MTH3030 Home

# 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.