Equations of Lines in 3D

April 30, 2008

Let l be the line represented by the symmetric equations

\displaystyle\frac{x-2}{4} = \frac{y+3}{-2}=\frac{z-1}{7}

1. Find the point of intersection of l and the plane 5x-y+2z=12.

2. Find the distance between P(0,1,0) and l.

3. The acute angle between l and the line m: \hspace{.3in} x=1+t; \hspace{.2in} y=3-2t; \hspace{.2in} z=4t.

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1 Comment | Calculus, Geometry | Tagged: , , | Permalink
Posted by ldvallejo

Area, Volume and Perimeter

March 23, 2008

Consider the regions \displaystyle \mathcal{R}_{1}\,\mathrm{and}\,\mathcal{R}_{2} enclosed by the curves \displaystyle y=x^{3}+\frac{1}{2} and \displaystyle x=-\left(y-\frac{1}{2}\right)^{2}. Set-up the integrals representing the following:

  1. The perimeter of \displaystyle \mathcal{R}_{2}.
  2. The total area of the regions \displaystyle \mathcal{R}_{1} and \displaystyle \mathcal{R}_{2}.
  3. The volume of the solid generated when \displaystyle \mathcal{R}_{1} is rotated about the line \displaystyle y=-\frac{1}{2} using the method of washers.
  4. The volume of the solid generated when \displaystyle \mathcal{R}_{2} is rotated about the line x=-1 using the method of cylindrical shells.

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5 Comments | Calculus | Tagged: , , , , | Permalink
Posted by wMw

Related Rates

March 23, 2008

A particle moves along the curve y=\ln x so that its abscissa is increasing at a rate of 2 units per second. At what rate is the particle moving away from the origin as it passes through the point (e,1)?

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Posted by wMw

Largest Rectangle in Some Region in the First Quadrant

March 23, 2008

Find the area of the largest rectangle that can be inscribed in the region bounded by the curve y=e^{-x}, the line x=0 and the positive x-axis.

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Posted by wMw

Normal Line Problem

March 23, 2008

Find the equation of the line normal to the curve \displaystyle y=\tanh \left[\sin^{-1} \left( x-\sqrt{2}\right) \right] at the point where \displaystyle x=\sqrt{2}.

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Posted by wMw

Integrals Involving Transcendental Functions

March 23, 2008

Evaluate the following integrals.

  1. \displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{\ln^{2}\left(\sin x\right)}{\tan x}\: dx
  2. \displaystyle \int\frac{2^{x}\: dx}{\left(2^{x}+1\right)\sqrt{4^{x}+2^{x+1}-8}}
  3. \displaystyle \int\frac{dx}{x+4\sqrt{x}+13}

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4 Comments | Calculus | Tagged: , , , , , | Permalink
Posted by wMw

Indeterminate Forms Involving Transcendental Functions

March 23, 2008

Find the limits.

  1. \displaystyle \lim_{x\rightarrow1^{+}}\left(\frac{1}{\ln x}-\frac{1}{\cosh^{-1}x}\right)
  2. \displaystyle \lim_{x\rightarrow0^{+}}\left(e^{x}+\sinh x\right)^{{\displaystyle \cot x}}

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Logarithmic and Implicit Differentiation

March 23, 2008

Solve for \displaystyle \frac{dy}{dx}.

  1. y=\displaystyle \sqrt[5]{\frac{4^{x}\coth\left(e^{-x}\right)}{\sin\left(x^{2}\ln x\right)}}
  2. \displaystyle e^{y}=\cosh\left(e^{x}+\log_{3}y\right)-2^{{\displaystyle x^{2}}}

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1 Comment | Calculus | Tagged: , , , , | Permalink
Posted by wMw

Continuity and L’Hospital’s Rule

March 22, 2008

Consider the function

f(x) =\Biggl\{ \begin{array}{lcl} \cos x, & \mathrm{if} & x=0,\\ \displaystyle x \ln x (1-x) ^{-1}, & \mathrm{if} & 0<x<1, \\ -1, & \mathrm{if} & x=1. \end{array}

Discuss the continuity of f on \left[0,1\right].

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Integrals of Some Transcendental Functions

March 21, 2008

Evaluate the following integrals:

1. \displaystyle\int \frac{x^2}{1-\cosh^2 (x^3)} \ dx
2. \displaystyle\int 5^x \cot \big( 5^x \big) \ dx
3. \displaystyle\int \frac{dx}{x\sqrt{9x^4-16}}
4. \displaystyle\int\frac{x^2-4x+3}{x^2-4x+5} \ dx

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2 Comments | Calculus | Tagged: , , , , , | Permalink
Posted by ldvallejo

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