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AP®︎ Calculus AB (2017 edition)
Course: AP®︎ Calculus AB (2017 edition) > Unit 6
Lesson 9: Connecting ƒ, ƒ’, and ƒ’’- The graphical relationship between a function & its derivative (part 1)
- The graphical relationship between a function & its derivative (part 2)
- Connecting f and f' graphically
- Visualizing derivatives
- Connecting f, f', and f'' graphically
- Connecting f, f', and f'' graphically (another example)
- Connecting f, f', and f'' graphically
- Curve sketching with calculus: polynomial
- Curve sketching with calculus: logarithm
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Connecting f and f' graphically
Given the graph of a function, we are asked to recognize the graph of its derivative. Created by Sal Khan.
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With the power rule giving us a declination of the exponent how can the derivative of something that looks like a squared function be a cubic? I agree that it is the right answer but my instincts would have told me that f'(x) would have gone to an exponential power of one which gives us a straight line. I want to stress that I see HOW to get the right answer in a problem like this I'm just wondering how it works numerically(7 votes)- If you have a graphing software, try inputing different even powers to see how they look:
x^2,x^4,x^6, etc. You will notice that even when they all follow the same pattern: they come from infinity, touch the x-axis only on 1 point and then go back to infinity, the 'curve' they follow is increasingly steep, with some practice you can recognise that the drawings are not quadratic, but at least forth power fucntions.(5 votes)
- Based upon what I've seen in this videos and previous videos, it appears as if you graph the derivative of a function, the leading term for the function of the derivative graph is always one power less than that of the actual function you are taking the derivative of. For example, if you have the equation f(x)=x^2, the graph of f'(x) would be f(x)=x. If you take the derivative of y=x^4, the graph of its derivative is y=x^3. Am I correct in saying that this holds true for every function (other than an undefined one). If so, is there some mathematical way of justifying it?
Thanks!(5 votes)
Before you proceed very far in calculus you'll encounter the power rule, which tells us that the derivative of x^n = nx^(n-1). For example, the derivative of x^4 is 4x^3 (not x^3, as you indicated). The justification will be provided in steps, because it's fairly easy to show that it applies when n is a non-negative integer, and proof that it applies also to negative and fractional exponents comes later.(3 votes)
Sorry, I made a typing error in the previous version of the question. The corrected question is as follows-
Is the derivative of this function (a part of a circle with radius 4) with equation f(x)=(16-x^2)^1/2, equal to -x/((16-x^2)^1/2)?(2 votes)
Well, the function does approximate a half circle, and that would be the correct function statement for it. To answer your question, what is the derivative of that function at x ~ -2.8? The derivative should be just about 1 (at that point on the surface of the circle, the tangent line forms a 45 degree angle).. Likewise, the derivative at x ~ 2.8 should be just about -1. With your equation, I get a very tiny amount (0.036) and -0.036, which are nearly horizontal and would not be correct for the slope of the tangent to a circle at those points. Hmmm.(6 votes)
So, what happens when you take the derivative of a function, then take the derivative of the derivative? Does the world explode, or is there a name for that kind of thing?(3 votes)
Double derivatives can be used to find a change in a function over time.
Ex.
you can find acceleration with a double derivative.
speed is distance with respect to time.
acceleration is speed with respect to time.
or
acceleration = distance w/ respect to time w/ respect to time(3 votes)
In this video, it looks like the graph of f(x) is basically a circle limited to the domain of [0, pi]. The corresponding derivative function (graph # 3) looks like the graph of the tangent function of a circle (though flipped vertically for some reason).(2 votes)- How about it being the plot of cotangent(x)?(1 vote)
how is infinity shown on a graph?(1 vote)- Great question!!
A HUMAN once said:
“...it's very much like your trying to reach infinity. You know that it's there, you just don't know where-but just because you can never reach it doesn't mean that it's not worth looking for.”
― Norton Juster, The Phantom Tollbooth
So the point is beautiful you cannot show infinity on a graph, but you can at least talk about it and show it by some sort of an identifier, an image, or a symbol.
Its almost like: nobody can tell you what apple tastes like, you have to be given the apple to find out but at least the people who have eaten an apple in their lives can talk among themselves and still know what they are talking about!
- Vedanta, Indian religious(véros science) text
I hope that was of help!!(2 votes)
How would you graph a derivative on a graphing calculator?(1 vote)- I use the (free) Desmos graphing calculator on my PC laptop, 'Droid tablet, and 'Droid phone. I'm able to graph derivatives by entering "d/dx" before the function. It's a pretty handy little widget, and powerful, too.
https://www.desmos.com/calculator(2 votes)
- At0:44sec in the video how is the slope coming from positive infinity at negative four?(1 vote)
At x = -4, the slope is vertical and has a positive infinity slope, thus, the graph is going from that positive infinity amount to 0 at y=4.(1 vote)
- If derivatives are graphing the slope of functions , does that mean the graphing of trigonometric values (sin, cos and tan) is the same as graphing the slope of a circle?(1 vote)
- No, to graph the slope of a circle, you would find the derivative of the equation of a circle, x²+y²=r². Check the section on "implicit differentiation" for more on that.(2 votes)
- The slope of our function is positive until it hits the y-axis, yet in the correct answer the derivative is negative?? i.e. if you were to draw tangents along the derivative they would all have a negative slope, throughout the derivative.(1 vote)
- do you have the answer to this know?(1 vote)
0:44Video transcript
I have a function
f of x here, and I want to think about
which of these curves could represent f prime
of x, could represent the derivative of f of x. Well, to think
about that, we just have to think
about, well, what is a slope of the tangent line
doing at each point of f of x and see if this
corresponds to that slope, if the value of these functions
correspond to that slope. So we can see when x
is equal to negative 4, the slope of the tangent
line is essentially vertical. So you could say it's
not really defined there. But as we go slightly to the
right of x equals negative 4, we just have a very,
very, very positive slope. So you could kind of
view it as our slope is going from infinity to very,
very positive to a little bit less positive to
a little bit less positive, to a little
bit less positive, to a little bit less positive. So which of these graphs
here have that property? Remember, this is trying
to graph the slope. So which of these functions down
here, which of these graphs, have a value that is essentially
kind of approaching infinity when x is equal to
negative 4, and then it gets less and less and less
positive as x goes to 0? So this one, it looks like it's
coming from negative infinity, and it's getting less and
less and less negative. So that doesn't seem to
meet our constraints. This one looks like it is
coming from positive infinity, and it's getting less and
less and less positive, so that seems to be OK. This has the same property. It's getting less and
less and less positive. This one right over here
starts very negative and gets less and less
and less negative. So we can rule that out. Now let's think about what
happens when x gets to 0. When x gets to 0, the
tangent line is horizontal. We're at a maximum point of
this curve right over here. The slope of a
horizontal line is 0. Remember, we're trying to look
for which one of these curves represent the value
of that slope. So which one of these curves
hit 0 when x is equal to 0? Well, this one doesn't. So the only candidate that
we have left is this one, and this one does hit
0 when x equals 0. And let's see if
it keeps satisfying what we need for f prime of x. So after that point,
it should start getting more and more negative. The slope should get more
and more and more negative, essentially approaching negative
infinity as x approaches 4. And we see that here. The value of this function is
getting more and more negative, and it's approaching negative
infinity as x approaches 4. So we'll go with this one. This looks like a pretty good
candidate for f prime of x.