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미적분학 및 연습 I - 생과대 (MATH 161) 제3차 시험 (2016년도 2학기)
학과 : 학번 : 이름 : 모든 풀이 과정은 알아볼 수 있도록 정성을 들여 작성한다. 칸이 부족한 경우, 각 문제 바로 뒷면에 이어 작성한다. 1. [12점] Let F i j k be a continuous vector function and v i j k a vector. Prove each. (a) v⋅
F
v⋅F [Hint: use definitions] (b)
F
≤
F (, : any constants)2. [13점] Use Lagrange multipliers to find the maximum and minimum values of the function
on the curve of intersection of the plane and the sphere
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3. [11점] Let r be a twice differentiable vector function satisfying r sin and r ′ cos . Show that r is orthogonal to r ″ at .
4. [14점] (a) Suppose that the function
ln
increases fastest at the point in the direction of u. Find the directional derivative of at the point in the direction of u.
(b) Find the equation of the tangent plane to the surface
ln
sin
at .3
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5. [13점] Suppose is a differentiable function of and , and sin cos
.
(a) Use the table of values to find the linear approximation of at the point .
(b) Use the linear approximation in (a) to estimate the value of .
6. [12점] Suppose and are functions of and such that
and ,
and suppose that
, and . Determine the value of .
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7. [11점] Evaluate the integral.
8. [14점] Evaluate the integrals.
(a)
if is the region in the first quadrant enclosed by the circle
and the lines and . (b)
