# 3.1 Defining the derivative  (Page 5/10)

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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 $70\text{°}\text{F}$ at $9$ p.m., reaches a low of $60\text{°}$ during the night, and rises back to $70\text{°}$ by $7$ a.m. the next morning. Suppose that the temperature in the house is given by $T\left(t\right)=0.4{t}^{2}-4t+70$ for $0\le t\le 10,$ 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 ${T}^{\prime }\left(3\right).$ Refer to [link] .

$\begin{array}{ccccc}\hfill {T}^{\prime }\left(3\right)& =\underset{t\to 3}{\text{lim}}\frac{T\left(t\right)-T\left(3\right)}{t-3}\hfill & & & \text{Apply the definition.}\hfill \\ & =\underset{t\to 3}{\text{lim}}\frac{0.4{t}^{2}-4t+70-61.6}{t-3}\hfill & & & \begin{array}{c}\text{Substitute}\phantom{\rule{0.2em}{0ex}}T\left(t\right)=0.4{t}^{2}-4t+70\phantom{\rule{0.2em}{0ex}}\text{and}\hfill \\ T\left(3\right)=61.6.\hfill \end{array}\hfill \\ & =\underset{t\to 3}{\text{lim}}\frac{0.4{t}^{2}-4t+8.4}{t-3}\hfill & & & \text{Simplify.}\hfill \\ & =\underset{t\to 3}{\text{lim}}\frac{0.4\left(t-3\right)\left(t-7\right)}{t-3}\hfill & & & =\underset{t\to 3}{\text{lim}}\frac{0.4\left(t-3\right)\left(t-7\right)}{t-3}\hfill \\ & =\underset{t\to 3}{\text{lim}}0.4\left(t-7\right)\hfill & & & \text{Cancel.}\hfill \\ & =-1.6\hfill & & & \text{Evaluate the limit.}\hfill \end{array}$

The instantaneous rate of change of the temperature at midnight is $-1.6\text{°}\text{F}$ per hour.

## Rate of change of profit

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

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

$R\left(x\right)=xp=x\left(-0.01x+400\right)=-0.01{x}^{2}+400x.$

Consequently,

$P\left(x\right)=-0.01{x}^{2}+300x-10,000.$

Therefore, evaluating the rate of change of profit gives

$\begin{array}{cc}\hfill {P}^{\prime }\left(10000\right)& =\underset{x\to 10000}{\text{lim}}\frac{P\left(x\right)-P\left(10000\right)}{x-10000}\hfill \\ & =\underset{x\to 10000}{\text{lim}}\frac{-0.01{x}^{2}+300x-10000-1990000}{x-10000}\hfill \\ & =\underset{x\to 10000}{\text{lim}}\frac{-0.01{x}^{2}+300x-2000000}{x-10000}\hfill \\ & =100.\hfill \end{array}$

Since the rate of change of profit ${P}^{\prime }\left(10,000\right)>0$ and $P\left(10,000\right)>0,$ the company should increase production.

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

${P}^{\prime }\left(3.25\right)=20>0;$ raise prices

## 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 $f\left(x\right)$ 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 $v\left(t\right)$ at time $t$ is the derivative of the position $s\left(t\right)$ at time $t.$ Average velocity is given by
${v}_{\text{ave}}=\frac{s\left(t\right)-s\left(a\right)}{t-a}.$

Instantaneous velocity is given by
$v\left(a\right)={s}^{\prime }\left(a\right)=\underset{t\to a}{\text{lim}}\frac{s\left(t\right)-s\left(a\right)}{t-a}.$
• We may estimate a derivative by using a table of values.

## Key equations

• Difference quotient
$Q=\frac{f\left(x\right)-f\left(a\right)}{x-a}$
• Difference quotient with increment $h$
$Q=\frac{f\left(a+h\right)-f\left(a\right)}{a+h-a}=\frac{f\left(a+h\right)-f\left(a\right)}{h}$
• Slope of tangent line
${m}_{\text{tan}}=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$
${m}_{\text{tan}}=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f\left(a\right)}{h}$
• Derivative of $f\left(x\right)$ at $a$
${f}^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$
${f}^{\prime }\left(a\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f\left(a\right)}{h}$
• Average velocity
${v}_{a\text{ve}}=\frac{s\left(t\right)-s\left(a\right)}{t-a}$
• Instantaneous velocity
$v\left(a\right)={s}^{\prime }\left(a\right)=\underset{t\to a}{\text{lim}}\frac{s\left(t\right)-s\left(a\right)}{t-a}$

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 $y=f\left(x\right).$

can you give me a problem for function. a trigonometric one
state and prove L hospital rule
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t=0), total online holiday sales were $42.3 billion, whereas in 2013 they were$48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t . Interpret the slope of the graph of S . Use part a. to predict the year when online shopping during Christmas will reach \$60 billion?
what is the derivative of x= Arc sin (x)^1/2
y^2 = arcsin(x)
Pitior
x = sin (y^2)
Pitior
differentiate implicitly
Pitior
then solve for dy/dx
Pitior
thank you it was very helpful
morfling
questions solve y=sin x
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
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
y=800
800
Bg
how do u factor the numerator?
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
2x^3+6xy-4y^2)^2 solve this
femi
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio