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台灣大學 105 年度 微積分(B)

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Multiple integrals 4 題Derivatives of multi-variable functions 3 題Applications of integration 2 題Differential equations 2 題

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Work need not be shown. Only answers will be graded. 5 points each and 100 points total.
第 1 題5 分Applications of differentiation
(A) Find the interval on which y=x3−3x2+2x+1y = x^3 - 3x^2 + 2x + 1 is both decreasing and concave downward.
第 2 題5 分Derivatives
(B) Find dydx=\frac{dy}{dx} = __(2)__ and d2ydx2=\frac{d^2y}{dx^2} = __(3)__ at the point x=1,y=1x = 1, y = 1 of the curve x3+xy−2y4=0x^3 + xy - 2y^4 = 0.
第 3 題5 分Techniques of integration
(C) Evaluate ∫233+2x−x2dx\int_2^3 \sqrt{3 + 2x - x^2} dx.
第 4 題5 分Applications of integration
(D) Find the volume of the solid generated by revolving the following region about the yy-axis: {(x,y):0≤x≤π\{(x,y) : 0 \leq x \leq \pi and 0≤y≤sin⁡x}0 \leq y \leq \sin x\}.
第 5 題5 分Applications of integration
(E) Find the arc length of the curve y=∫0xsin⁡tdty = \int_0^x \sqrt{\sin t} dt from x=0x = 0 to x=π/3x = \pi/3.
第 6 題5 分Differential equations
(F) Find the solution of the differential equation y′=(1+2x)(1+y2)y' = (1 + 2x)(1 + y^2) with the initial condition y(0)=1y(0) = 1.
第 7 題5 分Differential equations
(G) Find the solution of the differential equation y′+ytan⁡x=sec⁡xy' + y \tan x = \sec x with the initial condition y(0)=2y(0) = 2.
第 8 題5 分Infinite Series
(H) Find the first three nonzero terms of the McLaurin series of tan⁡x\tan x.
第 9 題5 分Parametric equations and polar coordinates
(I) Let u=u(x,y)u = u(x,y) be a function of x,yx,y. Express ∂u∂x\frac{\partial u}{\partial x} in terms of polar coordinates r,θr,\theta together with ∂u∂r\frac{\partial u}{\partial r} and ∂u∂θ\frac{\partial u}{\partial \theta}.
第 10 題5 分Derivatives of multi-variable functions
(J) Find the directional derivative of f(x,y,z)=xyz2f(x,y,z) = x^yz^2 at the point (e,e,1)(e,e,1) in the direction u=313i+1213j+413k\mathbf{u} = \frac{3}{13}\mathbf{i} + \frac{12}{13}\mathbf{j} + \frac{4}{13}\mathbf{k}. Answer:__(11)__
第 11 題5 分Derivatives of multi-variable functions
(K) Find the critical points of f(x,y)=x2+y4+3xy2−5xf(x,y) = x^2 + y^4 + 3xy^2 - 5x which are saddle points.
第 12 題5 分Derivatives of multi-variable functions
(L) Find the minimum of x2+y2+z2x^2 + y^2 + z^2 for (x,y,z)(x,y,z) on the intersection curve of the two surfaces y+2z=1y + 2z = 1 and 3x2+y2−z2=13x^2 + y^2 - z^2 = 1. Answer:__(13)__
第 13 題5 分Multiple integrals
(M) Evaluate ∫03∫x/31ey3dydx\displaystyle\int_0^3 \int_{\sqrt{x/3}}^1 e^{y^3} dy dx. Answer:__(14)__
第 14 題5 分Multiple integrals
(N) Let RR be the region in the first quadrant of the xyxy-plane bounded by xy=1xy = 1, xy=2xy = 2, y=xy = x and y=2xy = 2x. Evaluate ∬Rexydxdy\iint_R e^{xy} dx dy.
第 15 題5 分Multiple integrals
(O) Evaluate ∭Ωcos⁡x2+y2+z2x2+y2+z2dxdydz\iiint_{\Omega} \frac{\cos \sqrt{x^2 + y^2 + z^2}}{x^2 + y^2 + z^2} dx dy dz, where Ω\Omega is given by Ω={(x,y,z):1≤x2+y2+z2≤2}\Omega = \{(x,y,z) : 1 \leq x^2 + y^2 + z^2 \leq 2\}.
第 16 題5 分Multiple integrals
(P) Evaluate ∭Ωzex2+y2+3z2dxdydz\iiint_{\Omega} ze^{x^2+y^2+3z^2} dx dy dz, where Ω\Omega is the cylinder defined by Ω={(x,y,z):x2+y2≤1\Omega = \{(x,y,z) : x^2 + y^2 \leq 1 and 0≤z≤1}0 \leq z \leq 1\}.
第 17 題5 分Vector calculus
(Q) Let SS be the surface described by z=x2+y22z = x^2 + \frac{y^2}{2} with 4x2+y2≤14x^2 + y^2 \leq 1 oriented with normals with positive kk-components. Let F(x,y,z)=xi−yj+k\mathbf{F}(x,y,z) = x\mathbf{i} - y\mathbf{j} + \mathbf{k}. Evaluate ∬SF⋅dS.\iint_S \mathbf{F} \cdot d\mathbf{S}. Answer: __(18)__ Also, evaluate ∬SdS.\iint_S dS. Answer: __(19)__ .
第 18 題5 分Vector calculus
(R) Let CC be the counterclockwise oriented boundary of the region in the xyxy-plane enclosed by x2+y2−2x=0x^2 + y^2 - 2x = 0 and x2+y2−2y=0x^2 + y^2 - 2y = 0. Evaluate the line integral ∮C(y+e−x2)dx+(3x+sin⁡(y2))dy\oint_C (y + e^{-x^2}) dx + (3x + \sin(y^2)) dy.