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In this section, you will:
  • Find the derivative of a function.
  • Find instantaneous rates of change.
  • Find an equation of the tangent line to the graph of a function at a point.
  • Find the instantaneous velocity of a particle.

The average teen in the United States opens a refrigerator door an estimated 25 times per day. Supposedly, this average is up from 10 years ago when the average teenager opened a refrigerator door 20 times per day http://www.csun.edu/science/health/docs/tv&health.html Source provided. .

It is estimated that a television is on in a home 6.75 hours per day, whereas parents spend an estimated 5.5 minutes per day having a meaningful conversation with their children. These averages, too, are not the same as they were 10 years ago, when the television was on an estimated 6 hours per day in the typical household, and parents spent 12 minutes per day in meaningful conversation with their kids.

What do these scenarios have in common? The functions representing them have changed over time. In this section, we will consider methods of computing such changes over time.

Finding the average rate of change of a function

The functions describing the examples above involve a change over time. Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs.

A tangent line    to a curve is a line that intersects the curve at only a single point but does not cross it there. (The tangent line may intersect the curve at another point away from the point of interest.) If we zoom in on a curve at that point, the curve appears linear, and the slope of the curve at that point is close to the slope of the tangent line at that point.

[link] represents the function f ( x ) = x 3 4 x . We can see the slope at various points along the curve.

  • slope at x = −2 is 8
  • slope at x = −1 is –1
  • slope at x = 2 is 8

Graph of f(x) = x^3 - 4x with tangent lines at x = -2 with a slope of 8, at x = -3 with a slope of -1, and at x=2 with a slope of 8.
Graph showing tangents to curve at –2, –1, and 2.

Let’s imagine a point on the curve of function f at x = a as shown in [link] . The coordinates of the point are ( a , f ( a ) ) . Connect this point with a second point on the curve a little to the right of x = a , with an x -value increased by some small real number h . The coordinates of this second point are ( a + h , f ( a + h ) ) for some positive-value h .

Graph of an increasing function that demonstrates the rate of change of the function by drawing a line between the two points, (a, f(a)) and (a, f(a+h)).
Connecting point a with a point just beyond allows us to measure a slope close to that of a tangent line at x = a .

We can calculate the slope of the line connecting the two points ( a , f ( a ) ) and ( a + h , f ( a + h ) ) , called a secant line    , by applying the slope formula,

slope =  change in  y change in  x

slope =  change in  y change in  x

We use the notation m sec to represent the slope of the secant line connecting two points.

m sec = f ( a + h ) f ( a ) ( a + h ) ( a )         = f ( a + h ) f ( a ) a + h a

The slope m sec equals the average rate of change between two points ( a , f ( a ) ) and ( a + h , f ( a + h ) ) .

m sec = f ( a + h ) f ( a ) h

The average rate of change between two points on a curve

The average rate of change    (AROC) between two points ( a , f ( a ) ) and ( a + h , f ( a + h ) ) on the curve of f is the slope of the line connecting the two points and is given by

AROC = f ( a + h ) f ( a ) h

Questions & Answers

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
Practice Key Terms 7

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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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