + ezk] D) ex + z[(xy + y)i + xy2j + k] 2) Compute the gradient of the function at the given point. 24) f(x, y) = -x2-4y, p = (-6, 9) A) -08i - 6j B) 6i - 6j C) 6i - 4j D) -26i - 6j 24) Find the equation of the tangent plane (or tangent ʺhyperplaneʺ for a function of three variables) at the given point p. 25) f(x, y) = 5x, p = (, -, -5). 25) y A) x + y + 5z = -5 B) 5x + 5y + z = -5 C) 0x + 0y + z = -5 D) 5x + 5y + z = -0 26) f(x, y) = sin (xy), p = (π,, 0). A) x + πy + z = 2π B) πx + πy + z = 2π C) πx + πy + z = 0 D) x + πy + z = π 26) 2) Write parametric equations for the tangent line to the curve of intersection of the surfaces z = x2 + 6y2 and z = x + y + at the point (,, ). A) x = -t +, y = 5t +, z = 2t + B) x = -t +, y = 5t +, z = 2t + C) x = -t +, y = t +, z = 2t + D) x = -t +, y = t +, z = 2t + 2) Give an appropriate answer. 28) Find all points (x, y) at which the tangent plane to the graph z = x2 + 0x - 2y2 + 4y + 2xy is horizontal. 28) A) (-4, -) B) (4, ) C) (-4, ) D) (4, -)
Find the derivative of the function at the point p in the direction of a. 29) f(x, y) = e-8y cos x, p = (-2π, 0), a = i - j A) 8-5 B) - 2 C) - 4 D) 4 29) 0) f(x, y) = ln (-x + 5y), p = (-8, -6), a = 6i + 8j A) 0 B) - 4 65 C) 65 D) - 0 0) ) f(x, y, z) = -9x + 2y - 4z, p = (5, -4, -), a = i - 6 j - 2 k ) A) - 25 B) - C) - 2 D) - 2) f(x, y, z) = x4z - y2z2, p = (2, -5, ), a = 2i + 2j - k A) B) 50 C) 8 D) 8 2) ) For the function f(x, y) = e( - 2x + 6y) at the point p = (0, 0), find the unit vector for which f is increasing most rapidly. -2 A) 80 i + -6 80 j B) -2 40 i + 6 40 j C) -2 80 i + 6 80 j D) 2 40 i + -6 40 j ) 4) For the function f(x, y) = cos (6-6) at the point p =, f decreases most rapidly. A) C) 6π 2, find the unit vector for which -2 2 + 6π i + 6π 2 + 6π j B) 2 44 + 6π i + 6π 44 + 6π j -2 44 + 6π i + - 6π 44 + 6π j D) 2 2 + 6π i + - 6π 2 + 6π j 4) 5) Find the direction in which the function is increasing or decreasing most rapidly at the point p. f(x, y, z) = x y2 + z2, p = (-, -, -) 5) A) 6 i + j 6 + k 6 B) 6 i + j 6 - k 6 C) 6 i + j 6 - k 6 D) 6 i + j 6 + k 6 6) The Celsius temperature at a point (x,y) on a large metal plate is given by T(x, y) = 80 + (x + )2(y - 2)2 + (x-2)2. Find the direction of heat flow in the plate at the point (2, ). Heat flows in the direction of steepest temperature decrease. A) In the direction, - B) In the direction -2, 2 C) In the direction -, 2 D) In the direction -, 6) 4
Find dw by using the Chain Rule. Express your final answer in terms of t. dt ) w =x2 + xy + y2, x = e2t, y = t ) 8) w = ex cos y + ey cos x; x = -8t, y = t 8) Find dw dt by using the Chain Rule. Express your final answer in 9) w = ln (x + y) + ex; x = st, y = est A) s + e st st + est + e st B) s + e st st + est + te st C) s t + e st st + est + e st D) st + e st st + est + e st 9) 40) w = ex + yz; x = s2 + t2, y = s - t, z = s + t A) 0 B) C) e2s2 D) es2 - t2 40) Provide an appropriate answer. 4) Find w r when r = - and s = - if w(x, y, z) = xz + y 2, x = 5r + 6, y = r + s, and z = r - s. 4) A) w r = B) w r = C) w r = - D) w r = 0 42) Find w u when u = -6 and v = -4 if w(x, y, z) = xy 2 z, x = u, y = u + v, and z = u v. v A) w u = 0 2 B) w u = - 5 8 C) w u = - 5 4 D) w u = - 5 2 42) 4) Find x when u = and v = if z(x) = v x + and x = u v. 4) A) v = 2 2 6 B) v = 9 2(6)/2 C) v = 0 D) v = 2 2(6)/2 Use the Chain Rule or implicit differentiation to find the indicated derivative. 44) xey + x2 - y = 0. Find dy dx. 44) A) e y + 2y - xey B) ey + 2x - 2xey C) e y + 2x - xey D) e y + 2x + xey 45) x4 + yz2 + x2yz = 0. Find y. 45) A) - z 2y B) - z(y 2z + x2) y(2y2z + x2) C) - y 2z + x2 2y2z + x2 D) z(y 2z + x2) y(2y2z + x2) 5
