Calculus
3 Final Exam - Math
241 Fall, 2006
|
1. |
Find an equation for the plane if the plane contains the line (x – 1)/0 = (y – 1)/1 = (z – 1)/-1 and the plane is parallel to the line x = 11 + t, y = -3 + 2t, z = 17 - 2t Hint: Find normal to plane by cross product of vectors parallel to each line. Answer: -y
+ 2 -z
= 0 |
|
2. |
Suppose that w = z(u,v), where u = xy and v = x + y. Use the chain rule to express the partial derivative ∂2z/∂x∂y in terms of derivatives of z with respect to u and v . Hint: z is dependent variable, u and v intermediate variables, with x and y being the independent variables. Answer: ∂2z/∂x∂y = ∂z/∂u + u∂2z/∂u2 +v∂2z/∂u∂v + ∂2z/∂v2 |
|
3. |
Find the tangential and normal components of acceleration for the following path: r(t) = t i + t2 j + t3 k Hint: aT = a . T aN = √( a2 - aT2 ) Answer: aN = 2{√[(1 + 9t2 + 9t4) / (1 + 4t2 + 9t4)]} |
|
4. |
a. Let f(x,y) = sin(x2 + y2) and u = i – j. Find the directional derivative of f(x,y) in the direction of the vector, u . Hint: df/du = grad f . u 3z5x + 12 xz + cos(z + y) + 3 = 0 Answer: ∂z/∂x = -[(3z5 + 12z) / (15z4x + 12x - sin (z + y)] Answer: ∂z/∂y =
[sin (z + y) / (15z4x +
12x - sin (z
+ y)] |
|
5. |
You are sending a gift packed in a rectangular box with length x, width y, and depth, z. The shipping service requires that the sum of the length, width, and depth at most can be 108 inches. What are the dimensions of the largest box (largest volume) you can send? Constraint: x + y + z ≤ 108 Hint: Use Lagrange multiplier approach. |
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6. |
∫ ∫ (x + 2y2) dA D where D is the region in the first quadrant bounded by y = x, y = 3x, and x = 1. |