東京大学 新領域創成科学研究科 複雑理工学専攻 2021年8月実施 専門基礎科目 1.1 微分積分
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以下の問に答えよ。すべての定数と変数は実数、関数は実関数とする。導出の過程を省略し、答えのみ示せ。
問1
関数
(1)
(2)
問2
関数
(1)
(2) 微分方程式を解いて,
問3
次の不定積分を求め, 空欄に入る式を書け。
問4
(1) この楕円の接線の方程式を求めよ。ただし接点の座標を変数
(2) この接線が
問5
問
(1) 変数
(2) 上の重積分を計算せよ。
Answer the following questions. All constants and variables are real numbers. All functions are real functions. Omit the derivations and write only the answers.
(Q.1) Let y(x) be a function satisfying the differential equation
- (1) Obtain the solution for f(x) = 0. Use
and for arbitrary constants. - (2) Obtain the solution for
. e is the base of the natural logarithm. - (Q.2) Let y(x) be a function satisfying the differential equation
- (1) Changing variables as x + y = u, obtain the differential equation that x and u satisfy and y is not included.
- (2) Obtain the function f(u) that satisfies x = f(u) by solving the differential equation. Use C for an arbitrary constant.
(Q.3) Calculate the following indefinite integral and write down the expression that fills the blank space. a is a non-zero constant.
(Q.4) Consider an ellipse on a xy Cartesian coordinate system:
where a and b are non-zero positive constants.
- (1) Obtain the equation of the tangent line of the ellipse. The tangent point is expressed as
using a variable . - (2) Let the points where the tangent line intersects the x-axis and y-axis be A and B, respectively. Find the minimum length of the line segment AB.
- (Q.5) Consider the following multiple integral over the region D enclosed by the ellipse given in (Q.4):
(1) Obtain the Jacobian for the change of variables using r and
(2) Calculate the multiple integral above.