06Class 12 Maths

Application of Derivatives formulas

Rates of change, tangents, maxima and minima. Word problems here always come down to writing one function and differentiating it.

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All Maths formulas

Rates and tangents

Rate of change: $\displaystyle \frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$
Slope of the tangent: $\displaystyle m = \left.\frac{dy}{dx}\right|_{(x_0, y_0)}$
Tangent line: $\displaystyle y - y_0 = m(x - x_0)$
Normal line: $\displaystyle y - y_0 = -\frac{1}{m}(x - x_0)$

Increasing and decreasing

Increasing: $\displaystyle f'(x) > 0$
Decreasing: $\displaystyle f'(x) < 0$

Maxima and minima

Critical point: $\displaystyle f'(c) = 0 \ \text{or} \ f'(c) \ \text{does not exist}$
Local maximum: $\displaystyle f'(c) = 0, \ f''(c) < 0$
Local minimum: $\displaystyle f'(c) = 0, \ f''(c) > 0$
If f''(c) = 0, use the first derivative test instead.

Where students slip up

Forgetting endpoints when finding an absolute maximum on a closed interval.
Assuming f'(c) = 0 always means a maximum or minimum. x³ at 0 is neither.
Using the tangent's slope for the normal without taking the negative reciprocal.

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