Application of Derivatives

If a quantity y varies with another quantity x, i.e., y = f(x), then dy/dx represents the rate of change of y with respect to x. Similarly, if a quantity y varies with time t, then dy/dt represents the rate of change of y with respect to time t. These problems often involve relating the rates of change of two or more related quantities using the chain rule.

The derivative dy/dx at a point (x₀, y₀) on a curve y = f(x) gives the slope of the tangent to the curve at that point. The normal to the curve at that point is a line perpendicular to the tangent. \n\nIf the slope of the tangent is m_T, then the slope of the normal is m_N = -1/m_T (provided m_T ≠ 0). \n\nSpecial cases: \n1. If m_T = 0, the tangent is horizontal (equation y = y₀), and the normal is vertical (equation x = x₀). \n2. If m_T is undefined (i.e., dx/dy = 0), the tangent is vertical (equation x = x₀), and the normal is horizontal (equation y = y₀).

A function f(x) is said to be increasing on an interval (a, b) if for any x₁, x₂ ∈ (a, b), x₁ < x₂ implies f(x₁) ≤ f(x₂). It is strictly increasing if f(x₁) < f(x₂). \n\nA function f(x) is said to be decreasing on an interval (a, b) if for any x₁, x₂ ∈ (a, b), x₁ < x₂ implies f(x₁) ≥ f(x₂). It is strictly decreasing if f(x₁) > f(x₂). \n\nTest for increasing/decreasing functions: \n1. If f'(x) > 0 for all x in (a, b), then f(x) is strictly increasing on (a, b). \n2. If f'(x) < 0 for all x in (a, b), then f(x) is strictly decreasing on (a, b). \n3. If f'(x) = 0 for all x in (a, b), then f(x) is a constant function on (a, b).

A function f(x) has a local maximum at a point c if f(c) is the greatest value of f in some open interval containing c. Similarly, f(x) has a local minimum at c if f(c) is the least value of f in some open interval containing c. Points where f'(x) = 0 or f'(x) is undefined are called critical points.\n\nFirst Derivative Test: \n1. If f'(x) changes sign from positive to negative as x increases through c, then c is a point of local maximum. \n2. If f'(x) changes sign from negative to positive as x increases through c, then c is a point of local minimum. \n3. If f'(x) does not change sign as x increases through c, then c is neither a local maximum nor a local minimum (it's a point of inflection).\n\nSecond Derivative Test: \n1. Find f'(x) and f''(x). \n2. Find critical points by setting f'(x) = 0. Let c be a critical point. \n3. If f''(c) < 0, then c is a point of local maximum. \n4. If f''(c) > 0, then c is a point of local minimum. \n5. If f''(c) = 0, the test fails. Use the first derivative test.

For a continuous function f(x) on a closed interval [a, b], the absolute maximum (or global maximum) is the largest value of f(x) on that interval, and the absolute minimum (or global minimum) is the smallest value. \n\nSteps to find absolute maxima/minima: \n1. Find all critical points of f(x) in the open interval (a, b). \n2. Evaluate f(x) at these critical points and at the endpoints a and b. \n3. The largest value among these is the absolute maximum, and the smallest value is the absolute minimum.