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Section 3.4 / Limits at infinity; horizontal asymptotes

Overview

What happens to the height of a graph as we go very, very, very, very,

really far to the right? Or to the left? To make the question precise and get

answers, we need the concept of limits at infinity.

Details

discuss:

The symbol lim f (x) means, informally, the trend in the outx?8

put of f (x) as we let x grow indefinitely. Likewise, lim f (x) means the

x?-8

trend in the output of f (x) as we let x shrink (big negative) indefinitely. As

with limits at finite numbers, the possibilities are: no trend (no limit), infinite limits, or finite limits. To be thorough, lets have the formal definitions.

def: We say the limit of f (x) as x approaches infinity is L, and write

lim f (x) = L, if for any > 0 there exists some N > 0 such that |f (x)-L| <
x?8
whenever x = N .
We say the limit of f (x) as x approaches negative infinity is L, and
write lim f (x) = L, if for any > 0 there exists some N < 0 such that
x?-8
|f (x) - L| < whenever x = N .
We say that f has a limit of infinity as x approaches infinity, and write
lim f (x) = 8, if for any B > 0 there exists some N > 0 such that f (x) = B

x?8

whenever x = N .

We say that f has a limit of infinity as x approaches negative infinity,

and write lim f (x) = 8, if for any B > 0 there exists some N < 0 such
x?-8
that f (x) = B whenever x = N .
We say that f has a limit of negative infinity as x approaches infinity,
and write lim f (x) = -8, if for any B < 0 there exists some N > 0 such

x?8

that f (x) = B whenever x = N .

We say that f has a limit of negative infinity as x approaches negative

infinity, and write lim f (x) = -8, if for any B < 0 there exists some
x?-8
N < 0 such that f (x) = B whenever x = N .
1
def: We say the graph y = f (x) has a horizontal asymptote of y = L
whenever lim f (x) = L or lim f (x) = L.
x?8
x?-8
rk: What does it look like when a graph has a horizontal asymptote? The
graph gets closer and closer to the height of the asymptote as you go far
to the right (or left, or both). As before with limits at finite numbers, we
consider the phrase closer and closer to include a graph which is already
there. For example, technically any horizontal line is its own horizontal
asymptote.
ex: f (x) =
x2 -1
x2 +1
rk: It is false that a graph cannot touch its own asymptotes.
ex: f (x) = cos x
1
= 0. A similar statement holds for
fact: If lim f (x) = ±8, then lim f (x)
x?8
x?8
limits at negative infinity.
ex:
3x2 - x - 2
x?8 5x2 + 4x + 1
lim
def: The lead coefficient of a polynomial is the coefficient of the highest
power of the variable. That is, if the terms are ordered from greatest to
least power of the variable (as is custom), the lead coefficient is the leftmost
coefficient in the polynomial.
ex: The lead coefficient of 4x2 - 5 is 4. The lead coefficient of -2x3 + x - 17
is -2. The lead coefficient of 20x - 7x3 + 14 - 9x2 is -7.
discuss: When analyzing the limit at infinity of a rational function, there
are three main cases. Those are: (1) the numerator is of greater degree than
the denominator, (2) the numerator is of the same degree as the denominator, and (3) the numerator is of smaller degree than the denominator. In the
first case, the limit is either 8 or -8 (depending on the signs of the lead
coefficients). In the second case, the limit is the ratio of the lead coefficients
(numerator over denominator). In the third case, the limit is 0.
2
You can come up with similar rules for limits at negative infinity, but they
are a little more complicated, as you must look out for even/odd exponents
affecting the signs.
ex:
v
2x2 + 1
lim
x?8 3x - 5
ex:
v
lim x2 + 7x - x
x?8
Key vocab, techniques
vocab: limit at infinity, limit at negative infinity, horizontal asymptote
techniques: Compute limits at infinity. Given a limit which looks like
infinity divided by infinity, try dividing the top and bottom of the fraction by
the something which makes the limits of the top and bottom of the fraction
finite. Given a limit which looks like infinity minus infinity and a root is
involved, consider multiplying top and bottom by the conjugate.
Problems
1. Find the following limits if they exist. If they dont, say why not.
(a)
cos x
x?8
x
lim
(b)
lim
x?0
cos x
x
(c)
sin x
x?0 x
lim
2.
3x - 2
x?8 2x + 1
lim
3.
v
3
lim
x?-8
4.
x + x3
2x - x2
v
x + 3x2
lim
x?8 4x - 1
3
5.
v
lim ( 9x2 + x - 3x)
x?8
6.
v
lim
x?-8
7.
8.
1 + 4x6
2 - x3
x2
lim v
x?8
x4 + 1
v
lim ( x2 + x + 1 + x)

x?-8

4

Section 3.5 / Summary of curve sketching

Overview

We make a list of things to consider when graphing a function.

