1
\( y = x^3 + 3 x^2 \)
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2
\( y = 2 x^3 - 12 x^2 + 18 x \)
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3
\( y = x^4 - 4 x \)
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4
\( y = x^4 - 8 x^2 + 8 \)
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5
\( y = x (x - 4)^3 \)
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6
\( y = x^5 - 5 x \)
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7
\( y = \dfrac{1}{5} x^5 - \dfrac{8}{3} x^3 + 16 x \)
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8
\( y = (4 - x^2)^5 \)
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9
\( y = \dfrac{2 x + 3}{x + 2} \)
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10
\( y = \dfrac{x^2 + 5 x}{25 - x^2} \)
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11
\( y = \dfrac{x - x^2}{2 - 3 x + x^2} \)
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12
\( y = 1 + \dfrac{1}{x} + \dfrac{1}{x^2} \)
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13
\( y = \dfrac{x}{x^2 - 4} \)
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14
\( y = \dfrac{1}{x^2 - 4} \)
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15
\( y = \dfrac{x^2}{x^2 + 3} \)
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16
\( y = \dfrac{(x - 1)^2}{x^2 + 1} \)
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17
\( y = \dfrac{x - 1}{x^2} \)
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18
\( y = \dfrac{x}{x^3 - 1} \)
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19
\( y = \dfrac{x^3}{x^3 + 1} \)
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20
\( y = \dfrac{x^3}{x - 2} \)
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21
\( y = (x - 3) \sqrt{x} \)
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22
\( y = (x - 4) \sqrt[3]{x} \)
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23
\( y = \sqrt{x^2 + x - 2} \)
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24
\( y = \sqrt{x^2 + x} - x \)
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25
\( y = \dfrac{x}{\sqrt{x^2 + 1}} \)
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26
\( y = x \sqrt{2 - x^2} \)
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27
\( y = \dfrac{\sqrt{1 - x^2}}{x} \)
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28
\( y = \dfrac{x}{\sqrt{x^2 - 1}} \)
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29
\( y = x - 3 x^{\dfrac{1}{3}} \)
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30
\( y = x^{\dfrac{5}{3}} - 5 x^{\dfrac{2}{3}} \)
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31
\( y = \sqrt[3]{x^2 - 1} \)
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32
\( y = \sqrt[3]{x^3 + 1} \)
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33
\( y = \sin^3 x \)
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34
\( y = x + \cos x \)
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35
\(y = x \tan x\), \(-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}\)
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36
\(y = 2 x - \tan x\), \(-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}\)
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37
\(y = \sin x + \sqrt{3} \cos x\), \(-2 \pi \leq x \leq 2 \pi\)
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38
\(y = \csc x - 2 \sin x\), \(0 < x < \pi\)
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39
\( y = \dfrac{\sin x}{1 + \cos x} \)
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40
\( y = \dfrac{\sin x}{2 + \cos x} \)
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41
The graph of a function \(f\) is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function \(g\).
(a) The domains of \(g\) and \(g'\)
(b) The critical numbers of \(g\)
(c) The approximate value of \(g'(6)\)
(d) All vertical and horizontal asymptotes of \(g\)
\(g(x) = \sqrt{f(x)}\)
Enter your answer directly below each part above.
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42
The graph of a function \(f\) is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function \(g\).
(a) The domains of \(g\) and \(g'\)
(b) The critical numbers of \(g\)
(c) The approximate value of \(g'(6)\)
(d) All vertical and horizontal asymptotes of \(g\)
\(g(x) = \sqrt[3]{f(x)}\)
Enter your answer directly below each part above.
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43
The graph of a function \(f\) is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function \(g\).
(a) The domains of \(g\) and \(g'\)
(b) The critical numbers of \(g\)
(c) The approximate value of \(g'(6)\)
(d) All vertical and horizontal asymptotes of \(g\)
\(g(x) = |f(x)|\)
Enter your answer directly below each part above.
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44
The graph of a function \(f\) is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function \(g\).
