# 3.1 Defining the derivative  (Page 5/10)

 Page 5 / 10

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

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
show that lim f(x) + lim g(x)=m+l
list the basic elementary differentials
Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
by using integration product formula
Roha
find derivative f(x)=1/x
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
do u mean fundamental theorem ?
Roha
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