Details

fact: Even should a function f (x) approach infinity as x ? ±8, we can

discuss how the graph approaches infinity. We single out a special case, which

is that the graph becomes more and more linear as x increases or decreases

without bound.

def: If there is a line l(x) such that |f (x) – l(x)| ? 0 as x ? ±8, then

l(x) is said to be a slant asymptote for f .

rk: A rational function whose numerator is of degree one greater than the

denominator has a slant asymptote. To find it, long-divide.

ex: f (x) =

x3

x2 +1

discuss: To graph a function f (x):

(a) Find the domain of f .

(b) Identify any symmetry or periodicity f has.

(c) Find the x- and y-intercepts of f .

(d) Find vertical asymptotes.

(e) Chart (on a number line) where f is positive, negative, zero, and undefined.

(f) Find where f 0 is positive, negative, zero, and undefined.

(g) Locate cusps, terrace points, local extrema, and global extrema. If

global extrema do not exist, say so.

(h) Find where f 00 is positive, negative, and zero.

(i) Locate inflection points.

rk: All points of interest (local and global extrema, cusps, terrace points,

inflection points) should be labeled on the graph. Asymptotes should be

labeled with their equations.

ex:

1

x5/3 – x2/3

2×2

x2 -1

2

vx

x+1

cos x

2+sin x

Key vocab, techniques

vocab:

techniques:

Problems

v

1. Sketch the graph of f (x) = x – 4 x. Label all x- and y-intercepts.

Find where the graph is positive and negative, rising and falling, concave up

and concave down, and incorporate this information into your sketch. Label

any local and global extrema that the function has. If the function lacks a

global minimum and/or maximum, point this out.

2. Sketch the graph of f (x) = x3 – 9×2 + 15x – 135. Label all x- and

y-intercepts. Find where the graph is positive and negative, rising and

falling, concave up and concave down, and incorporate this information into

your sketch. Label any local and global extrema that the function has. If

the function lacks a global minimum and/or maximum, point this out.

5

2

3. Sketch the graph of f (x) = 5x 3 – 2x 3 . Label all x- and y-intercepts.

Find where the graph is positive and negative, rising and falling, concave up

and concave down, and incorporate this information into your sketch. Label

any local and global extrema that the function has. If the function lacks a

global minimum and/or maximum, point this out.

4. Sketch the graph of f (x) = 2×3 – 3×2 – 12x + 18. Label all x- and

y-intercepts. Find where the graph is positive and negative, rising and

falling, concave up and concave down, and incorporate this information into

your sketch. Label any local and global extrema that the function has. If

the function lacks a global minimum and/or maximum, point this out.

cos x

. Label all x- and y-intercepts. Find

5. Sketch the graph of f (x) = 2+sin

x

where the graph is positive and negative, rising and falling, concave up and

concave down, and incorporate this information into your sketch. Label any

local and global extrema that the function has. If the function lacks a global

minimum and/or maximum, point this out.

2

6. Sketch the graph of f (x) = x3 -x2 +x-1. Label all x- and y-intercepts.

Find where the graph is positive and negative, rising and falling, concave up

and concave down, and incorporate this information into your sketch. Label

any local and global extrema that the function has. If the function lacks a

global minimum and/or maximum, point this out.

v

7. Sketch the graph of f (x) = x2 + x – x. Label all x- and y-intercepts.

Find where the graph is positive and negative, rising and falling, concave up

and concave down, and incorporate this information into your sketch. Label

all local extrema, global extrema, and inflection points. If the function lacks

a global minimum and/or maximum, point this out.

8. Sketch the graph of f (x) = vxx2 -1 . Label all x- and y-intercepts. Find

where the graph is positive and negative, rising and falling, concave up and

concave down, and incorporate this information into your sketch. Label all

local extrema, global extrema, and inflection points. If the function lacks a

global minimum and/or maximum, point this out.

2

x

. Label all x- and y-intercepts. Find

9. Sketch the graph of f (x) = x-1

where the graph is positive and negative, rising and falling, concave up and

concave down, and incorporate this information into your sketch. Label all

local extrema, global extrema, and inflection points. If the function lacks a

global minimum and/or maximum, point this out.

3

Section 3.7 / Optimization problems

Overview

We apply our skills in finding extreme values to real life problems.

Details

discuss: An optimization problem asks us to maximize or minimize something subject to a constraint. Asking, Whats the largest cylindrical tank

you can build? is not helpful, because in theory there is no largest tank.

