Layered finals hub

• Intermediate Value Theorem (IVT): if $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$, then some $c\in(a,b)$ has $f(c)=N$. Use to prove roots exist.
• Extreme Value Theorem (EVT): if $f$ is continuous on closed $[a,b]$, then $f$ attains both an absolute max and min on that interval.
• Mean Value Theorem (MVT): if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c$ has $f'(c)=\dfrac{f(b)-f(a)}{b-a}$. Use for monotonicity/guarantees; special case: Rolle’s Theorem ($f(a)=f(b)$ gives $f'(c)=0$).

• Continuity at $x=a$: $\lim_{x\to a} f(x)=f(a)$; need both one-sided limits to match and equal the value.
• Derivative definition: $f'(a)=\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$; represents slope/instant rate.
• L'Hospital's Rule: if $\lim_{x\to a} f(x)=\lim_{x\to a} g(x)=0$ or $\pm\infty$ and $f,g$ differentiable near $a$ with $g'\neq0$, then $\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}$ (if latter limit exists or is $\pm\infty$).

If $G(x)=\int_a^x g(t)\,dt$ then $G'(x)=g(x)$ (chain rule when bound is $x^2$: $g(x^2)\cdot2x$).

Part 2: $\int_a^b f(x)dx = F(b)-F(a)$ for any antiderivative $F$. Signed area stays signed; change variables -> change bounds.

Average value: $f_{avg}=\tfrac{1}{b-a}\int_a^b f(x)dx$. Net change = integral of rate.

• Try substitution first; then parts; then algebra (trig IDs, completing square).
• Piecewise/absolute value: split intervals; keep sign in mind.
• Average value + total change problems pair FTC with context units.
• Series/alternating tests likely: be ready for ratio/root tests, absolute vs conditional convergence (if covered in your class).

Improper = infinite bounds or discontinuity. Replace bound with limit.

• Example: $\int_1^{\infty}x^{-2}dx = \lim_{t\to\infty}\int_1^t x^{-2}dx = 1$.
• If either side diverges (split at the discontinuity), the integral diverges.
• $p$-test: $\int_1^{\infty} x^{-p}dx$ converges if $p>1$, diverges if $p\le 1$. For $\int_0^1 x^{-p}dx$, converges if $p<1$, diverges if $p\ge1$.

• Compare to known convergent/divergent forms (e.g., $1/x^p$, $1/x$).
• At vertical asymptotes, split: $\int_a^b = \lim_{t\to c^-}\int_a^t + \lim_{t\to c^+}\int_t^b$; if either blows up, whole thing diverges.
• Log divergence benchmark: $\int_1^{\infty} 1/x\,dx$ diverges slowly.

Partition $[a,b]$: $\Delta x=\tfrac{b-a}{n}$, $x_i=a+i\Delta x$.

• Left: $L_n=\sum_{i=0}^{n-1} f(x_i)\Delta x$, Right: $R_n=\sum_{i=1}^n f(x_i)\Delta x$.
• Midpoint: $M_n=\sum f(\tfrac{x_i+x_{i+1}}{2})\Delta x$.
• Trapezoid: $T_n=\frac{\Delta x}{2}\Big(f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\Big)=\tfrac12(L_n+R_n).$

• If increasing: left under, right over. If decreasing: flip.
• Concave up: midpoint under, trapezoid over. Concave down: midpoint over, trapezoid under.
• Error scale: midpoint/trap ~ $1/n^2$ (smooth $f$); left/right ~ $1/n$. Increase $n$ or switch to mid/trap when bias is big.

Exponential: $y'=ky \Rightarrow y=y_0e^{kt}$. Logistic: $p' = r p (k-p)$, solution $p(t)=\tfrac{k}{1+Ae^{-rkt}}$, $A=\tfrac{k-p_0}{p_0}$.

• Inflection at $p=\tfrac{k}{2}$ (max growth).
• Long-run $p\to k$; if $p_0>k$ it decays down.
• Separation trick: rewrite $\tfrac{1}{p} + \tfrac{1}{k-p}$ to integrate quickly.

Iterative linearization for $y'=f(x,y)$ with $(x_0,y_0)$ and step $\Delta x$: [ x_{n+1}=x_n+\Delta x,\quad y_{n+1}=y_n+\Delta x\cdot f(x_n,y_n) ] Smaller $\Delta x$ -> better accuracy; mirror a slope field path. Use tables to avoid arithmetic slips.

1. Divide if numerator degree >= denominator.
2. Factor denominator (linear/irreducible quadratics).
3. Set coefficients: $\frac{A}{x+1}+\frac{B}{x-1}$ for distinct linear terms; solve linear system.
4. Integrate: linear factors -> logs; irreducible quadratics -> arctan/complete square. Example: $\frac{5x+1}{x^2-1}=\frac{2}{x+1}+\frac{3}{x-1} \Rightarrow 2\ln|x+1|+3\ln|x-1|+C$.

• Substitution: look for inner function + derivative factor.
• Parts: products where one simplifies on differentiation (logs, arctan) and one integrates easily (poly, exp).
• Trig: identities ($\sin^2+\cos^2=1$), half-angle to reduce powers.
• Rational: long division first if needed, then PFD; arctan for $a^2+x^2$ forms.

• Primary equation = quantity to max/min. Use constraints to reduce variables.
• Find critical points: derivative zero/undefined within domain; include endpoints.
• Justify: first derivative sign change, second derivative test, or candidates test on closed intervals.
• State answer with units/context; verify feasibility (dimensions positive, within constraints).

Related rates vs optimization: both differentiate, but optimization hunts extrema; related rates connects variable speeds.

Use linearization for quick approximations near a point; Euler repeats that idea to approximate DE solutions.