Differentiation - Solve Problems

Differentiation can be used and applied to solve various problems in Plane Geometry, Real functions, Optimization and Approximations. We know that if a quantity y (dependent variable) depends on the value of another quantity x (independent variable) and varies with the quantity x, then the rate of change of y with respect to x is dy/dx. If distance is given by x = f(t), then velocity is given by v = dx/dt .Differentiation is also used to find the equations of tangent and normal at a given point on the curve. The derivative of the curve (a given equation) at a point is nothing but the slope of the tangent at that point. The equation of the tangent can be found in the slope point form. The normal is perpendicular to the tangent at that point. It is also used to check if two curves cut orthogonally. When the product of their slopes at the point of intersection is -1, then the curves cut orthogonally. The concept of differentiation is used to find a point on the curve when tangent is parallel to x-axis( Rolle''s theorem) and the point where the tangent is parallel to the chord joining two points on the curve(Lagrange''s Mean Value theorem).It is also used to evaluate certain indeterminate forms of the type (0/0) (i.e.) Evaluating the limit f(x)/g(x) as x tends to a ,when f(a) and g(a)=0, using l'Hopital''s Rule.

Differential calculus has varied applications. It is useful to study the behaviour of a curve, from its equation. We can find in which interval the function is increasing and the interval in which the function is decreasing. Differential calculus has its application in optimization problems. Many practical problems may require minimizing or maximizing a quantity. We may require maximizing the area of a given figure or minimizing the perimeter or maximize the production within the given constraints and so on. Differentiation helps us to solve such problems by finding local maxima, local minima, absolute maxima and absolute minima of the function. The greatest task is to convert the life problem into a suitable function that is to be maximized or minimized.

Sometimes the point where the tangent is parallel to x-axis on the curve may not be a point of maximum or minimum. Such points are called points of inflection. At such points, the concavity or the convexity of the curve changes.