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pracmid2-2011

# pracmid2-2011 - the loan in 25 years(c Determine the total...

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Math 105 Practice Midterm2, Spring 2011 1. Short answer questions (1) Evaluate 1 arctan x x 2 +1 dx . (2). Solve the initial value problem dx dt = x 2 +5 x , x (0) = 1. (3). Evaluate lim ( x,y ) (0 , 0) x 2 y + xy 2 y 3 - x 3 . (4). Let z = x - y 2 - 1. Sketch the level curve through the point (6 , 1). Calculate the slope of the tangent line to this level curve at (6 , 1). 2. Find and classify the critical points of f ( x, y ) = x 4 + 2 y 2 - 4 xy . 3. A person purchased a home at the price \$300 , 000, paid a down pay- ment equal to 20% of the purchase price, and financed the remaining balance with a 25 year term mortgage. Assume that the person makes payments continuously at a constant annual rate A and that interest is compounded continuously at the rate of 5%. (a) Write down the differential equation that is satisfied by the amount y ( t ) of money owed on the mortgage at time t . (b) Determine A , the rate of annual payments, that is required to pay off
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Unformatted text preview: the loan in 25 years. (c) Determine the total interest paid during the 25 year term mortgage. 4. Consider the hill given by the function z = f ( x, y ) = √ 4-x 2-y 2 4 . (a) Compute f x and f y . (b) Find the unit vector that gives the direction of steepest ascent at the point P (1 , 2). Also ﬁnd a unit vector that gives the direction of no change at that point. Sketch these two vectors and the level curve through P . (c) Suppose you’re walking over the hill along the path that is right above the path ( x ( t ) , y ( t )) = ( t, t 2 + 1) in the xy-plane. As you pass the point (1 , 2 , f (1 , 2)), at what rate is your height changing? 5. Find the area of the shaded region bounded by the curves f ( x ) = x 3-4 x and g ( x ) =-x 2 + 2 x for x ≥ 0. 1...
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