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Rate of change of temperature

A homeowner sets the thermostat so that the temperature in the house begins to drop from 70 ° F at 9 p.m., reaches a low of 60 ° during the night, and rises back to 70 ° by 7 a.m. the next morning. Suppose that the temperature in the house is given by T ( t ) = 0.4 t 2 4 t + 70 for 0 t 10 , where t is the number of hours past 9 p.m. Find the instantaneous rate of change of the temperature at midnight.

Since midnight is 3 hours past 9 p.m., we want to compute T ( 3 ) . Refer to [link] .

T ( 3 ) = lim t 3 T ( t ) T ( 3 ) t 3 Apply the definition. = lim t 3 0.4 t 2 4 t + 70 61.6 t 3 Substitute T ( t ) = 0.4 t 2 4 t + 70 and T ( 3 ) = 61.6 . = lim t 3 0.4 t 2 4 t + 8.4 t 3 Simplify. = lim t 3 0.4 ( t 3 ) ( t 7 ) t 3 = lim t 3 0.4 ( t 3 ) ( t 7 ) t 3 = lim t 3 0.4 ( t 7 ) Cancel. = −1.6 Evaluate the limit.

The instantaneous rate of change of the temperature at midnight is −1.6 ° F per hour.

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Rate of change of profit

A toy company can sell x electronic gaming systems at a price of p = −0.01 x + 400 dollars per gaming system. The cost of manufacturing x systems is given by C ( x ) = 100 x + 10,000 dollars. Find the rate of change of profit when 10,000 games are produced. Should the toy company increase or decrease production?

The profit P ( x ) earned by producing x gaming systems is R ( x ) C ( x ) , where R ( x ) is the revenue obtained from the sale of x games. Since the company can sell x games at p = −0.01 x + 400 per game,

R ( x ) = x p = x ( −0.01 x + 400 ) = −0.01 x 2 + 400 x .

Consequently,

P ( x ) = −0.01 x 2 + 300 x 10,000 .

Therefore, evaluating the rate of change of profit gives

P ( 10000 ) = lim x 10000 P ( x ) P ( 10000 ) x 10000 = lim x 10000 −0.01 x 2 + 300 x 10000 1990000 x 10000 = lim x 10000 −0.01 x 2 + 300 x 2000000 x 10000 = 100 .

Since the rate of change of profit P ( 10,000 ) > 0 and P ( 10,000 ) > 0 , the company should increase production.

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A coffee shop determines that the daily profit on scones obtained by charging s dollars per scone is P ( s ) = −20 s 2 + 150 s 10 . The coffee shop currently charges $ 3.25 per scone. Find P ( 3.25 ) , the rate of change of profit when the price is $ 3.25 and decide whether or not the coffee shop should consider raising or lowering its prices on scones.

P ( 3.25 ) = 20 > 0 ; raise prices

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Key concepts

  • The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h .
  • The derivative of a function f ( x ) at a value a is found using either of the definitions for the slope of the tangent line.
  • Velocity is the rate of change of position. As such, the velocity v ( t ) at time t is the derivative of the position s ( t ) at time t . Average velocity is given by
    v ave = s ( t ) s ( a ) t a .

    Instantaneous velocity is given by
    v ( a ) = s ( a ) = lim t a s ( t ) s ( a ) t a .
  • We may estimate a derivative by using a table of values.

Key equations

  • Difference quotient
    Q = f ( x ) f ( a ) x a
  • Difference quotient with increment h
    Q = f ( a + h ) f ( a ) a + h a = f ( a + h ) f ( a ) h
  • Slope of tangent line
    m tan = lim x a f ( x ) f ( a ) x a
    m tan = lim h 0 f ( a + h ) f ( a ) h
  • Derivative of f ( x ) at a
    f ( a ) = lim x a f ( x ) f ( a ) x a
    f ( a ) = lim h 0 f ( a + h ) f ( a ) h
  • Average velocity
    v a ve = s ( t ) s ( a ) t a
  • Instantaneous velocity
    v ( a ) = s ( a ) = lim t a s ( t ) s ( a ) t a

For the following exercises, use [link] to find the slope of the secant line between the values x 1 and x 2 for each function y = f ( x ) .

Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
Andrew Reply
what is implicit of y³+x²y³=5 at (2,1)
Estelita Reply
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg
Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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