12 Lecture need to Solve Problems On Calculus 1

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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, let’s 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 don’t, 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, “What’s 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 “What’s 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),
you’ll 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 / Newton’s method
Overview
We learn an application of tangent lines: a technique to estimate otherwise
difficult-to-estimate quantities.
Details
discuss:
Suppose you’d like to estimate a difficult-to-find number (e.g.
v
6
2). Newton’s 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: Newton’s method is called iterative. This is because it works by
iterating on an initial guess to produce (hopefully) better guesses.
fact: In Newton’s method, xn+1 = xn –
f (xn )
f 0 (xn )
for each n = 1.
discuss: Sometimes Newton’s method fails to converge to the desired number. Sometimes it doesn’t converge at all.
Key vocab, techniques
vocab: Newton’s method
techniques: Use Newton’s 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 Newton’s 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 Newton’s 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 Newton’s method to estimate the solution to x3 = cos x. Using
x1 = 1, find x2 .
4. Apply Newton’s method to estimate the positive 8th root of 2. Using
x1 = 1, find x2 .
5. Suppose we are to apply Newton’s 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, we’ve considered derivatives of functions. It’s 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. What’s more, given any point in the plane,
there’s 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 Jill’s 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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