※ [本文转录自 NTU-Exam 看板]
作者: prolegend (黑诚) 看板: NTU-Exam
标题: [试题] 94下 张镇华 微积分乙下 期中考
时间: Tue Apr 11 14:30:55 2006
课程名称︰微积分乙下
课程性质︰大一分组编班
课程教师︰张镇华
开课系所︰医理公卫等
考试时间︰4/11(二)10:20~12:10am
试题 :
Examination 1 (Calculus B, second semester)
Each problem weights 10 points. 2006-4-11 (Gerard J. Chang)
1.(a) Use the partial fraction method to show that
du 1 │ u-a │
∫───── = ─ ln│───│+ C.
u^2-a^2 2a │ u+a │
(b) Use your result in (a) to find a solution of dy/dx = y^2 - 9 that
passes through
(i) (0,0) (ii) (0,3) (iii) (0,6).
2. SplendidLawn sells three types of lawn fertilizers, SL 24-4-8, SL 21-7-12,
and SL 17-0-0. The three numbers referee to the percentage of nitrogen,
phosphate, and potassium, in the order, of the contents. (For instance,
100g of SL 24-4-8 contains 24g of nitiogen.) Suppose that each year your
lawn require 500g nitrogen, 100g of phosphate, and 180g of potassium per
1000 square feet. How much of each of the three types of fertilizer should
you apply per 1000 square feet per year?
3. Recall that an n x n square matrix is said to be invertible if there is an
n x n matrix B such that AB = BA = I , where B is called the inverse of
-1 n
A and is denoted by A . -1 -1 -1
(a) Prove that if A is invertible, then so is A and (A ) = A.
(b) Prove that if A and B are invertible, then so is AB and
-1 -1 -1
(AB) = B A . ┌ ┐
│-1 0 -1│
(c) Find the inverse of │ 0 -2 0│.
│-1 1 2│
└ ┘ ┌ ┐
│2 4│
4. (a) Find the eigenvalues and eigenvectors of the matrix A =│6 4│.
20 ┌ ┐ └ ┘
(b) Find A x for x = │8│.
│7│ ┌ ┐
└ ┘ │1│
5. Given a plane through (2,1,5), which is perpendicular to│2│, and a line
│1│
└ ┘
through the points (1,-1,5) and (0,0,6). Where do the plane and the line
intersect?
x^2 - y^2
6. (a) Find the value of lim ────── , or show that it does not
(x,y)→(0,0) x^2 + y^2
exist.
x^3 - y^3
(b) Fine the value of lim ────── , or show that it does not
(x,y)→(0,0) x^2 + y^2
exist.
7. Let f be a function of three indepedent variables x, y, and z defind by:
xy
f(x,y,z) = e [sin(yz) + ln(zx)]. Find δf/δx , δf/δy , δf / δz ,
2 2 3
δ f / δx , δ f/δxδyδz. ┌ 2 ┐
│(x-y) │
8. Find a linear approximation to the function f(x,y) =│ 2 │at (2,-3),
│2x y │
2 2 └ ┘
where f: R → R . Use your result to find an approximation for f(1.9,-3.1).
9. Compute the directional derivative of f(x,y) =√(x^2 + 2y^2) at the point
┌ ┐
(-1,2) in the direction│-1│. In what direction does f(x,y) increase most
│ 3│
└ ┘
rapidly at (-1,2)?
10. (a) Suppose that w = F(x,y) is differentiable, and F(x,y) = 0 defines y
implicitly as a function of x. Prove that at any point where Fy≠0,
we have dy Fx
─── = - ── .
dx Fy
(b) Use (a) to find dy/dx for y = arc tanx.