(a) The domains of \(g\) and \(g'\)
(b) The critical numbers of \(g\)
(c) The approximate value of \(g'(6)\)
(d) All vertical and horizontal asymptotes of \(g\)
\(g(x) = \dfrac{1}{f}(x)\)
Enter your answer directly below each part above.
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45
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 \(v\).
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46
In the theory of relativity, the energy of a particle is \(E = \sqrt{m_0^2 c^4 + h^2 c^2 / \lambda^2}\) where \(m_0\) is the rest mass of the particle, \(\lambda\) is its wavelength, and \(h\) is Planck's constant. Sketch the graph of \(E\) as a function of \(\lambda\). What does the graph say about the energy?
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47
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 \(y = -\dfrac{W}{24 E I} x^4 + \dfrac{W L}{12 E I} x^3 - \dfrac{W L^2}{24 E I} x^2\) where \(E\) and \(I\) are positive constants. Sketch the graph of the deflection curve.
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48
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 a coordinate line and a particle with charge \(-1\) at a position \(x\) between them. 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?
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49
Find an equation of the slant asymptote. Do not sketch the curve.
\(y = \dfrac{x^2 + 1}{x + 1}\)
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50
Find an equation of the slant asymptote. Do not sketch the curve.
\(y = \dfrac{4 x^3 - 10 x^2 - 11 x + 1}{x^2 - 3 x}\)
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51
Find an equation of the slant asymptote. Do not sketch the curve.
\(y = \dfrac{2 x^3 - 5 x^2 + 3 x}{x^2 - x - 2}\)
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52
Find an equation of the slant asymptote. Do not sketch the curve.
\(y = \dfrac{-6 x^4 + 2 x^3 + 3}{2 x^3 - x}\)
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53
Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.
\(y = \dfrac{x^2}{x - 1}\)
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54
Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.
\(y = \dfrac{1 + 5 x - 2 x^2}{x - 2}\)
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55
Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.
\(y = \dfrac{x^3 + 4}{x^2}\)
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56
Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.
\(y = \dfrac{x^3}{(x + 1)^2}\)
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57
Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.
\(y = \dfrac{2 x^3 + x^2 + 1}{x^2 + 1}\)
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58
Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.
\(y = \dfrac{(x + 1)^3}{(x - 1)^2}\)
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59
Show that the curve \(y = \sqrt{4 x^2 + 9}\) has two slant asymptotes: \(y = 2 x\) and \(y = -2 x\). Use this fact to help sketch the curve.
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60
Show that the curve \(y = \sqrt{x^2 + 4 x}\) has two slant asymptotes: \(y = x + 2\) and \(y = -x - 2\). Use this fact to help sketch the curve.
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61
Show that the lines \(y = \left(\dfrac{b}{a}\right) x\) and \(y = -\left(\dfrac{b}{a}\right) x\) are slant asymptotes of the hyperbola \((x^2 / a^2) - (y^2 / b^2) = 1\).
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62
Let \(f(x) = (x^3 + 1)/x\). Show that \(\operatorname*{lim}\limits_{x \rightarrow \pm \infty} [f(x) - x^2] = 0\). This shows that the graph of \(f\) approaches the graph of \(y = x^2\), and we say that the curve \(y = f(x)\) is asymptotic to the parabola \(y = x^2\). Use this fact to help sketch the graph of \(f\).
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63
Discuss the asymptotic behavior of \(f(x) = (x^4 + 1)/x\) in the same manner as in Exercise 62. Then use your results to help sketch the graph of \(f\).
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64
Use the asymptotic behavior of \(f(x) = \cos x + 1/x^2\) to sketch its graph without going through the curve-sketching procedure of this section.
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65
Use the guidelines to sketch the curve \(y = \dfrac{2 x^2}{x^2 - 1}\).
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66
Sketch the graph of \(f(x) = \dfrac{x^2}{\sqrt{x + 1}}\).
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67
Sketch the graph of \(f(x) = \dfrac{\cos x}{2 + \sin x}\).
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68
Sketch the graph of \(f(x) = \dfrac{x^3}{x^2 + 1}\).
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Answered: 0 / 68