Calculus: Early Transcendentals

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$$
y=x^{3}+3 x^{2}
$$

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$ y = 2 + 3x^2 - x^3 $

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$ y = x^4 - 4x $

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$ y = x^4 - 8x^2 + 8 $

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$ y = x(x - 4)^3 $

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$ y = x^5 - 5x $

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$ y = \frac{1}{5} x^5 - \frac{8}{3} x^3 + 16x $

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$ y = (4 - x^2)^5 $

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$ y = \frac{x}{x - 1} $

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$ y = \frac{x^2 + 5x}{25 - x^2} $

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$ y = \frac{x - x^2}{2 - 3x + x^2} $

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$ y = 1 + \frac{1}{x} + \frac{1}{x^2} $

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$ y = \frac{x}{x^2 - 4} $

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$ y = \frac{x^2}{x^2 + 3} $

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$ y = \frac{(x - 1)^2}{x^2 + 1} $

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$ y = \frac{x - 1}{x^2} $

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$ y = \frac{x}{x^3 - 1} $

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$ y = \frac{x^3}{x^3 + 1} $

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$ y = \frac{x^3}{x - 1} $

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$ y = (x - 3)\sqrt{x} $

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$ y = (x - 4)\sqrt[3]{x} $

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$ y = \sqrt{x^2 + x - 1} $

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$ y = \sqrt{x^2 + x} - x $

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$ y = \frac{x}{\sqrt{x^2 + 1}} $

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$ y = x\sqrt{2 - x^2} $

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$ y = \frac{\sqrt{1 - x^2}}{x} $

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$ y = \frac{x}{\sqrt{x^2 - 1}} $

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$ y = x - 3x^{1/3} $

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$ y = x^{5/3} - 5x^{2/3} $

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$ y = \sqrt[3]{x^2 - 1} $

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$ y = \sqrt[3]{x^3 + 1} $

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$ y = \sin^3 x $

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$ y = x + \cos x $

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$ y = x\tan x $, $ -\pi /2 < x < \pi /2 $

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$ y = 2x - \tan x $, $ -\pi /2 < x < \pi /2 $

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$ y = \sin x + \sqrt{3} \cos x $, $ -2\pi \leqslant x \leqslant 2\pi $

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$ y = \csc x - 2\sin x $, $ 0 < x < pi $

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$ y = \dfrac {\sin x}{1 + \cos x} $

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$ y = \dfrac {\sin x}{2 + \cos x} $

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$ y = \arctan(e^x) $

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$ y = (1 - x)e^x $

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$ y = 1/(1 + e^{-x}) $

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$ y = e^{-x}\sin x $, $ 0 \leqslant x \leqslant 2\pi $

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$ y = \dfrac{1}{x} + \ln x $

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$ y = e^{2x} - e^x $

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$ y = (1 + e^x)^{-2} $

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$ y = e^x/x^2 $

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$ y = \ln(\sin x) $

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$ y = \ln(1 + x^3) $

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$ y = xe{-1/x} $

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$ y = \dfrac{\ln x}{x^2} $

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$ y = e^{\arctan x} $

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$ y = \tan^{-1} \left(\dfrac{x - 1}{x + 1}\right) $

In the theory of relativity, the mass of a particle is
$$ m = \dfrac{m_0}{\sqrt{1 - v^2/c^2}} $$
where $ m_0 $ is the rest mass of the particle, $ m $ is the mass when the particle moves with speed $ v $ relative to the observer, and $ c $ is the speed of light. Sketch the graph of $ m $ as a function of $ v $.

In the theory of relativity, the energy of a particle is
$$ E = \sqrt{m{_0}^2c^4 + h^2c^2/ \lambda^2} $$
where $ m_0 $ is the rest mass of the particle, $ \lambda $ is its wave length, and $ h $ is Planck's constant. Sketch the graph of $ E $ as a function of $ \lambda $. What does the graph say about the energy?

A model for the spread of a rumor is given by the equation
$$ p(t) = \dfrac{1}{1 + ae^{-kt}} $$
where $ p(t) $ is the proportion of the population that knows the rumor at the time $ t $ and $ a $ and $ k $ are positive constants.
(a) When will half the population have heard the rumor?
(b) When is the rate of spread of the rumor greatest?
(c) Sketch the graph of $ p $.

A model for the concentration at time $ t $ of a drug injected into the bloodstream is
$$ C(t) = K(e^{-at} - e^{-bt}) $$
where $ a $, $ b $, and $ K $ are positive constants and $ b > a $. Sketch the graph of the concentration function. What does the graph tell us about how the concentration varies as time passes?

The figure shows a beam of length $ L $ embedded in concrete walls. If a constant load $ W $ is distributed evenly along its length, the beam takes the shape of the deflection curve
$$ y = -\dfrac{W}{24EI} x^4 + \dfrac{WL}{12EI} x^3 - \dfrac{WL^2}{24EI} x^2 $$
where $ E $ and $ I $ are positive constants. ($ E $ is Young's modulus of elasticity and $ I $ is the moment of inertia of a cross-section of the beam.) Sketch the graph of the deflection curve.

Coulomb's Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge $ 1 $ located at positions $ 0 $ and $ 2 $ on the coordinate line and a particle with charge $ -1 $ at a position $ x $ between them. It follows from Coulomb's Law that the net force acting on the middle particle is
$$ F(x) = -\dfrac{k}{x^2} + \dfrac{k}{(x - 2)^2} $$ $$ 0 < x < 2 $$
where $ k $ is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?

Find an equation of the slant asymptote. Do not sketch the curve.

$ y = \dfrac{x^2 + 1}{x + 1} $

Find an equation of the slant asymptote. Do not sketch the curve.

$ y = \dfrac{4x^3 - 10x^2 - 11x + 1}{x^2 - 3x} $

Find an equation of the slant asymptote. Do not sketch the curve.

$ y = \dfrac{2x^3 - 5x^2 + 3x}{x^2 - x - 2} $

Find an equation of the slant asymptote. Do not sketch the curve.

$ y = \dfrac{-6x^4 + 2x^3 + 3}{2x^3 - x} $

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{x^2}{x - 1} $

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{1 + 5x - 2x^2}{x - 2} $

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{x^3 + 4}{x^2} $

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{x^3}{(x + 1)^2} $

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = 1 + \dfrac{1}{2}x + e^{-x} $

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = 1 - x + e^{1 + x/3} $

Show that the curve $ y = x - \tan^{-1} x $ has two slant asymptotes: $ y = x + \pi /2 $ and $ y = x - \pi /2 $. Use this fact to help sketch the curve.

Show that the curve $ y = \sqrt{x^2 + 4x} $ has two slant asymptotes: $ y = x +2 $ and $ y = -x - x $. Use this fact to help sketch the curve.

Show that the lines $ y = (b/a)x $ and $ y = -(b/a)x $ are slant asymptotes of the hyperbola $ (x^2/a^2) - (y^2/b^2) = 1 $.

Let $ f(x) = (x^3 + 1)/x $. Show that
$$ \displaystyle \lim_{x\to \pm \infty} [f(x) - x^2] = 0 $$
This shows that the graph of $ f $ approaches the graph of $ y = x^2 $, and we say that the curve $ y = x^2 $. Use this fact to help sketch the graph of $ f $.

Discuss the asymptotic behavior of $ f(x) = (x^4 + 1)/x $ in the same manner as in Exercise 74. Then use your results to help sketch the graph of $ f $.

Use the asymptotic behavior of $ f(x) = \sin x + e^{-x} $ to sketch its graph without going through the curve-sketching procedure of this section.