Real Analysis Live - Problem Solving ( check problem sheet here: https://tbsom.de/live )

TL;DR

Always determine the domain by checking where denominators (or other undefined expressions) could fail before computing derivatives.

Briefing Cornell Notes

Briefing

The session’s core takeaway is a practical toolkit for real analysis: before differentiating anything, lock down where the function is actually defined; then use the right differentiation rules (quotient, chain, and product) to compute derivatives reliably; and finally, use Taylor’s theorem to turn smooth functions into accurate rational approximations.

The first exercise starts with a function built from a fraction involving polynomials and exponentials. The key move is domain checking: the derivative rules only make sense where the expression is defined, so the denominator can’t vanish. By analyzing the denominator as a sum of a nonnegative term (a square) and a strictly positive exponential term, the denominator is shown never to be zero. That establishes the function’s domain as all real numbers, which removes any hidden “divide-by-zero” obstacle before any calculus begins.

With the domain secured, the derivative is computed using the quotient rule. The calculation is organized by splitting the numerator and denominator into separate differentiable components, then applying chain rule where exponentials and powers appear inside other functions. The result is an algebraically large expression—an explicit reminder that correct differentiation often produces “messy” formulas even when the underlying ideas are straightforward. The discussion then connects differentiability to continuity: on open intervals, differentiability implies continuity, and the quotient rule is not just a formula for derivatives but also a guarantee that differentiability is preserved when the numerator and denominator are differentiable and the denominator never hits zero.

The second major thread uses Taylor’s theorem to approximate an irrational number. The target is √(3/2), and the method is to rewrite it as a value of a function that can be expanded around a rational base point. A square-root function is expressed as (1+x)^{1/2}, expanded at x=0, and evaluated at x=1/2. The session computes a Taylor polynomial (up to third order) to produce a rational approximation, then uses the Taylor remainder term (involving the next derivative evaluated at an intermediate point) to bound the error. By estimating the remainder using worst-case bounds over the interval between 0 and 1/2, the approximation is shown accurate enough that the first few decimal digits are guaranteed “safe.”

After that, symmetry properties of functions are proved: if f is even and differentiable, then f′ is odd; if f is odd and differentiable, then f′ is even. The proofs lean on the chain rule by introducing a simple auxiliary map σ(x)=−x and rewriting even/odd definitions as compositions with σ. This turns symmetry into a calculus statement about how derivatives transform under reflection.

The final exercise uses mean value theorem logic to characterize functions from differential equations. If a differentiable function has derivative zero everywhere on a connected interval, it must be constant. Then the functional equation f′(x)=f(x) is handled by multiplying f by e^{−x} to create a new function whose derivative is zero; constancy forces f(x)=C·e^{x}. The argument works on connected intervals, with an explicit warning that disconnected domains can break the “constant everywhere” conclusion.

Cornell Notes

The session builds a workflow for real analysis problems: (1) verify the domain so denominators never vanish, (2) compute derivatives using the correct rule (quotient, chain, and product), and (3) use Taylor’s theorem to approximate values with guaranteed error bounds. A Taylor polynomial for √(3/2) is constructed by expanding (1+x)^{1/2} at x=0 and evaluating at x=1/2, then the remainder term is estimated to show the approximation is accurate enough to lock in specific decimal digits. Symmetry is proved via the chain rule: differentiating an even differentiable function yields an odd derivative, and differentiating an odd differentiable function yields an even derivative. Finally, mean value theorem reasoning shows that f′=0 implies f is constant, and f′(x)=f(x) forces f(x)=C·e^x on connected intervals.

Why is domain checking essential before differentiating the first quotient-type function?

Differentiation rules apply only where the function is defined. For a fraction, the denominator must never be zero. In the exercise, the denominator is x^2 + e^{x^2}. Since x^2 ≥ 0 and e^{x^2} > 0 for all real x, their sum is strictly positive, so the denominator can’t vanish. That lets the function be treated as a map from R to R without hidden singularities.

How does the quotient rule connect differentiability of the whole fraction to differentiability of numerator and denominator?

