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Chapter 4

Applications of Differentiation

Educators


Problem 1

Use the guidelines of this section to sketch the curve.
$$
y=x^{3}+3 x^{2}
$$

Bobby B.
University of North Texas

Problem 2

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 3

Use the guidelines of this section to sketch the curve.

$ y = x^4 - 4x $

Bobby B.
University of North Texas

Problem 4

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 5

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 6

Use the guidelines of this section to sketch the curve.

$ y = x^5 - 5x $

Bobby B.
University of North Texas

Problem 7

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 8

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 9

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 10

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 11

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 12

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 13

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 15

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 16

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 17

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 18

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 19

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 20

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 21

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 22

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 23

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 24

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 25

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 26

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 27

Use the guidelines of this section to sketch the curve.

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

Bobby B.
University of North Texas

Problem 28

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 29

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 30

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 31

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 32

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 33

Use the guidelines of this section to sketch the curve.

$ y = \sin^3 x $

Jamie M.
Numerade Educator

Problem 34

Use the guidelines of this section to sketch the curve.

$ y = x + \cos x $

Jamie M.
Numerade Educator

Problem 35

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 36

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 37

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 38

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 39

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 40

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 41

Use the guidelines of this section to sketch the curve.

$ y = \arctan(e^x) $

Jamie M.
Numerade Educator

Problem 42

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 43

Use the guidelines of this section to sketch the curve.

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

Carson M.
Numerade Educator

Problem 44

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 45

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 46

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 47

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 48

Use the guidelines of this section to sketch the curve.

$ y = e^x/x^2 $

Jamie M.
Numerade Educator

Problem 49

Use the guidelines of this section to sketch the curve.

$ y = \ln(\sin x) $

Jamie M.
Numerade Educator

Problem 50

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 51

Use the guidelines of this section to sketch the curve.

$ y = xe{-1/x} $

Clarissa N.
Numerade Educator

Problem 52

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 53

Use the guidelines of this section to sketch the curve.

$ y = e^{\arctan x} $

Jamie M.
Numerade Educator

Problem 54

Use the guidelines of this section to sketch the curve.

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

Jamie M.
Numerade Educator

Problem 55

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 $.

Jamie M.
Numerade Educator

Problem 56

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?

Jamie M.
Numerade Educator

Problem 57

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 $.

Bobby B.
University of North Texas

Problem 58

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?

Jamie M.
Numerade Educator

Problem 59

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.

Carson M.
Numerade Educator

Problem 60

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?

Jamie M.
Numerade Educator

Problem 61

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

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

Jamie M.
Numerade Educator

Problem 62

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

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

Jamie M.
Numerade Educator

Problem 63

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

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

Jamie M.
Numerade Educator

Problem 64

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

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

Jamie M.
Numerade Educator

Problem 65

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} $

Jamie M.
Numerade Educator

Problem 66

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} $

Jamie M.
Numerade Educator

Problem 67

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} $

Jamie M.
Numerade Educator

Problem 68

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} $

Jamie M.
Numerade Educator

Problem 69

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} $

Jamie M.
Numerade Educator

Problem 70

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} $

Carson M.
Numerade Educator

Problem 71

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.

Bobby B.
University of North Texas

Problem 72

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.

Carson M.
Numerade Educator

Problem 73

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 $.

Bobby B.
University of North Texas

Problem 74

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 $.

Bobby B.
University of North Texas

Problem 75

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 $.

Bobby B.
University of North Texas

Problem 76

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.

Bobby B.
University of North Texas