1. Table of Derivatives: Calculate $f, f', f'', f'''$ first.
2. Evaluate at Center ($a$): Find the "Heart" of each derivative at the center.
3. Apply the Factorial: Divide term $n$ by $n!$.
4. The Shift: Remember to write each power as $(x-a)^n$.
Construct the 3rd-degree Maclaurin Polynomial ($T_3$) for each function.
$f(x) = e^{2x}$
$f(x) = \sin(x)$
$f(x) = \frac{1}{1-x}$
Construct the 2nd-degree Taylor Polynomial centered at the given point.
$f(x) = \sqrt{x}$ centered at $a = 1$.
$f(x) = \ln(x)$ centered at $a = 1$.
Compare the Taylor series for $e^x$ and $\cos x$. Why does $\cos x$ only have even powers ($x^2, x^4...$)? What happened to the odd powers like $x$ and $x^3$? (Hint: What is the value of $\sin(0)$?)
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Use your $T_3$ for $e^{2x}$ from Part I to estimate the value of $e^{0.2}$.
1. Set $x = 0.1$ (since $2 \cdot 0.1 = 0.2$).
2. Calculate the sum.
Take the Maclaurin series for $\sin x$: $x - x^3/6 + x^5/120 - ...$
Task: Differentiate every term in the series.
Does your new series look like the series for $\cos x$?
Does this prove that $\frac{d}{dx} [\sin x] = \cos x$ in the world of polynomials?
Objective: Explain Taylor Polynomials to a younger student using a hidden drawing.
The Activity:
1. Draw a large curve on a hidden paper.
2. Show them only a tiny 1-inch section of the line.
3. Ask: "Based on this tiny piece, can you guess where the line is going next?"
The Lesson: "Calculus helps us make the 'Smartest Guess' possible by looking at the first step very closely."
Response: ___________________________________________________________