Calculus I · Building a Function From Its Derivatives

Where this is heading

Three doors open from here, and you can already see through each.

The second-derivative test (Chapter 8). At a critical point, where $f'(a) = 0$, the quadratic approximation collapses to $f(x) \approx f(a) + \tfrac12 f''(a)(x-a)^2$ — a pure parabola. If $f''(a) > 0$ it opens upward and $a$ is a minimum; if $f''(a) < 0$ it opens downward and $a$ is a maximum. The whole second-derivative test is just reading the leading Taylor term at a critical point.

Convexity (Chapter 8). The error of the tangent-line approximation is $R_1 = \tfrac12 f''(c)(x-a)^2$. When $f'' \ge 0$, that remainder is never negative, so $f$ never dips below its tangent line — it stays above all of them. A function that stays above all its tangents is convex, and convex is exactly the condition under which gradient descent is guaranteed to find the true bottom. The inequality $e^x \ge 1+x$ you proved in Chapter 6 was this story’s first instance.

Newton’s quadratic convergence (Chapter 9). Newton’s method replaces $f$ with its tangent line $T_1$ and solves. The error it leaves behind is governed by the $T_1$ remainder, $\tfrac12 f''(c)(x-a)^2$ — proportional to the square of how far off you were. That square is the reason Newton’s correct digits double each step.

A curve is a line, then a parabola, then a polynomial. Once you can choose how close, the rest of calculus is deciding how close you need.

Problems

Six tiers. Fluency and Practice are doing* — compute Taylor polynomials until the machine is automatic, then apply and estimate. Concept & counterexample and ◆ Diamond are thinking — the convergence subtleties and the proofs (the remainder is the real content). Teach it is teaching. Looking ahead opens Chapters 8 and 9. Do all of Fluency; the Concept tier here guards against the most common Taylor mistakes.*

Fluency — do these fast, until the machine is automatic

1. Find the degree-2 Taylor polynomial at $0$ for: (a) $e^x$; (b) $e^{2x}$; (c) $e^{-x}$; (d) $e^{x/3}$.
2. Write the first three nonzero terms at $0$ for: (a) $\sin x$; (b) $\cos x$; (c) $\sin(2x)$; (d) $\cos\tfrac{x}{2}$.
3. Find the degree-3 Taylor polynomial at $0$ for: (a) $\tfrac{1}{1-x}$; (b) $\tfrac{1}{1+x}$; (c) $\tfrac{1}{1-2x}$. State the coefficient pattern in each.
4. Write the degree-2 polynomial that matches each at $0$ (compute $f(0),f'(0),f''(0)$ first): (a) $\sqrt{1+x}$; (b) $(1+x)^{1/3}$; (c) $\ln(1+x)$; (d) $\ln(1-x)$.
5. By substituting or multiplying known series, write the first three nonzero terms at $0$ for: (a) $\tfrac{1}{1+x^2}$; (b) $e^{x^2}$; (c) $x\,e^{x}$.
6. Compute $f^{(k)}(0)$ for $k=0,1,2,3$ and write the coefficient $c_k$ for: (a) $e^x$; (b) $\cos x$.
7. Estimate with the first few series terms, then compare to a calculator: (a) $e^{0.2}$; (b) $\cos(0.1)$; (c) $\sin(0.3)$; (d) $\sqrt{1.1}$ via $(1+x)^{1/2}$.
8. Taylor at a center other than $0$, so $(x-a)$ powers matter: find the degree-2 Taylor polynomial of (a) $e^x$ at $a=1$; (b) $\sqrt{x}$ at $a=4$.

Practice — apply, estimate, and bound the error

1. Find the general-term series of $e^{-x}$ at $0$ and check it against $e^x$’s with the signs flipped.
2. Derive the geometric series as the Taylor series of $\tfrac{1}{1-x}$ by computing $f^{(k)}(0)$, and state where the equality is valid.
3. Find the first three nonzero terms of $\ln(1+x)$, and explain in one line how the leading term explains the Chapter 1 fact $\tfrac{\ln(1+t)}{t}\to 1$.
4. Estimate $e^{0.1}$ with $T_2$, then bound the error using $|f'''|\le 1.2$ on $[0,0.1]$. Confirm the true value falls inside your bound.
5. Estimate $\cos(0.2)$ with $1 - \tfrac{x^2}{2}$. Then bound the error using the next nonzero term $\tfrac{x^4}{24}$, and explain why — for an even function like $\cos$ — the error jumps by two powers, not one.
6. The arctangent series. Given $\tfrac{d}{dx}\arctan x = \tfrac{1}{1+x^2} = 1 - x^2 + x^4 - \cdots$, find the first three nonzero terms of $\arctan x$ at $0$ (each term is an antiderivative of one above; match $\arctan 0 = 0$). Set $x=1$ to glimpse $\tfrac{\pi}{4} = 1 - \tfrac13 + \tfrac15 - \cdots$.
7. Find the first three terms of the binomial series for $(1+x)^{1/2}$ and $(1+x)^{-1}$, and check the second against the geometric series.
8. Multiply series: find the first three nonzero terms of $e^x\cos x$ by multiplying the two series. (Notice which power drops out.)
9. The Gaussian. By substituting $-x^2$ into the series for $e^u$, find the first three nonzero terms of $e^{-x^2}$ — the bell curve at the heart of probability and machine learning. Is it an even or odd function, and where is it largest?
10. (Notebook) Plot $\sin x$ with its $1$-, $3$-, $5$-, and $7$-term Taylor polynomials on $[-2\pi, 2\pi]$, and describe how many humps each captures before peeling off.