In real life, we deal with questions like Whats the largest cylindrical tank

you can build for a thousand bucks? Home Depot is selling suitably thick

sheet metal for $12 per square foot. Here, the condition that must be met

is called the constraint. If you think about the Extreme Value Theorem for a

moment (while considering very tall thin tanks and very short wide tanks),

youll see that there must be a largest tank we can build with a thousand

dollars.

ex: Fencing in a rectangular pen given a fixed amount of fence.

ex: Fencing in a rectangular pen given a fixed amount of fence and using a

river as one wall.

ex: Find the dimensions of a cylindrical can (with lid) with minimal surface

area and volume 1000 cubic centimeters.

ex: Find the point on the parabola y 2 = 2x closest to the point (1, 4).

ex: Crossing a river to a point downstream, and you can run faster than

you can row.

ex: Drawing a rectangle of maximum area inside of a semicircle, assuming

that one side of the triangle lies along the diameter of the circle.

Key vocab, techniques

vocab: constraint

techniques: How to solve most optimization problems: start by identifying

a quantity to be optimized (maximized or minimized) (e.g. maximizing area).

Express that quantity as a function of some parameter(s) (e.g. length and

1

width of the pen). If necessary, express one of the parameters as a function

of the other; this step often uses a constraint given in the problem (e.g.

expressing length of the pen in terms of width using the fact that the amount

of fencing is fixed). Differentiate the function, and find the global extremum.

Problems

1. Find the point on the graph y =

v

x closest to (2, 0).

2. Find the point on the graph of x = 1 +

Hint: The point is not on the graph!

v

y closest to the point (0, 1).

3. You build your pet octopus a cylindrical tank with no lid. The circular

floor will be constructed of stylish black onyx at $100 per square foot, and

the walls of carnelian quartz at $200 per square foot. The tank must have a

volume of at least 250p cubic feet. How cheaply can it be built?

4. Find the rectangle of greatest area that can be inscribed in a right triangle with side lengths 5, 12, and 13. Assume that two sides of the rectangle

must lie along the two legs of the triangle.

5. Find the area of the largest rectangle that can be inscribed in the ellipse

2

x2

+ yb2 = 1.

a2

6. Find the volume of the largest cylinder which can be inscribed in a sphere

of radius r.

2

Section 3.8 / Newtons method

Overview

We learn an application of tangent lines: a technique to estimate otherwise

difficult-to-estimate quantities.

Details

discuss:

Suppose youd like to estimate a difficult-to-find number (e.g.

v

6

2). Newtons method is as follows. First, declare a function so that the

value you seek is zero of the function. For example, let f (x) = x6 – 2, so

that we seek the positive zero of f . Make a close guess to the true value; call

this guess x1 . Find the tangent line to f (x) at x1 , then find the zero of this

tangent line. Call the zero of the tangent line x2 . Repeat the process, finding x3 , x4 , and so on. If things are working correctly, you should be getting

closer and closer to the actual value of the quantity you are trying to estimate.

discuss: Newtons method is called iterative. This is because it works by

iterating on an initial guess to produce (hopefully) better guesses.

fact: In Newtons method, xn+1 = xn –

f (xn )

f 0 (xn )

for each n = 1.

discuss: Sometimes Newtons method fails to converge to the desired number. Sometimes it doesnt converge at all.

Key vocab, techniques

vocab: Newtons method

techniques: Use Newtons method to perform estimations. In most problems, without a calculator it takes a while to get beyond the first two iterations.

Problems

1. Try applying Newtons method to find the zero of

v

x

for x = 0

v

f (x) =

– -x for x < 0
1
using an initial guess of a > 0. What goes wrong?

2. Try applying Newtons method to find the zero of

f (x) = x3 – 3x + 6

using an initial guess of x1 = 1. What goes wrong? As a lesson from this

example, for x1 to be a reasonable first guess, what must be true about f 0 (x1 )?

3. Apply Newtons method to estimate the solution to x3 = cos x. Using

x1 = 1, find x2 .

4. Apply Newtons method to estimate the positive 8th root of 2. Using

x1 = 1, find x2 .

5. Suppose we are to apply Newtons method to cos x. Suppose 0 = x0 = 2p

and xn = x0 for n even, xn = 2p – x0 for n odd. Find all possible values of

cot x0 .

2

Section 3.9 / Antiderivatives

Overview

To date, weve considered derivatives of functions. Its a different question

to ask: Given a function, what other function has the given one as its derivative? We introduce the antiderivative and record some basic facts about

antiderivatives.

Details

def: If F 0 (x) = f (x), then F is called an antiderivative of f .

ex: An antiderivative of 3×2 is x3 . Another antiderivative of 3×2 is x3 + 4.

