Ch 7Applications of Differential Calculus — Class 12 Mathematics, concept-first.

Differential calculus was originally called infinitesimal calculus, because its core idea is to break something into infinitesimally small parts to see how it changes.

The main aim of differential calculus is to break a quantity into infinitesimally small parts to study how it changes -- which is why it was originally called infinitesimal calculus.

Before applying derivatives to any problem, it helps to fix the two readings of that recur throughout the chapter: the derivative as the slope of a curve, and the derivative as a rate of change of one…

Slope of a line. For a non-vertical line , take any horizontal segment starting on and the vertical segment from its end back to ; the ratio (vertical length)/(horizontal length) is always the same co…

The derivative also measures the rate of change of one variable with respect to another — population growth rates, production rates, water-flow rates, velocity and acceleration are everyday instances.

A related-rates problem involves at least two quantities that are changing with time and are linked to each other through some equation; knowing the rate at which some of the quantities change lets yo…

Definition 7.1 (Tangent). The tangent line to a plane curve at a given point is the straight line that just touches the curve at that point.

Definition 7.3. The angle between two curves, if they intersect, is defined as the acute angle between the tangent lines to the two curves at the point of intersection.

The Mean Value Theorem establishes the existence of a point, strictly between two given points, at which the tangent to the curve is parallel to the secant joining those two points.

Theorem 7.2 (Rolle's Theorem). Let be continuous on the closed interval and differentiable on the open interval . If , then there is at least one point where .

Theorem 7.3 (Lagrange's Mean Value Theorem). Let be continuous on the closed interval and differentiable on the open interval (where are not necessarily equal).

This section applies the Mean Value Theorem to genuine physical and inequality-proving problems — the pattern exercised fully in Exercise 7.3.

Taylor's series and Maclaurin's series expand a function that is infinitely differentiable as an infinite power series.

This section discusses various indeterminate forms that arise when computing , and the systematic method — l'Hôpital's Rule — for evaluating such limits.

While computing for certain functions , direct substitution can produce one of the following forms: These are said to have "the form of a number," but no value can be assigned to them consistently wit…

l'Hôpital's Rule. Suppose and are differentiable functions with .

Direct and applications. For a ratio that is already or at the point in question, differentiate numerator and denominator separately (not as a quotient — do not use the quotient rule) and take the lim…

The forms all arise from an expression where the base and exponent separately approach values that make the combination indeterminate. Each is resolved by the same three-step logarithm procedure:

Using the first derivative, a function can be tested for its monotonicity (increasing or decreasing) at a point, and its local extrema (maxima or minima) on a domain can be located.

Monotonicity describes a function's behaviour of increasing or decreasing.

The absolute maxima and minima describe the largest and smallest values a function takes on an interval.

has a relative (local) maximum at if there is an open interval containing on which is the largest value; similarly, has a relative (local) minimum at if there is an open interval containing on which i…

Once the intervals on which a function is increasing or decreasing are known, locating its relative extrema is straightforward using the following test.

The second derivative of a function is used to determine its concavity, convexity, its points of inflection, and (via the Second Derivative Test) its local extrema.

A graph is said to be concave down (convex up) at a point if the tangent line there lies above the graph in the vicinity of the point; it is concave up (convex down) at a point if the tangent line lie…

The Second Derivative Test relates critical points, extreme values, and concavity into a single practical tool for classifying whether a critical point is a relative minimum or maximum.

Optimization is the process of finding an extreme value (either maximum or minimum) of some quantity under given conditions.

Two further properties of a curve — its symmetry and its asymptotic behaviour at infinity — round out the toolkit needed before sketching a curve in the next section.

If an image or curve is the mirror reflection of itself with respect to a line, the curve is said to be symmetric with respect to that line (the line of symmetry).

An asymptote for the curve is a straight line which is, informally, "a tangent at infinity" to the curve — the distance between the curve and the line tends to as the point on the curve runs off to in…

When sketching the graph of a function — by hand or with software — only a part of the true (often infinite) graph can ever actually be shown.

Sample papers and previous-year board questions for this subject.