In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.
This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable
x .
Consider the graph in
[link] of the function
$y=\text{sin}\phantom{\rule{0.1em}{0ex}}x+\text{cos}\phantom{\rule{0.1em}{0ex}}x.$ Describe its overall shape. Is it periodic? How do you know?
Using a graphing calculator or other graphing device, estimate the
$x$ - and
$y$ -values of the maximum point for the graph (the first such point where
x >0). It may be helpful to express the
$x$ -value as a multiple of π.
Now consider other graphs of the form
$y=A\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+B\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$ for various values of
A and
B . Sketch the graph when
A = 2 and
B = 1, and find the
$x$ - and
y -values for the maximum point. (Remember to express the
x -value as a multiple of π, if possible.) Has it moved?
Repeat for
A = 1,
B = 2. Is there any relationship to what you found in part (2)?
Complete the following table, adding a few choices of your own for
A and
B :
A
B
x
y
A
B
x
y
0
1
$\sqrt{3}$
1
1
0
1
$\sqrt{3}$
1
1
12
5
1
2
5
12
2
1
2
2
3
4
4
3
Try to figure out the formula for the
y -values.
The formula for the
$x$ -values is a little harder. The most helpful points from the table are
$\left(1,1\right),\left(1,\sqrt{3}\right),\left(\sqrt{3},1\right).$ (
Hint : Consider inverse trigonometric functions.)
If you found formulas for parts (5) and (6), show that they work together. That is, substitute the
$x$ -value formula you found into
$y=A\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+B\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$ and simplify it to arrive at the
$y$ -value formula you found.
Key concepts
For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
For a function
$f$ and its inverse
${f}^{\mathrm{-1}},f\left({f}^{\mathrm{-1}}\left(x\right)\right)=x$ for all
$x$ in the domain of
${f}^{\mathrm{-1}}$ and
${f}^{\mathrm{-1}}\left(f\left(x\right)\right)=x$ for all
$x$ in the domain of
$f.$
Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
The graph of a function
$f$ and its inverse
${f}^{\mathrm{-1}}$ are symmetric about the line
$y=x.$
For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.
$f\left(x\right)={x}^{2}-4,x\ge 0$
a.
${f}^{\mathrm{-1}}\left(x\right)=\sqrt{x+4}$ b. Domain
$\text{:}\phantom{\rule{0.2em}{0ex}}x\ge \mathrm{-4},\text{range}\text{:}\phantom{\rule{0.2em}{0ex}}y\ge 0$
marginal rate of substitution in economics for an example is an implicit differentiation function.
It is a proportion of comparison. It could be expressed as the area of a triangle of the old value proportional to the new, and then the next value and so on.
James
The new shapes form a line with a derivative curve
James
That curve could be expressed mathematically.
A good real life example is the proportion at which people barter in pawn shops.
"How about 100?"
"What are you trying to do? rob me? 50!"
"No way chap. 75."
"Ill give you 62. deal?"
"68. Nothing more nothing less."
"deal"
James
the proportion of what someone was trying to get for their product, versus what they were offered to the new price they wanted for their product and what they were offered
James
The proportion of the differentiating triangles would be somewhat 1:2.
and since there is little variation to the curve then it looks more like a straight line.
James
By the way This is one of the hardest subjects for me. I have a really hard time expressing things in such a way. I'm trying to have more exact calculations which is why I still study the subject.
Luke 17:21 nor will they say, See here or See there For indeed, the kingdom of God is within you.
You've never 'touched' anything. The e-energy field created by your body has pushed other electricfields. even our religions tell us we're the gods. We live in energies connecting us all. Doa/higgsfield
Scott
if you have any calculus questions many of us would be happy to help and you can always learn or even invent your own theories and proofs. math is the laws of logic and reality. its rules are permanent and absolute. you can absolutely learn calculus and through it better understand our existence.
Scott
ya doubtless
Bilal
help the integral of x^2/lnxdx
Levis
also find the value of "X" from the equation that follow
(x-1/x)^4 +4(x^2-1/x^2) -6=0
please guy help
Levis
Use integration by parts. Let u=lnx and dv=x2dx Then du=1xdx and v=13x3.
∫x2lnxdx=13x3lnx−∫(13x3⋅1x)dx
∫x2lnxdx=13x3lnx−∫13x2dx
∫x2lnxdx=13x3lnx−19x3+C
Bilal
itz 1/3 and 1/9
Bilal
now you can find the value of X from the above equation easily
Bilal
Pls i need more explanation on this calculus
usman
usman
from where do you need help?
Levis
thanks Bilal
Levis
integrate e^cosx
Uchenna
-sinx e^x
Leo
Do we ask only math question?
or ANY of the question?
bilal kumhar you are so biased
if you are an expert what are you doing here lol😎😎😂😂 we are here to learn
and beside there are many questions on this chat which you didn't attempt
we are helping each other stop being naive and arrogance
so give me the integral of x^x
Levis
Levis
I am sorry
Bilal
Bilal
it okay buddy
honestly i am pleasured to meet you
Levis
x^x ... no anti derivative for this function... but we can find definte integral numerically.
Bilal
thank you Bilal Kumhar
then how we may find definite integral let say x^x,3,5?
Levis
evaluate 5-×square divided by x+2
find x as limit approaches infinity
if your asking for derivative
dy/dz=x^2/2(lnx-1/2)
Levis
iam sorry
f(x)=x^x
it means the output(range ) depends to input(domain) value of x by the power of x
that is to say if x=2 then x^x would be 2^2=4
f(x) is the product of X to the power of X
its derivatives is found by using product rule
y=x^x
introduce ln each side we have
lny=lnx^x
=lny=xlnx
A maxima in a curve refers to the maximum point said curve. The maxima is a point where the gradient of the curve is equal to 0 (dy/dx = 0) and its second derivative value is a negative (d²y/dx² = -ve).
Leo we don't just do like that buddy!!!
use first principle
y+∆y=x+∆x
∆y=x+∆x-y
∆y=(x+∆x)^2+(x+∆x)-x^2+x
on solving it become
∆y=3∆x+∆x^2 as ∆x_>0 limit=3 if you do by calculator say plugging any value of x=0.000005 which approach 0 you get 3