C
ongratulations! Today is your day. You’re off to take the AP exam! You’re off and away.
You have the stars on your elbows. And the brains in your head. You will get a 5 Because of the Stewart you’ve read!
But what if we asked you To graph ex?
But you only know graphing For polynomials, at best.
So what do you do here? Are you stuck in a ditch?
No, you can do something. Let a Taylor Polynomial be your fix!
So how does this work? You’ll learn this today. First let us choose 0 As our value for a.
This means that our series Is centered at a. And when a equals 0, Taylor and Maclaurin say...
HOORAY!
So let us go back To lovely ex, Which must be graphed Before the next test!
So now that we’re centred around our value for a, Let’s make a polynomial Out of our function today! y 1
x
So next let us find The value of c.
y
1
So, let c= a= f(0 0 )= 0 e=
1
x
And c would just be f(a), indeed!
We want our polynomial To have the same derivative As ex at a. Oh wow, we’re riveted!
“Why do we do this?”
And f’(x) is The same as the f. f(x)
e
x
ex
f’(x)
Since ex will not die, It’s always what’s left.
So let’s make a rule, For each of the terms.
Rules this way!
Rules in here!
And then we can sum them And graph what we’ve learned.
1
y x
Each term is xn multiplied By the nth derivative of f(a). Put that all over n factorial, Oh you’ll be set for the sums... hip hip hooray!
y ex
x
1 + x + x2/2! +x3/3!
So now we can graph Our function, f. With a 3rd degree Taylor polynomial, We are close-ish to ex.
And then… will you succeed? Yes! You will, indeed! 5 on the AP exam, guaranteed!
Kid, you’ll move mountains!
So… Be your name
Sarah
or Sherry or Sam