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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Posted by wMw

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

Logarithmic Differentiation and Derivatives of Some Transcendental Functions

March 21, 2008

Find \frac{dy}{dx}.

1. y = \displaystyle\frac{x^{\sec x} 2^{(x+1)^2}}{x\sin^3 (1+\sqrt{x})}

2. e^{xy} + \cosh^{-1} y = \tan^{-1}(\tanh x)

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Posted by ldvallejo

Some Limits of Indeterminate Forms

March 21, 2008

Evaluate the following limits:

1. \displaystyle\lim\limits_{x\rightarrow 1} \frac{2x^3-3x^2+1}{3x^3-4x^2-x+2}

2. \displaystyle\lim\limits_{x\rightarrow \infty} \big( xe^{-\frac{1}{x}}-x \big)

3. \displaystyle\lim\limits_{x\rightarrow \infty} \frac{\ln(x+e^x)}{3x}
4. \displaystyle\lim\limits_{x\rightarrow 1^+} (2-x)^{\tan(\frac{\pi}{2} x)}
5. \displaystyle\lim\limits_{x\rightarrow 0^+} \frac{1}{\ln(\cos x)} \int_0^x \tan^{-1} t \ dt

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

Equation of a Line in 2D

March 18, 2008

Find an equation of the line through the point (2,4) and is perpendicular to the line whose equation is x - 5y + 10 = 0.

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Leave a Comment » | Algebra | Tagged: , , , , | Permalink
Posted by Tina

Finding the equation of a plane

March 12, 2008

Find an equation of the plane containing the point P(6,2,4) and the line \displaystyle \ell :\ \frac{1}{5}(x-1) = \frac{1}{6}(y+2) = \frac{1}{7}(z-3).

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Leave a Comment » | Analytic Geometry | Tagged: , , , | Permalink
Posted by Tina

Distance of a Point from a Line in 3D

March 11, 2008

Find the distance of the point P(1,2,1) from the line \ell :\ x = 3 + t,\ y = 2 + t,\ z = 1 + t,\ t \in \mathbb{R}.

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7 Comments | Analytic Geometry | Tagged: , , , | Permalink
Posted by Tina

Phasors

March 6, 2008

Find the average power consumption of an alternating-current circuit if the current intensity I and voltage V are expressed by the following formulas, respectively:

I=I_{0}\cos\left(\omega t+\alpha \right)

V=V_{0}\cos\left(\omega t+\alpha+\phi\right),

where \phi is the constant phase shift of the voltage as compared with the current intensity.

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Posted by wMw

Indefinite Integrals: Substitution Rule

March 5, 2008

Evaluate the following integrals:

  1. \displaystyle \int \frac{x^3}{\sqrt{x^2 + 25}}\ dx
  2. \displaystyle \int \tan^2 \theta \sec^4 \theta\ d \theta
  3. \displaystyle \int \sqrt[3]{x^2 - 2x + 1}\ dx

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Posted by Tina

Derivatives of Integrals

March 5, 2008

Evaluate the following.

  1. \displaystyle \frac{d}{dx} \int_{x}^{0} \sqrt{1+t^{4}}\ dt
  2. \displaystyle \frac{d}{dx} \int_{\frac{1}{x}}^{\sqrt{x}} \cos t^{2}\ dt,\; x>0
  3. \displaystyle \frac{d^{2}}{dx^{2}} \int_{0}^{x} \left( \int_{1}^{\sin t} \sqrt{1+u^{4}}\ du \right) dt

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

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