12.4 Derivatives (Page 2/18)

Precalculus

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Finding the average rate of change

Find the average rate of change connecting the points $(2, −6)$ and $(−1, 5) .$

We know the average rate of change connecting two points may be given by

AROC = \frac{f (a + h) - f (a)}{h} .

If one point is $(2, - 6),$ or $(2, f (2)),$ then $f (2) = −6.$

The value $h$ is the displacement from $2$ to $- 1,$ which equals $- 1 - 2 = −3.$

For the other point, $f (a + h)$ is the y -coordinate at $a + h,$ which is $2 + (−3)$ or $−1,$ so $f (a + h) = f (−1) = 5.$

\begin{array}{l} AROC = \frac{f (a + h) - f (a)}{h} \\ = \frac{5 - (- 6)}{- 3} \\ = \frac{11}{- 3} \\ = - \frac{11}{3} \end{array}

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Find the average rate of change connecting the points $(- 5, 1.5)$ and $(- 2.5, 9) .$

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Understanding the instantaneous rate of change

Now that we can find the average rate of change, suppose we make $h$ in [link] smaller and smaller. Then $a + h$ will approach $a$ as $h$ gets smaller, getting closer and closer to 0. Likewise, the second point $(a + h, f (a + h))$ will approach the first point, $(a, f (a)) .$ As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at $x = a,$ and the slope of the secant line will get closer and closer to the slope of the tangent at $x = a .$ See [link] .

Graph of an increasing function that contains a point, P, at (a, f(a)). At the point, there is a tangent line and two secant lines where one secant line is connected to Q1 and another secant line is connected to Q2. — The connecting line between two points moves closer to being a tangent line at $x = a .$

Because we are looking for the slope of the tangent at $x = a,$ we can think of the measure of the slope of the curve of a function $f$ at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change , or the derivative of the function at $x = a .$ Both can be found by finding the limit of the slope of a line connecting the point at $x = a$ with a second point infinitesimally close along the curve. For a function $f$ both the instantaneous rate of change of the function and the derivative of the function at $x = a$ are written as $f' (a),$ and we can define them as a two-sided limit that has the same value whether approached from the left or the right.

f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}

The expression by which the limit is found is known as the difference quotient .

A General Note

Definition of instantaneous rate of change and derivative

The derivative , or instantaneous rate of change , of a function $f$ at $x = a,$ is given by

f' (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}

The expression $\frac{f (a + h) - f (a)}{h}$ is called the difference quotient.

We use the difference quotient to evaluate the limit of the rate of change of the function as $h$ approaches 0.

Derivatives: interpretations and notation

The derivative of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.

The derivative of a function $f (x)$ at a point $x = a$ is the slope of the tangent line to the curve $f (x)$ at $x = a .$ The derivative of $f (x)$ at $x = a$ is written $f^{'} (a) .$
The derivative $f^{'} (a)$ measures how the curve changes at the point $(a, f (a)) .$
The derivative $f^{'} (a)$ may be thought of as the instantaneous rate of change of the function $f (x)$ at $x = a .$
If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time $t = a .$

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

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bill

-24m+3+3mÁ^2

Susan

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Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

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Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

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Abubakar

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Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

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Method

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Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

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Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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