Important Points on Applications of Derivatives

Identifying Increasing and Decreasing Functions

Derivatives are employed to determine if a given function is increasing, decreasing, or constant, especially in graphical representations. If a function f is continuous within an interval [p, q] and differentiable in the open interval (p, q), then:

f is increasing in [p, q] if f'(x) > 0 for all x ∈ (p, q)

f is decreasing in [p, q] if f'(x) < 0 for all x ∈ (p, q)

f is a constant function in [p, q] if f'(x) = 0 for all x ∈ (p, q)

Tangent and Normal Lines

Tangent lines touch a curve at a single point without crossing it, while normal lines are perpendicular to tangents.

For a point P(x1, y1) on the curve, the equation of the tangent is given as:

y – y1 = f'(x1)(x – x1)

The equation of the normal is:

y – y1 = [-1 / f'(x1)](x – x1)

Maxima and Minima

Derivatives are pivotal in determining the highest and lowest points on a curve, also known as maxima and minima. These points are vital in understanding the turning points of a function.

Monotonicity

Functions can be classified as monotonic if they are either continuously increasing or decreasing across their entire domain. Functions that exhibit both increasing and decreasing behavior are considered non-monotonic.

Approximation or Finding Approximate Values

Derivatives are instrumental in estimating very small variations or changes in a quantity. This involves using the delta symbol (△) to represent approximate values.

Points of Inflection

For continuous functions, points of inflection occur where the second derivative changes sign while the first derivative exists. These points signify changes in the curvature of the curve.

In this article, we are going to see solved questions and also practice questions for a better understanding of the concept of the applications of derivatives.