How many antiderivatives do you suppose a function has?

rk: It is not always easy to find an antiderivative of a given function. For

example, what is the antiderivative of x sin x? A significant part of MA242

is techniques for finding antiderivatives.

fact: It is an consequence of the Mean Value Theorem that if

everywhere then F (x) = c for some real number c.

d

F (x)

dx

=0

fact: Let F and G be two continuous functions whose domain is [a, b]. If

F 0 (x) = G0 (x) on (a, b), then F (x) = G(x) + c for some constant c. In words,

two functions with the same derivatives must differ by a constant.

def: Given an antiderivative of a function f , one can always find lots of

antiderivatives. The set of all antiderivatives of a function f is called the general antiderivative of f . The general antiderivative of a function is always an

infinite family of functions. In light of the above, if F is any antiderivative

of f , then the general antiderivative is the set of all functions of the form

F (x) + C where C is a real number.

def: The act of finding the antiderivative of a function is sometimes called

antidifferentiation.

fact: The power rule for antidifferentiation.

ex: Find the general antiderivative of g 0 (x) = 4 sin x +

1

v

2×5 – x

.

5

discuss: If you begin with a function f , find the general antiderivative,

and then graph a few of the members of that family, you might notice some

things. The graphs of any two antiderivatives are vertical shifts of one another, so they cannot intersect. Whats more, given any point in the plane,

theres an antiderivative which passes through it (so long as there is no domain issue). This sets us up for the following type of problem.

def: Suppose f is a function with antiderivative F . Let D be the domain

of F . If (a, b) is any point in the plane where a is in D, then there is an

antiderivative of f which passes through (a, b). The problem of finding this

antiderivative is called an initial value problem.

ex: Find the antiderivative of f (x) = x2 – 4x + 7 which passes through the

point (0, 4).

ex: Find vthe antiderivative of f (x) = sin(2x) which passes through the

point ( p8 , – 2).

ex: A rock is thrown upward with initial velocity v0 from height h0 . The

acceleration is constant at a. Find things out about the trajectory of the rock.

Key vocab, techniques

vocab: antiderivative, general antiderivative, antidifferentiation, initial value

problem

techniques: Find the general antiderivative of a function. Find the antiderivative of a function satisfying a given initial condition.

Problems

1. A rock is dropped from the top of the moon castle. At the same time, a

second rock is thrown upward from the ground below with a speed of 20m/s.

The rocks collide in air. At what height did they collide? Assume the moon

castle is 400m tall. The acceleration due to gravity on the moon is -1.6m/s2 .

2. A rock is tossed upward at 6m/s from a height 10m above the surface of

Callisto. The acceleration due to gravity on Callisto is -1.2m/s2 . Determine:

(a) the equation which gives the height of the rock t seconds after release,

2

(b) the time at which the rock reaches its peak,

(c) the time at which the rock hits the ground, and

(d) the velocity of the rock when it hits the ground.

3.

An object is launched up from a height of 10m with initial velocity

40m/s. Assume the acceleration due to gravity is -10m/s2 . Find:

(a) the equation which gives the height of the object as a function of time

since launch,

(b) the time when the object peaks,

(c) the time when the object hits the ground, and

(d) the velocity upon impact with the ground.

4. A rock is tossed upward at 7m/s from a height 10m above the surface of

Titan. The acceleration due to gravity on Titan is -1.4m/s2 . Determine:

(a) the equation which gives the height of the rock t seconds after release,

(b) the time at which the rock reaches its peak,

(c) the time at which the rock hits the ground, and

(d) the velocity of the rock when it hits the ground.

5. A ball is launched upward from the surface of Mars at a speed of 37m/s.

Ten seconds later, a second ball is launched upward at a speed of 74m/s.

The balls collide in air. At what height do they collide? The acceleration

due to gravity on Mars is -3.7m/s2 .

6. Jack stands on the roof of a 96 foot tall building. Jill stands on the

ground below. Jack throws a rock downward and Jill throws a rock upward.

(a) If the rocks are each thrown at the same time with speed 32ft/sec,

where do they collide?

(b) If the rocks are thrown at the same time with the same speed and

collide as Jills rock is at the top of its arc, how fast were the rocks

thrown?

(c) If the rocks are thrown at the same time and collide at a height of 80ft

and Jack threw his rock with speed 16ft/sec, how fast did Jill throw

her rock?

3

(d) If the rocks are each thrown with speed 64ft/sec and collide at a height

of 64ft, how long after Jill threw her rock did Jack throw his?

7.

v

x sec x – 7 x cos x

Let f (x) =

.

x cos x

(a) Find the general antiderivative of f (x).

(b) Find an antiderivative of f (x) which contains …

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