yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

2015 AP Calculus BC 5a | AP Calculus BC solved exams | AP Calculus BC | Khan Academy


3m read
·Nov 11, 2024

Consider the function ( f(x) ) is equal to ( \frac{1}{x^2} - Kx ) where ( K ) is a nonzero constant. The derivative of ( f ) is given by, and they give us this expression right over here. It's nice that they took the derivative for us.

Now part A, let ( K ) equal 3 so that ( f(x) ) is equal to ( \frac{1}{x^2} - 3x ). So they said ( K ) equal to three. Write an equation for the line tangent to the graph of ( F ) at the point whose x-coordinate is four.

To find an equation for a line, the equation of a line is going to be of the form ( y = mx + b ) where ( m ) is the slope of the line and ( b ) is the y-intercept. The slope of the line right over here, this needs to be equal to the derivative evaluated when ( x ) is equal to 4.

So we could say ( y = ) or let me write it this way, we could say that ( m ) is going to be equal to ( F' ) when ( x ) is equal to 4. So ( F' ) of 4 which is equal to, well we know that ( K ) is equal to three. They gave us ( F' ) of ( x ), so it's going to be ( 3 - \frac{2 \cdot 4}{4^2 - 3 \cdot 4} ) squared.

Now, this is an eight right over here. All I did is ( F' ) of ( x ) when ( K ) is equal to 3 is going to be ( 3 - \frac{2x}{x^2 - 3x} ), and all of that squared. I want to evaluate what ( F' ) of four is. So every place where I saw an ( x ), I substitute it with a four. Where I saw the ( k ), ( k ) is three, and so this is going to be equal to the numerator ( 3 - 8 ) is (-5) over, this is ( 16 - 12 ) which is going to be ( 4 ).

So ( 16 - 12 ) is ( 4 ), and then we square it, so it's going to be ( \frac{-5}{4} ) squared. And so let me write this way: ( m = \frac{-5}{16} ).

So how do we figure out ( b )? Now, what are the coordinates when ( x ) is equal to 4? What is ( y ) going to be equal to? Well, ( Y = f(x) ), so we know that ( y ) on the curve, we know that ( Y ) is going to be equal to ( f(4) ), so before we evaluated ( f' ) of four, now we're going to evaluate ( y ) as being ( f(4) ), which is equal to ( \frac{1}{4^2} - 3 \cdot 4 ).

That is equal to ( \frac{1}{16 - 12} ) which is ( \frac{1}{4} ). So this point right here when ( x ) is 4, then ( y ) is equal to ( \frac{1}{4} ).

So we can use that information to solve for ( b ) when ( y ) is ( \frac{1}{4} ). So we're going to say ( y = m \frac{-5}{16} x + b ). Well, when ( y = \frac{1}{4} ) and ( x = 4 ), then plus ( b ).

So I can now solve for ( b ). All I did is I used ( F' ) of ( x ) to figure out ( m ) when ( x ) is equal to 4. Then I said, okay, well what is the value of ( y ) when ( x ) is equal to 4? So if I know ( y ), ( m ), and ( x ), then I can solve for ( b ).

So let's just do that: ( \frac{1}{4} = 4 \cdot \frac{-5}{16} + b ). I can add ( \frac{5}{4} ) to both sides, and I get ( \frac{5}{4} + \frac{1}{4} = b ) or ( b = \frac{6}{4} ) which you could say, well there's a bunch of ways you could write this.

We could just say this is equal to ( 1.5 ). So our equation is ( y = \frac{-5}{16} x + 1.5 ) or if we wanted to write everything as a fraction, we could say ( y = \frac{-5}{16} x + \frac{3}{2} ).

And there you go.

More Articles

View All
Sergey Brin | All-In Summit 2024
They wondered if there was a better way to find information on the web. On September 15th, 1997, they registered Google as a website. One of the greatest entrepreneurs of our times, someone who really wanted to think outside the box, if that sounds like i…
Should You Go To University?
I would just say that if you are self-aware enough to realize that you’re sort of middle of the road and you’re not that good, then sure, go to university, get your stamp, try not to be brainwashed, and use it to at least get your first job. But if you’r…
COVID-19, Humans, and Wildlife: What Do We Know? | National Geographic
Hi YouTube, my name is Natasha Daley and I am a staff writer at National Geographic. We have a fantastic panel for you today on the intersection of COVID-19, humans, and wildlife. I’m gonna be joined by three wonderful Nat Geo explorers to talk all about …
A Mysterious Sinking | Lawless Oceans
[music playing] KARSTEN (VOICEOVER): I’ve asked my friend Lugs to help me take a look at the Ping Shin 101’s last journey. KARSTEN: Let’s just go through this together because there are a couple of things I need some verification on. Ready? KARSTEN (VO…
Complex exponentials spin
In the last video, we did a quick review of the exponential and what it means. Then we looked and figured out what the magnitude of an exponential is. The magnitude is equal to one. Now we’re going to look closely at this complex exponential as it represe…
Introduction to price elasticity of supply | APⓇ Microeconomics | Khan Academy
We’ve done many videos on the price elasticity of demand. Now we’re going to focus on the price elasticity of supply, and it’s a very similar idea; it’s just being applied to supply. Now, it’s a measure of how sensitive our quantity supplied is to percen…