Equation of a tangent line Finding the equation of the line tangent to f(x)=xe^x when x=1
Equation of a tangent line
- I've told you multiple times that the derivative of a curve
- at a point is the slope of the tangent line, but our
- friend [? Akosh ?]
- sent me a problem where it actually wants you to find the
- equation of the tangent line.
- And I realize, I've never actually done that.
- So it's worthwhile.
- So let's do that.
- So it says, find the equation of the tangent line to the
- function f of x is equal to x e to the x at x is equal to 1.
- So let's just get an intuition of what we're even looking for.
- So this function is going to look something like, I actually
- graphed it, because it's not a trivial function to graph.
- So this is x e to the x, this is what it looks like.
- I'm just using a graphing calculator, and you can
- see, I just typed it in.
- And what this is asking us, is ok.
- At the point, x is equal to 1.
- So this is the point x is equal to one.
- So f of x is going to be someplace up here, and
- actually, f of x is going to be equal to e, right?
- Because f of 1 is equal to what?
- 1 times e to the 1.
- So it equals e.
- So we're saying at the point, at the point 1 comma e, so at
- the point 1 comma 2.71, whatever, whatever.
- So that's what point?
- That's this point.
- So it's right here.
- 2 point, this is e right here, the point 1 comma e.
- So we want to do is figure out the equation of the
- line tangent to this point.
- So what we're going to do, is we're going to solve it by
- figuring out its slope, which is just the derivative
- at that point.
- So we have to figure out the derivative at
- exactly this point.
- And then we use what we learned from algebra 1 to figure out
- its equation, and we'll graph it here, just to confirm that
- we actually figured out the equation of the tangent line.
- So the first thing we want to know is the slope of the
- tangent line, and that's just the derivative at this point.
- When x is equal to 1, or at the point 1 comma e.
- So what's the derivative of this?
- So f prime of x.
- f prime of x is equal to, well, this looks like a
- job for the product rule.
- Because we know how to figure out the derivative of x, we
- know how to figure out the derivative of e to the x, and
- they're just multiplying by each other.
- So the product rules help us.
- The derivative of this thing is going to be equal to the
- derivative of the first expression of the
- first function.
- So the derivative of x is just 1, times the second function,
- times e to the x, plus the first function, x, times the
- derivative of the second function.
- So what's the derivative of e to the x?
- And that's what I find so amazing about the number e, or
- the function e to the x, is that the derivative of e
- to the x is e to the x.
- The slope at any point of this curve is equal to the
- value of the function.
- So this is the derivative.
- So what is the derivative of this function at the point x
- is equal to 1, or at the point 1 comma e?
- So we just evaluate it.
- We say f prime of 1 is equal to 1 time e to the 1 plus 1 times
- e to the 1, well, that's just equal e plus e.
- And that's just equal to 2 e.
- And you know, we could figure out what that number, e is just
- a constant number, but we write e because it's easier to write
- e than 2.7 et cetera, and an infinite number of digits,
- so we just write 2e.
- So this is the slope of the equation, or this is the slope
- of the curve when x is equal to one, or at the point
- 1e, or 1 f of 1.
- So what is the equation of the tangent line?
- So let's go ahead and take this form, the equation's going to
- be y is equal to, I'm just writing it in the, you know,
- not the point slope, the mx plus b form that you
- learned in algebra.
- So the slope is going to be 2e.
- We just learned that here.
- That's the derivative when x is equal to 1.
- So 2e times x plus the y-intercept.
- So if we can figure out the y-intercept of this
- line, we are done.
- We have figured out the equation of the tangent line.
- So how do we do that?
- Well, if we knew a y or an x where this equation
- goes through, we could then solve for b.
- And we know a y and x that satisfies this equation.
- The point 1 comma e.
- The point where we're trying to find the tangent line, right?
- So this point, 1 comma e, this is where we want to
- find the tangent line.
- And by definition, the tangent line is going to
- go through that point.
- So let's substitute those points back in here, or this
- point back into this equation, and then solve for b.
- So y is equal to e, is equal to 2 e, that's just the slope at
- that point, times x, times 1, plus b.
- It might confuse you, because e, you'll say, oh, e,
- is that a variable?
- No, it's a number, remember, it's like pi.
- It's a number.
- You can substitute 2.7 whatever there, but we're not doing
- that, because this is cleaner.
- And let's solve.
- So you get e is equal to 2e plus b.
- Let's subtract 2e from both sides.
- You get b is equal to e minus 2e.
- b is equal to minus e.
- Now we're done.
- What's the equation of the tangent line?
- It is y is equal to 2 times e x plus b.
- But b is minus e, so it's minus e.
- So this is the equation of the tangent line.
- If you don't like these e's there, you could replace that
- with the number 2.7 et cetera, and this would become 5 point
- something, and this would just be minus 2.7 something.
- But this looks neater.
- And let's confirm.
- Let's use this little graphing calculator to confirm that that
- really is the equation of the tangent line.
- So let me type it in here.
- So it's 2, 2 times e times x, right, that's 2ex minus e.
- And let us graph this line.
- There we go.
- It graphed it.
- And notice that that line, that green line, I don't know if you
- can, maybe I need to make this bigger for it to
- show up, bolder.
- I don't know if that helps.
- But if you look here, so this red, this is our original
- equation, x e to the x, that's this curve.
- We want to know equation of the tangent line
- at x is equal to 1.
- So it's the point x is equal to 1.
- And when x is equal to 1, f of x is e, right, you can just
- substitute back into the original equation to get that.
- So this is the point, 1 comma e.
- So the equation of the tangent line, its slope is going to be
- the derivative at this point.
- So we solved the derivative of this function, and evaluated
- it at x is equal to 1.
- That's what we did here.
- We figured out the derivative, evaluated x equals 1.
- And so we said, OK, the slope.
- The slope at when x is equal to 1 and y is equal to e, the
- slope at that point is equal to 2e.
- And we figured that out from the derivative.
- And then we just used our algebra 1 skills to figure out
- the equation of that line.
- And how did we do that?
- We knew the slope, because that's just the derivative
- at that point.
- And then we just have to solve for the y-intercept.
- And the way we did that is we said, well, the point 1 comma e
- is on this green line as well.
- So we substituted that in, and solve for our y-intercept,
- which we got as minus e, and notice that this line
- intersects the y-axis at minus e, that's about minus
- 2.7 something.
- And there we have it.
- We have shown that, and visually, it shows that
- this is the tangent line.
- Anyway, hope you found that vaguely useful.
- If you did, you should thank [? Akosh ?]
- for being unusually persistent, and having me do this problem.
- See you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
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This is great, I finally understand quadratic functions!
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