3.1 Defining the derivative  (Page 3/10)

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Finding the equation of a tangent line

Find the equation of the line tangent to the graph of $f\left(x\right)=1\text{/}x$ at $x=2.$

We can use [link] , but as we have seen, the results are the same if we use [link] .

$\begin{array}{ccccc}\hfill {m}_{\text{tan}}& =\underset{x\to 2}{\text{lim}}\frac{f\left(x\right)-f\left(2\right)}{x-2}\hfill & & & \text{Apply the definition.}\hfill \\ & =\underset{x\to 2}{\text{lim}}\frac{\frac{1}{x}-\frac{1}{2}}{x-2}\hfill & & & \text{Substitute}\phantom{\rule{0.2em}{0ex}}f\left(x\right)=\frac{1}{x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.1em}{0ex}}f\left(2\right)=\frac{1}{2}.\hfill \\ & =\underset{x\to 2}{\text{lim}}\frac{\frac{1}{x}-\frac{1}{2}}{x-2}·\frac{2x}{2x}\hfill & & & \begin{array}{c}\text{Multiply numerator and denominator by}\phantom{\rule{0.2em}{0ex}}2x\phantom{\rule{0.2em}{0ex}}\text{to}\hfill \\ \text{simplify fractions.}\hfill \end{array}\hfill \\ & =\underset{x\to 2}{\text{lim}}\frac{\left(2-x\right)}{\left(x-2\right)\left(2x\right)}\hfill & & & \text{Simplify.}\hfill \\ & =\underset{x\to 2}{\text{lim}}\frac{-1}{2x}\hfill & & & \text{Simplify using}\phantom{\rule{0.2em}{0ex}}\frac{2-x}{x-2}=-1,\text{for}\phantom{\rule{0.2em}{0ex}}x\ne 2.\hfill \\ & =-\frac{1}{4}\hfill & & & \text{Evaluate the limit.}\hfill \end{array}$

We now know that the slope of the tangent line is $-\frac{1}{4}.$ To find the equation of the tangent line, we also need a point on the line. We know that $f\left(2\right)=\frac{1}{2}.$ Since the tangent line passes through the point $\left(2,\frac{1}{2}\right)$ we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation $y=-\frac{1}{4}x+1.$ The graphs of $f\left(x\right)=\frac{1}{x}$ and $y=-\frac{1}{4}x+1$ are shown in [link] .

Find the slope of the line tangent to the graph of $f\left(x\right)=\sqrt{x}$ at $x=4.$

$\frac{1}{4}$

The derivative of a function at a point

The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the derivative    . The process of finding a derivative is called differentiation    .

Definition

Let $f\left(x\right)$ be a function defined in an open interval containing $a.$ The derivative of the function $f\left(x\right)$ at $a,$ denoted by ${f}^{\prime }\left(a\right),$ is defined by

${f}^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$

provided this limit exists.

Alternatively, we may also define the derivative of $f\left(x\right)$ at $a$ as

${f}^{\prime }\left(a\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f\left(a\right)}{h}.$

Estimating a derivative

For $f\left(x\right)={x}^{2},$ use a table to estimate ${f}^{\prime }\left(3\right)$ using [link] .

Create a table using values of $x$ just below $3$ and just above $3.$

$x$ $\frac{{x}^{2}-9}{x-3}$
$2.9$ $5.9$
$2.99$ $5.99$
$2.999$ $5.999$
$3.001$ $6.001$
$3.01$ $6.01$
$3.1$ $6.1$

After examining the table, we see that a good estimate is ${f}^{\prime }\left(3\right)=6.$

For $f\left(x\right)={x}^{2},$ use a table to estimate ${f}^{\prime }\left(3\right)$ using [link] .

$6$

Finding a derivative

For $f\left(x\right)=3{x}^{2}-4x+1,$ find ${f}^{\prime }\left(2\right)$ by using [link] .

Substitute the given function and value directly into the equation.

$\begin{array}{ccccc}\hfill {f}^{\prime }\left(x\right)& =\underset{x\to 2}{\text{lim}}\frac{f\left(x\right)-f\left(2\right)}{x-2}\hfill & & & \text{Apply the definition.}\hfill \\ & =\underset{x\to 2}{\text{lim}}\frac{\left(3{x}^{2}-4x+1\right)-5}{x-2}\hfill & & & \text{Substitute}\phantom{\rule{0.2em}{0ex}}f\left(x\right)=3{x}^{2}-4x+1\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}f\left(2\right)=5.\hfill \\ & =\underset{x\to 2}{\text{lim}}\frac{\left(x-2\right)\left(3x+2\right)}{x-2}\hfill & & & \text{Simplify and factor the numerator.}\hfill \\ & =\underset{x\to 2}{\text{lim}}\left(3x+2\right)\hfill & & & \text{Cancel the common factor.}\hfill \\ & =8\hfill & & & \text{Evaluate the limit.}\hfill \end{array}$

Revisiting the derivative

For $f\left(x\right)=3{x}^{2}-4x+1,$ find ${f}^{\prime }\left(2\right)$ by using [link] .

Using this equation, we can substitute two values of the function into the equation, and we should get the same value as in [link] .

$\begin{array}{ccccc}\hfill {f}^{\prime }\left(2\right)& =\underset{h\to 0}{\text{lim}}\frac{f\left(2+h\right)-f\left(2\right)}{h}\hfill & & & \text{Apply the definition.}\hfill \\ & =\underset{h\to 0}{\text{lim}}\frac{\left(3{\left(2+h\right)}^{2}-4\left(2+h\right)+1\right)-5}{h}\hfill & & & \begin{array}{c}\text{Substitute}\phantom{\rule{0.2em}{0ex}}f\left(2\right)=5\phantom{\rule{0.2em}{0ex}}\text{and}\hfill \\ f\left(2+h\right)=3{\left(2+h\right)}^{2}-4\left(2+h\right)+1.\hfill \end{array}\hfill \\ & =\underset{h\to 0}{\text{lim}}\frac{3{h}^{2}+8h}{h}\hfill & & & \text{Simplify the numerator.}\hfill \\ & =\underset{h\to 0}{\text{lim}}\frac{h\left(3h+8\right)}{h}\hfill & & & \text{Factor the numerator.}\hfill \\ & =\underset{h\to 0}{\text{lim}}\left(3h+8\right)\hfill & & & \text{Cancel the common factor.}\hfill \\ & =8\hfill & & & \text{Evaluate the limit.}\hfill \end{array}$

The results are the same whether we use [link] or [link] .

For $f\left(x\right)={x}^{2}+3x+2,$ find ${f}^{\prime }\left(1\right).$

${f}^{\prime }\left(1\right)=5$

Velocities and rates of change

Now that we can evaluate a derivative, we can use it in velocity applications. Recall that if $s\left(t\right)$ is the position of an object moving along a coordinate axis, the average velocity    of the object over a time interval $\left[a,t\right]$ if $t>a$ or $\left[t,a\right]$ if $t is given by the difference quotient

find the domain and range of f(x)= 4x-7/x²-6x+8
find the range of f(x)=(x+1)(x+4)
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
d/dx{1/y - lny + X^3.Y^5}
How to identify domain and range
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
sorry
Dr
Dr
:(
Shun
was up
Dr
hello
is it chatting app?.. I do not see any calculus here. lol
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