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AP®︎ Calculus AB (2017 edition)
Unit 3: Lesson 4
Derivatives of negative and fractional powers with power rule- Differentiating integer powers (mixed positive and negative)
- Power rule (negative & fractional powers)
- Fractional powers differentiation
- Power rule (with rewriting the expression)
- Radical functions differentiation intro
- Differentiate integer powers (mixed positive and negative)
- Worked example: Tangent to the graph of 1/x
- Power rule review
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Worked example: Tangent to the graph of 1/x
A practice problem for finding the y-intercept of a tangent line to f(x) = 1/x. Created by Sal Khan.
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Why is y = 1/k and x = k ?
How is that possible ?(7 votes)
That's how the function is defined. At the beginning of the video Sal said we are dealing with the function y = 1/x. The specific question we are answering is what is the y-intercept when x = k. When x = k, y is equal to 1/k.(27 votes)
At3:45, why is the derivative of the curve the same as the slope of its tangent line?(0 votes)
Because the derivative was defined to be exactly that.(38 votes)
What is Local linearization?(4 votes)
The tangent line at a point on a curve can approximate points close to that value.(5 votes)
what about tangent lines using a circle ?(1 vote)- A circle is not a function, and so isn't applicable here. A function must have no more than one y value for any given x value. To understand the concept of a function better, watch the videos in this section: http://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-relationships-functions/cc-8th-function-intro/v/testing-if-a-relationship-is-a-function.
However, you can find the tanget line of a circle by using "implicit" differentiation and providing a specific point on the circle. Think about a circle for a minute: for most of the x values on a circle, there are two corresponding y values, one above the center point and one below the center point. You can't determine the tanget line of a circle with just an x value because it matters which y value corresponding to the x value you're choosing. With an ellipse or circle, the slope is determined by BOTH the x and the y variable (unlike a function, whose slope is defined in terms of x only). Sal has so good videos on the topic here: http://www.khanacademy.org/math/calculus/differential-calculus/implicit_differentiation/e/implicit-differentiation.
That said, once you find the derivative of a circle through implicit differentiation, finding the y-intercept of a line tanget to the circle, given an arbitrary k where x = k, works just the same as it does in this video.(8 votes)
- This is kind of the same thing as what he does in this video but it is sort of backwards and I'm having trouble with it. If you have the equation of a curve and are asked to find the points where the tangent line has a y-intercept of 5, how do you go about that?(3 votes)
- Let us consider the facts we know about tangent lines:
1. They must intersect the function at the point of tangency.
2. They must have the same slope as does the function at the point of tangency.
Using the point-slope form of the line, this means:
y-y₁ = m(x-x₁)
Where (x₁, y₁) is the point of tangency.
Now, we need to replace m with the first derivative of the curve, since the first derivative can be interpreted as the slope of the curve (as a function instead of a constant).
y-y₁ = (f')(x-x₁)
where f' is the first derivative of the function.
We are now given an additional piece of information, the y=intercept of the tangent line. in other words, the tangent line goes through the point (0, 5). We plug that in for x and y:
5-y₁ = (f')(-x₁)
Note: we would use the original function (not its derivative) to give us enough information to solve for two unknowns. We plug in the point of tangency (x₁,y₁) into the original function. This would give us two equations and two unknowns, which would allow us to solve for (x₁,y₁) (there may be multiple solutions).(3 votes)
- The slope of a line can be written as tan = y/x. if at k f(x) is 1/k then tan = (1/k)/k = 1/(k^2) but when derivative is taken it gives us -1/(k^2) why is that?(2 votes)
- Notice that the slope of the line you are finding gives the slope from the origin to (k, f(k)) for some some x = k. But the derivative of f(x) at x=k gives the slope exactly at x =k. These slope from the origin is a secant line, and the second is an instantaneous slope. Thus, their values are not the same in this example.(3 votes)
Are there more examples? I don't understand what he did to make x to the -1 turn into a -1 times x to the -2(1 vote)- When using the power rule you should multiply the x coefficient (1) by the exponent (-1), and then reduce the exponent by 1.(3 votes)
- at 2.40, why is the exponent of -1 in f(x) became -2 in f ' (x)?(1 vote)
The equation is f(x)=x^-1
Since the power rule states that
d/dx (x^n) = nx^(n-1)
Thus, f'(x) = -1x^(-1-1) = -1x^-2(2 votes)
- I have a silly question, In the derivative introduction videos Sal said that for a function to be differentiable it must be continuous, Defined, has no vertical asymptotes and the limit of the derivative must exist if there are any other rules please tell me
my question is how can a function like 1/x not be continuous and has a vertical asymptote still has a derivative of -x^-2 ?(1 vote) - Why is the y-value 1/k ?(1 vote)
- Remember that the function is defined as f(x) = 1/x, where x does not equal 0. So if f(x) = 1/x, then any number we plug into the function is defined as 1 over that number.