Concept & counterexample — think; true/false with justification, or build the example

1. 1. $T_n$ matches $f$ and its first $n$ derivatives at the center. (b) A larger $n$ always gives a better approximation at every $x$. (c) The Taylor series of a polynomial is itself. (d) If two functions share the same Taylor polynomial of every degree at $a$, they’re equal. (Part (d) is subtle — see problem 31.)
2. Find the error. A student writes: “$\tfrac{1}{1-x} = 1 + x + x^2 + \cdots$, so at $x = 2$, $\;-1 = 1 + 2 + 4 + \cdots$.” Identify the precise mistake.
3. Explain why $\sin x$’s series has only odd powers and $\cos x$’s has only even powers, by connecting each to the function’s symmetry.
4. Construct a function whose degree-1 Taylor approximation at $0$ is exactly $1 + 3x$, then a different one with the same approximation. What do all such functions share, and where do they differ?
5. In one or two sentences, explain why the factorial appears, by reference to what $k$ derivatives do to $(x-a)^k$.
6. Is it possible? A function whose tangent-line error shrinks like $(x-a)^2$, and another whose error shrinks only like $|x-a|$. What property separates them?
7. The series for $\ln(1+x)$ is $x - \tfrac{x^2}{2} + \cdots$. Without computing the radius of convergence formally, predict the value of $x$ at which it must start to fail, and why. (Where does $\ln(1+x)$ itself misbehave?)

◆ Diamond — the remainder, and the deep identities, are the real content

1. ◆ Derive the coefficient formula: for $P(x)=\sum c_k(x-a)^k$, show $P^{(k)}(a)=k!\,c_k$, hence $c_k=\tfrac{f^{(k)}(a)}{k!}$. (The “everything but one term dies” argument.)
2. ◆ Prove the quadratic Taylor polynomial is the unique parabola matching $f$’s value, slope, and curvature at $a$.
3. ◆ Show the $n=0$ case of Taylor’s remainder is exactly the Mean Value Theorem, and the $n=1$ case is the second-order MVT from Chapter 6.
4. ◆ Prove $\cos x$ is an even function and $\sin x$ an odd function directly from their series (every power even, resp. odd).
5. ◆◆ Derive Euler’s formula $e^{ix}=\cos x + i\sin x$ by substituting $x\to ix$ into $e^x$’s series, using $i^2=-1$ to sort the terms into real and imaginary groups. Then set $x=\pi$ to obtain $e^{i\pi}=-1$.
6. ◆◆ Prove Taylor’s theorem with Lagrange remainder by the auxiliary-function method: fix $x$, choose $M$ with $f(x)=T_n(x)+M(x-a)^{n+1}$, define $g(t)=f(x)-\sum_{k=0}^n\tfrac{f^{(k)}(t)}{k!}(x-t)^k-M(x-t)^{n+1}$, show $g(a)=g(x)=0$, apply Rolle, and finish via the telescoping derivative.
7. ◆◆ Prove $\sin x$’s Taylor series converges to $\sin x$ for every $x$, using $|f^{(n+1)}|\le 1$ to bound $|R_n(x)|\le\tfrac{|x|^{n+1}}{(n+1)!}\to 0$. Note why this is cleaner than the $e^x$ case.
8. ◆◆ The famous warning. Let $f(x)=e^{-1/x^2}$ for $x\ne0$, $f(0)=0$; accept that $f^{(k)}(0)=0$ for all $k$. Write its Taylor series at $0$, note it converges everywhere but not to $f$, and explain what this proves about 19(d).

Teach it — explain it, write it up, or fix it

1. In three or four sentences to a friend, teach “the derivative is the best linear approximation,” and how Taylor extends it to the best quadratic, cubic, and beyond.
2. Debug your classmate. They write: “$e^x = 1 + x + x^2 + x^3 + \cdots$, since every derivative of $e^x$ is $e^x$.” Pinpoint the missing ingredient and give the correct series, with a one-line reason for the $\tfrac1{k!}$.
3. Write a complete worked solution estimating $\ln(1.1)$ from a Taylor polynomial of $\ln(1+x)$, with an error bound, narrated so a peer could redo it for $\ln(1.2)$.
4. Design a problem. Choose a function (try a trig one) and a center $a$, and write a problem asking for its degree-2 (or first three nonzero) Taylor terms, with a clean key.
5. In a few sentences, teach Euler’s formula to someone who knows the three series but has never seen them combined — what single substitution does all the work, and why is the result surprising?
6. In a few sentences, teach the difference between a Taylor polynomial (finite, carrying a remainder) and a Taylor series (infinite, equal to the function only where it converges and the remainder vanishes).

Looking ahead — Chapters 8 and 9 in your hands early

1. At a critical point ($f'(a)=0$), the quadratic approximation is $f(x)\approx f(a)+\tfrac12 f''(a)(x-a)^2$. Predict: $f''(a)>0$ — min or max? $f''(a)<0$? (You’ve derived the second-derivative test.)
2. The tangent-line error is $R_1=\tfrac12 f''(c)(x-a)^2$. Predict the sign condition on $f''$ that makes $R_1\ge0$ always — i.e., makes $f$ stay above all its tangents. (That’s convexity, Chapter 8.)
3. Newton’s method (Chapter 9) replaces $f$ by its tangent line and solves. Using the $T_1$ remainder, predict why the error after a Newton step is proportional to the square of the error before.
4. In problem 17 you found the series of the Gaussian $e^{-x^2}$. Its cousin $e^{-x^2/2}$ is the bell curve of probability (Book 4). From the series, predict its symmetry and its single maximum — and notice you now understand the most important curve in statistics as nothing but a reshuffled exponential.

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