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Find the equation of a tangent line to the curve of the function f ( x ) = 5 x 2 x + 4 at x = 2.

y = 19 x 16

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Finding the instantaneous speed of a particle

If a function measures position versus time, the derivative measures displacement versus time, or the speed of the object. A change in speed or direction relative to a change in time is known as velocity . The velocity at a given instant is known as instantaneous velocity .

In trying to find the speed or velocity of an object at a given instant, we seem to encounter a contradiction. We normally define speed as the distance traveled divided by the elapsed time. But in an instant, no distance is traveled, and no time elapses. How will we divide zero by zero? The use of a derivative solves this problem. A derivative allows us to say that even while the object’s velocity is constantly changing, it has a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance.

Instantaneous velocity

Let the function s ( t ) represent the position of an object at time t . The instantaneous velocity    or velocity of the object at time t = a is given by

s ( a ) = lim h 0 s ( a + h ) s ( a ) h

Finding the instantaneous velocity

A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after t seconds is given by s ( t ) = −16 t 2 + 36 t + 200 , find the instantaneous velocity of the ball at t = 2.

First, we must find the derivative s ( t ) . Then we evaluate the derivative at t = 2 , using s ( a + h ) = 16 ( a + h ) 2 + 36 ( a + h ) + 200 and s ( a ) = 16 a 2 + 36 a + 200.

s ( a ) = lim h 0 s ( a + h ) s ( a ) h          = lim h 0 16 ( a + h ) 2 + 36 ( a + h ) + 200 ( 16 a 2 + 36 a + 200 ) h          = lim h 0 16 ( a 2 + 2 a h + h 2 ) + 36 ( a + h ) + 200 ( 16 a 2 + 36 a + 200 ) h          = lim h 0 16 a 2 32 a h 16 h 2 + 36 a + 36 h + 200 + 16 a 2 36 a 200 h          = lim h 0 16 a 2 32 a h 16 h 2 + 36 a + 36 h + 200 + 16 a 2 36 a 200 h          = lim h 0 32 a h 16 h 2 + 36 h h          = lim h 0 h ( 32 a 16 h + 36 ) h          = lim h 0 ( 32 a 16 h + 36 )          = 32 a 16 0 + 36   s ( a ) = 32 a + 36   s ( 2 ) = 32 ( 2 ) + 36          = 28
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A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after t seconds is given by s = 16 t 2 + 60 t 12. What is its instantaneous velocity after 4 seconds?

–68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.

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Key equations

average rate of change AROC = f ( a + h ) f ( a ) h
derivative of a function f ( a ) = lim h 0 f ( a + h ) f ( a ) h

Key concepts

  • The slope of the secant line connecting two points is the average rate of change of the function between those points. See [link] .
  • The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See [link] , [link] , and [link] .
  • The difference quotient is the quotient in the formula for the instantaneous rate of change:
    f ( a + h ) f ( a ) h
  • Instantaneous rates of change can be used to find solutions to many real-world problems. See [link] .
  • The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See [link] .
  • Instantaneous rates of change can be interpreted to describe real-world situations. See [link] and [link] .
  • Some functions are not differentiable at a point or points. See [link] .
  • The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See [link] .
  • Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time.
  • Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See [link] .

Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
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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