Application of Derivatives — NDA Mathematics

Formulas (2)

• Tangent and normal to a curve · Tangent & normal at a point
  $$m_{\text{tangent}}=\left.\frac{dy}{dx}\right|_{(x_1,y_1)} \qquad m_{\text{normal}}=-\frac{1}{dy/dx} \qquad y-y_1=m(x-x_1)$$
• Rates of change and small-change approximation · Related rates & small-change approximation
  $$\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt} \qquad \Delta y\approx f'(x)\,\Delta x \qquad dy=f'(x)\,dx$$

Watch out for (3)

• The normal slope is the NEGATIVE reciprocal
  → Tangent and normal to a curve
• "Total change" is NOT the differential estimate
  → Rates of change and small-change approximation
• Related rates need the CHAIN RULE
  → Rates of change and small-change approximation

Watch out for (5)

• Monotonicity is decided by the sign of $f'$, not $f$
  → Increasing and decreasing intervals
• $f''>0$ is a MINIMUM, not a maximum
  → Critical points and the derivative tests
• $f'=0$ is NECESSARY, not sufficient, for an extremum
  → Critical points and the derivative tests
• On a closed interval, ALWAYS test the endpoints
  → Greatest and least value on an interval
• Count genuine SIGN-CHANGES of $f'$, not just roots of $f'$
  → Conditions for no extremum / counting extrema

Formulas (2)

• Trigonometric maxima without a derivative · Range of a sin x + b cos x
  $$-\sqrt{a^2+b^2}\ \le\ a\sin x+b\cos x\ \le\ \sqrt{a^2+b^2}, \qquad \sin x\cos x=\tfrac12\sin 2x\in\left[-\tfrac12,\tfrac12\right]$$
• AM-GM: the calculus-free shortcut · AM-GM
  $$\frac{u+v}{2}\ge\sqrt{uv}\quad(u,v>0),\ \text{equality at } u=v$$

Watch out for (2)

• Differentiating $a\sin x+b\cos x$ works but wastes a minute
  → Trigonometric maxima without a derivative
• Collapse BEFORE asking about intervals or periods
  → Trigonometric maxima without a derivative