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Finding instantaneous rates of change

Many applications of the derivative involve determining the rate of change at a given instant of a function with the independent variable time—which is why the term instantaneous is used. Consider the height of a ball tossed upward with an initial velocity of 64 feet per second, given by s ( t ) = −16 t 2 + 64 t + 6 , where t is measured in seconds and s ( t ) is measured in feet. We know the path is that of a parabola. The derivative will tell us how the height is changing at any given point in time. The height of the ball is shown in [link] as a function of time. In physics, we call this the “ s - t graph.”

Graph of a negative parabola with a vertex at (2, 70) and two points at (1, 55) and (3, 55).

Finding the instantaneous rate of change

Using the function above, s ( t ) = −16 t 2 + 64 t + 6 , what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight?

The velocity at t = 1 and t = 3 is the instantaneous rate of change of distance per time, or velocity. Notice that the initial height is 6 feet. To find the instantaneous velocity, we find the derivative    and evaluate it at t = 1 and t = 3 :

f ( a ) = lim h 0 f ( a + h ) f ( a ) h          = lim h 0 16 ( t + h ) 2 + 64 ( t + h ) + 6 ( 16 t 2 + 64 t + 6 ) h Substitute  s ( t + h )  and  s ( t ) .          = lim h 0 16 t 2 32 h t h 2 + 64 t + 64 h + 6 + 16 t 2 64 t 6 h Distribute .          = lim h 0 32 h t h 2 + 64 h h Simplify .          = lim h 0 h ( 32 t h + 64 ) h Factor the numerator .          = lim h 0 32 t h + 64 Cancel out the common factor  h . s ( t ) = 32 t + 64 Evaluate the limit by letting  h = 0.

For any value of t , s ( t ) tells us the velocity at that value of t .

Evaluate t = 1 and t = 3.

s ( 1 ) = −32 ( 1 ) + 64 = 32 s ( 3 ) = −32 ( 3 ) + 64 = −32

The velocity of the ball after 1 second is 32 feet per second, as it is on the way up.

The velocity of the ball after 3 seconds is −32 feet per second, as it is on the way down.

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The position of the ball is given by s ( t ) = −16 t 2 + 64 t + 6. What is its velocity 2 seconds into flight?

0

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Using graphs to find instantaneous rates of change

We can estimate an instantaneous rate of change at x = a by observing the slope of the curve of the function f ( x ) at x = a . We do this by drawing a line tangent to the function at x = a and finding its slope.

Given a graph of a function f ( x ) , find the instantaneous rate of change of the function at x = a .

  1. Locate x = a on the graph of the function f ( x ) .
  2. Draw a tangent line, a line that goes through x = a at a and at no other point in that section of the curve. Extend the line far enough to calculate its slope as
    change in  y change in  x .

Estimating the derivative at a point on the graph of a function

From the graph of the function y = f ( x ) presented in [link] , estimate each of the following:

  1. f ( 0 )
  2. f ( 2 )
  3. f ' ( 0 )
  4. f ' ( 2 )

Graph of an odd function with multiplicity of two and with two points at (0, 1) and (2, 1).

To find the functional value, f ( a ) , find the y -coordinate at x = a .

To find the derivative    at x = a , f ( a ) , draw a tangent line at x = a , and estimate the slope of that tangent line. See [link] .

Graph of the previous function with tangent lines at the two points (0, 1) and (2, 1). The graph demonstrates the slopes of the tangent lines. The slope of the tangent line at x = 0 is 0, and the slope of the tangent line at x = 2 is 4.
  1. f ( 0 ) is the y -coordinate at x = 0. The point has coordinates ( 0 , 1 ) , thus f ( 0 ) = 1.
  2. f ( 2 ) is the y -coordinate at x = 2. The point has coordinates ( 2 , 1 ) , thus f ( 2 ) = 1.
  3. f ( 0 ) is found by estimating the slope of the tangent line to the curve at x = 0. The tangent line to the curve at x = 0 appears horizontal. Horizontal lines have a slope of 0, thus f ( 0 ) = 0.
  4. f ( 2 ) is found by estimating the slope of the tangent line to the curve at x = 2. Observe the path of the tangent line to the curve at x = 2. As the x value moves one unit to the right, the y value moves up four units to another point on the line. Thus, the slope is 4, so f ( 2 ) = 4.
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Questions & Answers

A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
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meena
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meena
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
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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