So if we said that x = 1, then f(1) = 1/1. If x = 2, then f(2) = 1/2. If x = pi, then f(pi) = 1/pi. And so on for any number that is not zero. In this example, x = k (k is an arbitrary constant), and therefore f(k) = 1/k.
Now, keep in mind that f(x) is just a fancy way of saying y. We can define our x,y ordered pairs in terms of (x,y). Or, we can define them in terms of (x,f(x)). It's the same thing. But here, we know from the problem that x = k (note that Sal double-underlined that fact). So we can substitue k for x, which yields (k,f(k)). Since we know that the rule for f(x) = 1/x, and f(x) = y, and x = k, we know that f(x) = 1/x is the same as saying for a constant, k, f(k) = 1/k or as saying y = 1/k.(2 votes)
3:45Video transcript
In terms of k, where
k does not equal 0, what is the y-intercept of
the line tangent to the curve f of x is equal to 1/x
at the point of the curve where x is equal to k? So let's just think about
what they're asking. So if I were to
draw myself-- let me draw some quick
axes right over here. So that's my y-axis. This is my x-axis
right over here. And the graph f of
x is equal to 1/x would look something like this. So it looks something like this. So it kind of spikes up
there, and then it comes down, and then it goes like this. And I'm just doing a
rough approximation of it. So it would look
something like that. And then on the negative side,
it looks something like this. So this is my hand-drawn
version of roughly what this graph looks like. So this right over here
is f of x is equal to 1/x. Now, we are concerning ourselves
with the point x equals k. So let's say-- I mean, it could
be anything that's non-zero, but let's just say that
this is k right over here. So that is the point k 1/k. We can visualize the line
tangent to the curve there. So it might look
something like this. And we need to figure
out its y-intercept. Where does it
intercept the y-axis? So we need to figure out
this point right over here. Well, the best way to do
it, if we can figure out the slope of the
tangent line here, the slope of the tangent
line is just the derivative of the line at that point. If we could figure out the
slope of the tangent line, we already know that
line contains the point. Let me do this in
a different color. We know it contains
the point k comma 1/k. So if we know its slope, we
know what point it contains, we can figure out what
its y-intercept is. So the first step is
just, well, what's the slope of the tangent line? Well, to figure out the
slope of the tangent line, let's take the derivative. So if we write f of x,
instead writing it as 1/x, I'll write it as x to
the negative 1 power. That makes it a little
bit more obvious that we're about to use
the power rule here. So the derivative
of f at any point x is going to be equal
to-- well, it's going to be the exponent
here is negative 1. So negative 1 times
x to the-- now we decrement the exponent
to the negative 2 power. Or I could say it's negative
x to the negative 2. Now, what we care about is
the slope when x equals k. So f prime of k is going
to be equal to negative k to the negative 2 power. Or another way of thinking about
it, this is equal to negative 1 over k squared. So this right over
here is the slope of the tangent
line at that point. Now, let's just think
about what the equation of the tangent line is. And we could think about
it in slope-intercept form. So we know the equation of a
line in slope-intercept form is y is equal to mx plus
b, where m is the slope and b is the y-intercept. So if we can get it in this
form, then we know our answer. We know what the
y-intercept is going to be. It's going to be b. So let's think about
it a little bit. This equation, so
we could say y is equal to our m, our slope
of the tangent line, when x is equal to k, we just
figure out to be this business. It equals this thing
right over here. So let me write that in blue. Negative 1 over k
squared times x plus b. So how do we solve for b? Well, we know what y is
when x is equal to k. And so we can use
that to solve for b. We know that y is equal to
1/k when x is equal to k. So this is going to be
equal to negative 1-- that's not the same color. Negative 1 over k
squared times k plus b. Now, what does this simplify to? See, k over k squared is
the same thing as 1/k, so this is going
to be negative 1/k. So this part, all of this
simplifies to negative 1/k. So how do we solve for b? Well, we could just
add 1/k to both sides and we are left with-- if
you add 1/k here-- actually, let me just do that. Plus 1/k, left-hand side,
you're left with 2/k, and on the right-hand side,
you're just left with b, is equal to b. So we're done. The y-intercept of
the line tangent to the curve when x equals
k is going to be 2/k. If we wanted the equation
of the line, well, we've done all the work. Let's write it out. It'll be satisfying. It's going to be y is equal
to negative 1/k squared x plus our
y-intercept, plus 2/k. And we're done.