Location of Roots 1. If both roots of equation ax² + bx + c = 0 represent opposite signs then (i) D = b² – 4ac > 0 (ii) Product of roots of equation is less than zero 2. If both of equation ax² + bx + c = 0 have same sign then two possibilities Read more about Location of Roots[…]
Partial Fractions – Proper Fraction 1. An expression of the form, f(x) = aₒ + a₁ x + a₂ x + … an xⁿ where n is a non-negative integer and aₒ, a₁, a₂, … an are real numbers such that an ≠ 0, is called a polynomial in x of degree n. 2. Division Algorithm: Read more about Partial Fractions – Proper Fraction[…]
Sign of the Quadratic Expression Sign of the Expression ax² + bx + c: 1. Sign of the expression ax² + bx + c is same as that of ‘a’ for all value of x if b² – 4ac ≤ 0 i, e. if the roots of ax² + bx+ c = 0 are imaginary Read more about Sign of the Quadratic Expression[…]
Domain & Range of a Real Function Domain: Generally real functions in calculus are described by some formula and their domains are not explicitly stated. In such cases to find the domain of a function f (say) we use the fact that the domain is the set of all real numbers x for which f Read more about Domain & Range of a Real Function[…]
Intervals & Real Functions Closed Interval: Let a and b be two given real numbers such that a < b Then the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b]. i.e., [a, b] = {x ϵ R| a ≤ x Read more about Intervals & Real Functions[…]
Properties of Periodic Functions Periodic Functions: A function f (x) is said to be a periodic function if there exists a positive real number T such that f (x + T) = f (x) for all x ϵ R. for all values of x in the domain. If there exists a least positive real number Read more about Properties of Periodic Functions[…]
Evaluation of Limits of the Form 1∞ To evaluate the exponential limits of the form 1∞, we use the following result. Result: If such that exists. Then . Proof: Let A . . . . . Remark: The above result can also be restated in the following form: If and such that exists. Then . Particular Read more about Evaluation of Limits of the Form 1∞[…]
Evaluation of Algebraic Limits Let f(x) be an algebraic function and a be any real number, then is known as an algebraic limit. Example: a) . b) . The limit of algebraic functions can be find by the following methods. 1. Method of Direct Substitution: can be evaluated by method of direct substitution, if f(x) Read more about Evaluation of Algebraic Limits[…]
Complex Numbers Any number of the form x + iy where x, y ϵ R and i² = -1 is called a complex number. In the complex number x + iy, x is called the real part and y is called the imaginary part of the complex number. A complex number is said to be Read more about Complex Numbers[…]
Hyperbolic Functions Formulas: i. ii. iii. iv. v. vi. vii. viii. ix. x. xi. xii. xiii. xiv. xv. xvi. xvii. Example: If the prove that cosh μ = secθ. Solution: , cosh μ = secθ , , , , , , , , , , , = secθ Hence proved cosh μ = secθ. Share Read more about Hyperbolic Functions[…]