Slope of tangent and normal: The slope of the tangent to a continuous curve y = (x) at point p(x₁, y₁) is = . If the tangent of curve makes an angle θ with x-axis then .If tangent is parallel to x – axis then θ = 0. If it is perpendicular to x-axis (i.e. parallel Read more about Tangents and Normal[…]
Strictly increasing function: A function f(x) is said to be a strictly increases function on (a, b) if x₁ < x₂ ⇒ f (x₁) < f (x₂) ” x₁, x₂ ϵ (a, b).Ex: aˣ is strictly increasing function on R for a > 1. F(x) = eˣ is increasing on R. Strictly decreasing function: A Read more about Monotonic function[…]
(i) Algebraic functions: a) , . b) , . c) , . (ii) Exponential functions: a) , . b) , .(iii) Logarithmic functions: a) , . b) , . (iv) Trigonometric functions: a) d/dx (sin x) = cosx b) d/dx (cos x) = – sin x c) d/dx (tan x) = sec²x d) d/dx (cot Read more about Differentiation Types:[…]
Rule (I): Differentiation of a constant function is zero i.e., d/dx (c) = 0 Rule (II): Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that , This is the derivative of a constant times a function is the constant times the derivative of the function. Read more about Fundamental rules for differentiation[…]
If y = f(x), then dy/dx the derivative of y with respect to x, is itself, in general or function of x and can be differentiated again. To fix up the idea, we shall call dy/dx as the first order derivative of y with respect to x and the derivative of dy/dx w.r.t. x as Read more about Higher Order Derivatives[…]
Evaluation of trigonometric limits: In order to evaluate trigonometric limits the following results are very useful: sinx/x = 1 tanx/x = 1 sin⁻¹x/x = 1 tan⁻¹x/x = 1 sinx⁰/x = π/180 sin (x – a)/(x – a) = 1 tan (x – a)/(x – a) = 1 |sinx|/x does not exist |tanx|/x does not exist Read more about Exponential and Logarithmic Limits[…]
i. If, f, g ate two continuous functions at a point a of their common domain D, then f ± g. fg are continuous at a point, then f/g is also continuous at a.ii. If f is continuous at a and f(a) ≠ 0, then there exists an open interval (a – δ, a + Read more about Properties of Continuous functions[…]
To evaluate the exponential limits of the form1∞, 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 cases: . . . Read more about Evaluation of limits of the form1∞[…]
Various methods of evaluations of algebraic limits: To evaluate algebraic limits, we have the following methods: i. Direct substitution method ii. Factorization method iii. Rationalisation method iv. Using some standard results v. Method of evaluating limits when variable tends to ∞ or – ∞ Direct substitution method: If by direct substitution of the point in Read more about Evaluation and Algebric of Limits – I[…]
The algebra of limits: Let f and g be two real functions with domain D. We define four new functions f ± g, fg, f/g on domain D by setting (f ± g) (x) = f(x) ± g(x), (fg) (x) = f(x) g(x) (f/ g)(x) = f(x)/ g(x), if g(x) ≠ 0 for any x Read more about Evaluation and Algebric of Limits[…]