# Calculus 微积分代写：Mathematics and Statistics MATH2069 代写

## 微积分代写介绍

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### 微积分代写类型

• Differential Calculus 微分学
• Integral Calculus 积分学
• Derivatives 导数
• Definite Integral 定积分
• Indefinite Integral 不定积分
• Function 函数
• Limit Theory 极限理论
• Ordinary Differential Equation 常微分方程
• Partial Differential Equation 偏微分方程

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## 微积分代写案例：MATH2069 Calculus Test for MATH2A

1. i) a) The curve

r(t) = ti+t2(2 j−k)

intersects the plane

x+ 2y+ 3z= 0

at (0,0,0) and at one other point. Find this other point.

b) Write a parametric vector equation for the tangent line to the curve

at each point of intersection.

c) Find the cosine of the angle between the curve and the normal direc-

tion to the plane at each point of intersection.

ii) a) Find an equation for the tangent plane to the sphere

x2+y2+z2−4y−2z+ 2 = 0

at the point (1,1,2).

b) Show that the sphere is perpendicular to the paraboloid

3×2+ 2y2−2z= 1

at the point of intersection (1,1,2).

(Hint: show that the normal vectors are perpendicular.)

iii) Let f(x, y) = x3−2y2.

a) Find a unit vector in the xy-plane which points in the direction of

greatest increase of fat the point (2,1).

b) Find the directional derivative of fat the point (2,1) in the direction

of the vector i−j

iv) Use the method of Lagrange multipliers to find the maximum and mini-

mum values of the function xy2on the circle x2+y2= 1.

2. i) a) Sketch the region of integration for the iterated integral

b) Evaluate the integral above by reversing the order of integration.

ii) Use spherical coordinates to find the volume of the region bounded by

the cone

z=rx2+y2

3

and the sphere

x2+y2+z2= 4.

iii) a) Sketch the region Ω in the first quadrant bounded by the curves

y=x,y= 4x,xy = 1, and xy = 4.

b) Use the transformation u=xy, v =x

yto find the area of Ω.

3. i) Let Fbe the vector field

F(x, y, z) = (yzexyz +z)i+xzexyz j+ (xyexyz +x)k.

a) Show that Fis conservative by computing its curl.

b) Find a scalar potential function φfor F.

c) Calculate

ZC

F(r)·dr,

where Cis the curve

C:r(t) = t3i−t2j+tk,0≤t≤1.

ii) Let Sbe the part of the unit sphere in the first octant (x, y, z ≥0). Let

Fbe the vector field F(x, y, z) = xi.

a) Write down a parameterization of S.

b) Find the flux of Fout of S.

iii) Use Gauss’ Divergence Theorem to find the flux of the vector field

F(x, y, z) = cos zi+yj+ex27y2k,

out of the solid bounded by the paraboloid z= 4 −x2−y2and the xy-plane.

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