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Find the first partial derivatives of the function. Find the first partial derivatives of the function.

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Suppose (1, 1) is a critical point of a function f with continuous second derivatives.In the case of Suppose (1, 1) is a critical point of a function f with continuous second derivatives.In the case of , , what can you say about f ? A) f has a saddle point at (1,1) B) f has a local minimum at (1,1) C) f has a local maximum at (1,1) , Suppose (1, 1) is a critical point of a function f with continuous second derivatives.In the case of , , what can you say about f ? A) f has a saddle point at (1,1) B) f has a local minimum at (1,1) C) f has a local maximum at (1,1) , Suppose (1, 1) is a critical point of a function f with continuous second derivatives.In the case of , , what can you say about f ? A) f has a saddle point at (1,1) B) f has a local minimum at (1,1) C) f has a local maximum at (1,1) what can you say about f ?


A) f has a saddle point at (1,1)
B) f has a local minimum at (1,1)
C) f has a local maximum at (1,1)

D) All of the above
E) A) and C)

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Find an equation of the tangent plane to the given surface at the specified point. Find an equation of the tangent plane to the given surface at the specified point.

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Suppose that over a certain region of space the electrical potential V is given by Suppose that over a certain region of space the electrical potential V is given by .Find the rate of change of the potential at in the direction of the vector . A) 20 B) 44 C) D) -2.91 E) .Find the rate of change of the potential at Suppose that over a certain region of space the electrical potential V is given by .Find the rate of change of the potential at in the direction of the vector . A) 20 B) 44 C) D) -2.91 E) in the direction of the vector Suppose that over a certain region of space the electrical potential V is given by .Find the rate of change of the potential at in the direction of the vector . A) 20 B) 44 C) D) -2.91 E) .


A) 20
B) 44
C) Suppose that over a certain region of space the electrical potential V is given by .Find the rate of change of the potential at in the direction of the vector . A) 20 B) 44 C) D) -2.91 E)
D) -2.91
E) Suppose that over a certain region of space the electrical potential V is given by .Find the rate of change of the potential at in the direction of the vector . A) 20 B) 44 C) D) -2.91 E)

F) A) and B)
G) A) and C)

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Find three positive numbers whose sum is Find three positive numbers whose sum is and whose product is a maximum. A) B) C) D) E) and whose product is a maximum.


A) Find three positive numbers whose sum is and whose product is a maximum. A) B) C) D) E)
B) Find three positive numbers whose sum is and whose product is a maximum. A) B) C) D) E)
C) Find three positive numbers whose sum is and whose product is a maximum. A) B) C) D) E)
D) Find three positive numbers whose sum is and whose product is a maximum. A) B) C) D) E)
E) Find three positive numbers whose sum is and whose product is a maximum. A) B) C) D) E)

F) B) and C)
G) A) and B)

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The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E) in. and the height is The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E) in.?


A) The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E)
B) The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E)
C) The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E)
D) The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E)
E) The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is in. and the height is in.? A) B) C) D) E)

F) D) and E)
G) A) and B)

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A contour map for a function f is shown. Use it to estimate the value of A contour map for a function f is shown. Use it to estimate the value of . A) 35 B) 28 C) 48 D) E) 78 . A contour map for a function f is shown. Use it to estimate the value of . A) 35 B) 28 C) 48 D) E) 78


A) 35
B) 28
C) 48
D) A contour map for a function f is shown. Use it to estimate the value of . A) 35 B) 28 C) 48 D) E) 78
E) 78

F) None of the above
G) B) and E)

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Use the Chain Rule to find Use the Chain Rule to find where . where Use the Chain Rule to find where . . Use the Chain Rule to find where .

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Use implicit differentiation to find Use implicit differentiation to find A) B) C) D) Use implicit differentiation to find A) B) C) D)


A) Use implicit differentiation to find A) B) C) D)
B) Use implicit differentiation to find A) B) C) D)
C) Use implicit differentiation to find A) B) C) D)
D) Use implicit differentiation to find A) B) C) D)

E) A) and C)
F) None of the above

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Find the critical points of the function. Find the critical points of the function. A) B) C) D) E)


A) Find the critical points of the function. A) B) C) D) E)
B) Find the critical points of the function. A) B) C) D) E)
C) Find the critical points of the function. A) B) C) D) E)
D) Find the critical points of the function. A) B) C) D) E)
E) Find the critical points of the function. A) B) C) D) E)

F) All of the above
G) B) and E)

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The length l, width w and height h of a box change with time. At a certain instant the dimensions are The length l, width w and height h of a box change with time. At a certain instant the dimensions are and , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the surface area is changing. and The length l, width w and height h of a box change with time. At a certain instant the dimensions are and , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the surface area is changing. , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the surface area is changing.

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Find the shortest distance from the point Find the shortest distance from the point to the plane . A) B) C) D) E) to the plane Find the shortest distance from the point to the plane . A) B) C) D) E) .


A) Find the shortest distance from the point to the plane . A) B) C) D) E)
B) Find the shortest distance from the point to the plane . A) B) C) D) E)
C) Find the shortest distance from the point to the plane . A) B) C) D) E)
D) Find the shortest distance from the point to the plane . A) B) C) D) E)
E) Find the shortest distance from the point to the plane . A) B) C) D) E)

F) A) and B)
G) A) and C)

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Find three positive numbers whose sum is Find three positive numbers whose sum is and whose product is a maximum. and whose product is a maximum.

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