Math 292 Diablo Valley College Parametric Equations of The Line of Intersection of The Planes Exam Math 292 final exam with 8 questionsMath 292 final exam

Math 292 Diablo Valley College Parametric Equations of The Line of Intersection of The Planes Exam Math 292 final exam with 8 questionsMath 292 final exam with 8 questionsMath 292 final exam with 8 questions Math 292: Calculus III
Final Exam – SU20
Name__________________
Show all work on exam paper and give exact answers to get credit for problems!
Use only methods that were covered in class or the textbook! Simplify answers!
Final Exam is worth 200 points, each problem is worth 25 points.
1. a) Find parametric equations of the line of intersection of the planes 3x − 2y + z = 0
and 8x + 2y + z −11 = 0 .
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b) Find the equation of the plane through the points: (2,4,5), (1,5,7), (−1,6,8) .
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2. The radius and height of a right-circular cylinder are measured with errors at most
0.1 inches. If the height and radius are measured to be 10 inches and 2 inches,
respectively, use differentials to approximate the maximum possible error in the
calculated value of the volume.
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3. Find the gradient ∇f and the directional derivative at the point P (1,−1,2) in the
direction a = 2 , − 1,1 for the function f (x,y,z) = x 3 z − yx 2 + z2 . In which direction is
the directional derivative at P decreasing most rapidly and what is its value?
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4. Find all relative maxima, relative minima, and saddle points for the function:
f ( x,y ) = 2 x 3 + xy 2 + 5 x 2 + y 2 .
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5. Use a double integral in polar coordinates to find the area inside the graph of
and outside the circle r = 1 .
r = 2 cos θ
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6. Evaluate the line integral
∫ (x
C
2
)
+ y 2 dx − x dy along the curve C: x 2 + y 2 = 1
counterclockwise from (1,0) to (0,1).
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7. Use Stokes’ Theorem to evaluate
∫ ( x + y )dx + (2x − 3)dy + ( y + z )dz over the
C
boundary of the triangle with vertices (2,0,0), (0,3,0), (0,0,6) traversed in the counter
clockwise direction.
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8. Use the divergence theorem to evaluate the flux integral
 
F
∫∫ ⋅ n dS where
S


2
3
n is the outward unit normal
F( x,y,z ) = x 3 + cot an −1 ( y 2 z 3 ), y 3 − e x + z , z 3 + ln( x − y ) , 
to S, and S is the surface of the solid enclosed in the hemisphere z = a 2 − x 2 − y 2
and the plane z=0.
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