Advanced Calculus Problems

Advanced Calculus Problems Calculus is a very important branch of mathematics. The different branches of mathematics have been enriched a lot by the application of calculus. Branches of Advance calculus: - 1) Differential calculus Real number Function Limit Continuity Differentiation Second order derivative 2) Integral calculus Indefinite integral Method of substitution Integration by parts Integrals of some special form of functions Definite integral 3) Differential equation Order and degree of differential equations Differential equations of the first order and of the first degree Linear second order differential equations with constant coefficients 4) Application of calculus Significance of derivatives Tangent and normal Maxima and minima Definite integral as an area Velocity and acceleration Few examples of advance calculus problem: - Example 1: - Find the derivatives of y w.r.t. x when y = (x + 1) (x 1) Solution: - Differentiating both sides w.r.t. x, dy / dx = (x + 1) d / dx ( x 1) + (x 1) d/dx (x + 1) = (x + 1) [ d / dx (x) d / dx (1)] + (x 1) [d/dx (x) + d/dx (1)] = (x + 1) (1 0) + (x 1) (1 0) = (x+1) + (x -1) = x + 1+ x 1 = 2x Therefore dy /dx = 2x Example 2: Evaluate x ^4 dx Solution: - x ^4 dx = x^ (4 + 1) / (4+ 1) = x^5 / 5 Therefore x ^4 dx = x^5 / 5