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calc3midterm

# calc3midterm - paths to approach the limit point(4(10...

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Calculus III – Midterm March 4 2009 The examination is 75 minutes. No notes, books or calculators are permitted. Remember to show all your working and read questions carefully. (1) (10 points) Show by either geometric arguments or by the properties of the dot and cross product that the following is always true: -→ a ± -→ a × -→ b ² = 0 where -→ a and -→ b are any vectors. (2) (10 points) Suppose we have two planes with equations x + y + z = 1 2 x - y - z = 1 1. Write down normal vectors for both planes. 2. What is the angle between the planes? 3. Find a vector that points along the line that is the intersection of the two planes. (3) (10 points) Prove that the following limit does not exist lim ( x,y ) (1 , 1) x 2 + y 2 - 2( x + y ) + 2 x 2 - y 2 - 2( x - y ) Hint: Choose two diﬀerent but appropriate
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Unformatted text preview: paths to approach the limit point. (4) (10 points) Suppose a bullet is ﬁred vertically with speed 100 ms-1 starting at the point (0 , , 0) at t = 0 1. Write down the equation for the displacement vector. 2. After how long does it return to its starting point? Assume that the acceleration due to gravity g = 10 ms-2 . (5) (10 points) Suppose that an object has the following displacement vector-→ r ( t ) = ³ ln t, ln t 2 , ln t 3 ´ for t > . 5. How far along the space curve does the object travel between times t = 1 and t = 20? What type of space curve is this? (6) (10 points) Sketch a series of level curves of the function z = 2 x 2 + y 1...
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