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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

can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
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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