The quotient rule isn’t only an algebraic derivative formula; it also relies on a differentiability premise: if U and V are differentiable and V(x) ≠ 0, then U/V is differentiable and its derivative is given by (U′V − UV′)/V^2. The session repeatedly uses this logic: once U and V are built from differentiable pieces (polynomials, exponentials, sums), the quotient inherits differentiability wherever the denominator stays nonzero.

What is the Taylor-polynomial strategy used to approximate √(3/2) with a rational number?

Rewrite the target as a function value that can be expanded around a rational point. The session uses √(3/2) = √(1 + 1/2) = (1 + x)^{1/2} at x = 1/2. It computes derivatives of f(x)=(1+x)^{1/2} at x=0, forms a Taylor polynomial T3 (up to the cubic term), and evaluates T3(1/2) to get a rational approximation.

How does the remainder term guarantee accuracy (safe digits) rather than just giving a number?

Taylor’s theorem bounds the error between the true value f(h) and the polynomial T3(h) using the next derivative (here the 4th derivative) at some intermediate point c between 0 and h. The session estimates the remainder term by taking a worst-case bound over c∈(0,1/2), ensuring the absolute error is below a threshold like 0.0003. That implies the first few decimal digits cannot change under the true value.

Why does differentiating an even function produce an odd derivative (and vice versa)?

Even means f(x)=f(−x), odd means f(x)=−f(−x). Using σ(x)=−x, the session rewrites these as compositions with σ and applies the chain rule. For even f, differentiating f(x)=f(σ(x)) gives f′(x)=f′(−x)·σ′(x)=f′(−x)·(−1), so f′(−x)=−f′(x), which is the definition of oddness. A similar chain-rule computation for odd f yields that f′ is even.

How does f′(x)=f(x) force f(x)=C·e^x?

Create a new function whose derivative is zero. Define g(x)=f(x)e^{−x}. By the product rule, g′(x)=f′(x)e^{−x} + f(x)(−e^{−x}) = (f′(x)−f(x))e^{−x}. If f′(x)=f(x), then g′(x)=0 everywhere on the connected interval. Mean value theorem reasoning implies g is constant, so f(x)=C·e^x.

Review Questions

Given a quotient U(x)/V(x), what two conditions must hold at a point x0 for the quotient rule to apply there?
In the Taylor approximation of √(3/2), what role does the intermediate point c in the remainder term play, and how is it handled in the error estimate?
If f is differentiable on a connected interval and f′(x)=0 for all x, what theorem ensures f is constant, and why does connectedness matter?

Key Points

1
Always determine the domain by checking where denominators (or other undefined expressions) could fail before computing derivatives.
2
For quotient functions, differentiability of the whole fraction follows from differentiability of numerator and denominator together with the condition that the denominator never equals zero.
3
Chain rule and quotient rule often produce large expressions; keeping U and V separate helps prevent mistakes.
4
Taylor polynomials can approximate irrational values by rewriting the target as f(h) for a function f expanded around a rational base point.
5
Taylor’s theorem plus a remainder bound turns approximation into a guaranteed error statement, which can lock in specific decimal digits.
6
Even/odd symmetry of differentiable functions transfers predictably to derivatives: even→odd and odd→even, provable cleanly via the chain rule with σ(x)=−x.
7
A differentiable function with derivative zero everywhere on a connected interval must be constant; this supports solving f′(x)=f(x) by reducing it to a constant via multiplication by e^{−x}.

Highlights

The denominator x^2 + e^{x^2} is strictly positive for all real x, so the first quotient function is defined on the entire real line—no hidden singularities.

The Taylor approximation of √(3/2) is not just computed; it’s certified using the remainder term, ensuring the first decimal digits are provably correct.

Symmetry proofs for even/odd functions use the chain rule by composing with σ(x)=−x, turning reflection properties into derivative identities.

From f′(x)=f(x), defining g(x)=f(x)e^{−x} forces g′(x)=0, and mean value theorem reasoning yields f(x)=C·e^x on connected intervals.

Topics

Domain Checking
Quotient Rule
Taylor Polynomial
Even/Odd Functions
Mean Value Theorem
Differential Equation

Mentioned

Fabian