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MTH 3030: Midterm 2 Practice 1

Problem 1

Identify the object represented by the equation \( \displaystyle x^2+9y^2+8x-6y+17=0 \).

Problem 2

If \( \displaystyle f(x,y)=6x^2+7x^2y+5y^3 \), determine \( \displaystyle \frac{\partial^2 f}{\partial x\partial y}(x,y) \).

Problem 3

Determine parametric equations of the plane that passes through the point \( A(10,-2,-5) \) and is parallel to the vectors \( \overrightarrow u=\langle 2,-5,7\rangle \) and \( \overrightarrow v=\langle -7,5,-2\rangle \).

Problem 4

Determine the length of the curve given parametrically by \( \displaystyle \overrightarrow r(t)=\left\langle 2\cos t, 2\sin t, t\right\rangle \) for \( 4\leq t\leq 7 \).

Problem 5

Determine the cartesian coordinates of the point whose polar coordinates are \( \displaystyle\left(8,\frac{\pi}3\right) \).

Problem 6

Assume that the functions \( f(x,y) \) and \( g(x,y) \) are given in the following way \( f(x,y)=3x^2-12y \) and \( g(x,y)=3x-y \). Determine \( \displaystyle g\left(f(5,4),g(4,7)\right) \).

Problem 7

Assume that the plane \( \alpha \) is given with its symmetric equation \( 5x-4y+4z=-91 \). Let \( A \) be the point with coordinates \( (3,3,5) \). Determine the coordinates of the point \( P \) in the plane \( \alpha \) such that the vector \( \overrightarrow{AP} \) is orthogonal to \( \alpha \).

Problem 8

The line \( \ell \) is given with its parametric equation \( \displaystyle \overrightarrow r(t)=\left\langle 4t+8, -t-6,5t+5\right\rangle \). The plane \( \alpha \) is given with its symmetric equation \( x+2y-4z=3 \). Determine a symmetric equation of the plane that contains \( \ell \) and is orthogonal to \( \alpha \).

Problem 9

Determine parametric equations of the curve in the \( xy \)-plane whose equation in polar coordinates is \( \displaystyle r=\frac{1}{3\sin\theta+4\cos\theta} \). Sketch the curve.

Problem 10

Let \( \gamma \) be the curve consisting of all points in the \( xy \) plane whose distance from the point \( A(0,21) \) is twice as large as its distance from the point \( B(3,21) \). Determine the parametric equations and the length of the curve \( \gamma \).

Problem 11

The curve \(\gamma\) consists of all points whose polar coordinates \((r,\theta)\) satisfy \(r=\theta|\cos\theta|\) for \(\displaystyle \theta\in\left[0,\frac{5\pi}2\right]\).

  • (a) Sketch the curve \(\gamma\).

  • (b) Determine the area of the portion of the curve that lies inside its largest loop but outside of its smallest loop.