46) zexy + yz cos x = 0. Find. 46) A) exy - y cos x yz(sin x + exy) B) exy + y sin x yz(cos x + exy) C) exy + z cos x yz(sin x + exy) D) exy + y cos x yz(sin x + exy) 4) The radius r and height h of a cylinder are changing with time. At the instant in question, r = 2 cm, h = cm, dr/dt = 0.0 cm/sec and dh/dt = -0.02 cm/sec. At what rate is the cylinderʹs volume changing at that instant? 4) A) 0. cm/sec B).8 cm/sec C) 0.88 cm/sec D) 0.44 cm/sec Write a chain rule formula for the following derivative. 48) w for w = f(p, q); p = g(x, y), q = h(x, y) A) w = w C) w = w p p + w p + w q q q B) w = w p p D) w = w p + w q 48) 49) w t for w = f(x, y, z); x = g(r, s), y = h(t), z = k(r, s, t) 49) A) w t = w y + w B) w t = dy dt + t C) w t = w y dy dt + w t D) w t = w t Find the equation of the tangent plane to the given surface at the indicated point. 50) ex sin(yz) - 5x = 0, (0, π, ) A) 5x - (y - π) = -π(z - ) B) -5x - (y - π) - π(z - ) = 0 C) 5x - (y - π) - (z - ) = 0 D) -5x + (y - π) + π(z - ) = 0 50) 5) x2 + 2yz + y2 - xz - z2-8 = 0, (, -, -) A) 5(x + ) - 8(y - ) - (z - ) = 0 B) 5(x - ) - 8(y + ) = -(z + ) C) 5(x - ) - 8(y + ) - (z + ) = 0 D) 5(x - ) - 8(y + ) + (z + ) = 8 5) 52) z = ex2 + y2, (0, 0, ) A) z = 2 B) z = - C) z = D) z = 0 52) Use the total differential dz to approximate the change in z as (x, y) moves from P to Q. Then use a calculator to find the corresponding exact change Δz (to the accuracy of your calculator). 5) x2 + xy - y2; P(2, 2), Q(2.02,.9) A) dz = 0.24; Δz = 0.29 B) dz = 0.8; Δz = 0.80 C) dz = 0.8; Δz = 0.89 D) dz = -0.40; Δz = -0.96 5) 6
54) ln (xy2); P(5, -), Q(4.95, -.99) A) dz = -0.0; Δz = -0.005006 B) dz = -0.; Δz = -0.05006 C) dz = 0.0; Δz = 0.005006 D) dz = 0.; Δz = 0.05006 54) 55) If the length, width, and height of a rectangular solid are measured to be 9, 4, and 4 inches respectively and each measurement is accurate to within 0. inch, estimate the maximum percentage error in computing the volume of the solid. 55) A) 5.50% B) 6.% C).% D) 4.89% Give an appropriate answer. 56) Find all points (x, y) at which the tangent plane to the graph z = x2 + 0x - 2y2 + 4y + 2xy is horizontal. 56) A) (-4, -, -22) B) (-4,, -0) C) (4,, 66) D) (4, -, 42) Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle 5) f(x, y) = 2xy + 2x + 6y 5) 58) f(x, y) = x + y - 48x - 4y + 8 58) Find the global maxima and minima of the function on the given domain. 59) f(x, y) = x2 + xy + y2 on the square -6 x, y 6 59)
Answer Key Testname: MTH 22 TEST 2 REVIEW ) 2) ) B 4) B 5) D 6) B ) Answers will vary. One possibility is Path : x = t, y = t ; Path 2: x = t, y = t2 8) Answers will vary. One possibility is Path : x = t, y = t ; Path 2: x = 0, y = t 9) B 0) A ) fx(x, y) = 2) fx(x, y) = - y (x + y)2 ; f y(x, y) = - x (x + y)2 x () /2 ; f y(x, y) = - y () /2 ) fx(x, y) = ye-y; fy(x, y) = xe-y( - y) 4) fxx(x, y) = 2 - ex+y; fyy(x, y) = - ex+y; fyx(x, y) = fxy(x, y) = -ex+y 5) B 6) fxx(x, y) = -y4 cos xy2; fyy(x, y) = - 2x[2xy2 cos (xy2) + sin (xy2)]; fyx(x, y) = fxy(x, y) = - 2y[xy2 cos (xy2) + sin (xy2)]; ) B 8) B 9) C 20) C 2) B 22) D 2) D 24) C 25) B 26) A 2) C 8
Answer Key Testname: MTH 22 TEST 2 REVIEW 28) A 29) D 0) D ) D 2) D ) B 4) C 5) A 6) D ) 4e4t + (2t + )e2t + 2t 8) e-8t(-8 cos t - sin t) et( cos -8t + 8 sin -8t) 9) A 40) A 4) A 42) C 4) D 44) C 45) B 46) D 4) C 48) C 49) C 50) B 5) C 52) C 5) C 54) A 55) B 56) A 5) -, - ; saddle point 58) (4, ); local minimum; (4, -); saddle point; (-4, ); saddle point; (-4, -); local maximum 59) Global maximum: 08 at (6, 6) and (-6, -6); global minimum: 0 at (0, 0) 9