Derivative Applications Calculator

None Answer only.

Summary Methods used as pills.

Detailed Full accordion with before→after, theorem, and explanation per step.

c — Critical Points

f'(c)=0, classify min/max via f''

IP — Inflection Points

f''(c)=0 with sign change

L(x) — Linear Approx

Tangent line at a; optional eval at x

x* — Newton's Method

Root finding: xₙ₊₁ = xₙ − f/f'

opt — Optimization

EVT: endpoints + critical points

MVT — Mean Value Thm

Find c: f'(c) = average slope

∫∫ — Curve Sketching

f', f'', CP, IP, mono, concavity, EVT

dy/dx — Implicit Diff

dy/dx = −∂H/∂x / ∂H/∂y

Critical pt \(f'(c)=0\text{ or undefined}\)

2nd Deriv Test \(f''(c)>0\Rightarrow\text{min},\;f''(c)<0\Rightarrow\text{max}\)

Inflection \(f''(c)=0,\;f'''(c)\neq 0\)

Linear Approx \(L(x)=f(a)+f'(a)(x-a)\)

Newton's \(x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\)

MVT \(\exists c:\,f'(c)=\frac{f(b)-f(a)}{b-a}\)

EVT \(f\text{ cont. on }[a,b]\Rightarrow\exists\min,\max\)

Rolle's Thm \(f(a)=f(b)\Rightarrow\exists c:\,f'(c)=0\)

Implicit \(\frac{dy}{dx}=-\frac{F_x}{F_y}\)

Concave up \(f''(x)>0\)

Concave down \(f''(x)<0\)

Critical/Inflection/MVT/Sketch/Optimize: set [a,b] search interval
Linear Approx: set center a; optionally enter x to compare L(x)≈f(x)
Newton: enter an initial guess x₀ near the root
Implicit: enter F(x,y) and G(x,y) (e.g. x²+y² and 25 for the unit circle); optionally enter point (x,y) to evaluate dy/dx
All roots found by bisection on 400 sub-intervals

f'Derivative Applications