How to Learn Calculus of One Variable: v. 1 by J.D. Ghosh, Anwarul Haque

By J.D. Ghosh, Anwarul Haque

Intends to convey out essentially the most vital suggestions affiliates with the theoretical features. This ebook can assist the coed to benefit the tools of Calculus and theoretical points.

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Extra resources for How to Learn Calculus of One Variable: v. 1

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Y = − a f a f ⇒ a x − 1f a x − 3f ≥ 0 ⇒ x−3 − x−3 ≥0 2 ⇒ x −4≤0 1 10. e; y = remember the following facts: af af f x ) one must g x 29 Function values of x for which af af af af f x ) consists of those g x 1. The domain of y ( = af af f x ≥0 g x af af af f x ≥ 0 ⇔ f x ≥ 0 , g x > 0 , or f x ≥ 0 , 2. g x g (x) < 0. 3. FG x − α IJ ≤ 0 ⇔ α ≤ x < β H x − βK or β < x ≤ α ac- cording as α < β or β < α . 4. FG x − α IJ ≥ 0 ⇔ x ≥ α or x < β if β < α and ⇔ H x −βK LM N Hence, domain = R − −3 , LM 2 , + ∞I N3 K I a K f 2 = −∞ , − 3 ∪ 3 or, alternatively: 2 x− 3x − 2 3 ≥ 0 ⇔ x < − 3 or ≥0⇔ x+3 2x + 6 f LMN IK 2 Hence, domain = a−∞ , − 3f ∪ L , + ∞I MN 3 K x≥ a 2 2 ⇔ x ∈ −∞ , − 3 ∪ ,+∞ 3 3 x −1 x +1 x ≤ α or x > β if α < β .

2. When a function defined by a single formula y = f (x) does not become imaginary or undefined for any value of independent variable x, the domain of the function y = f (x) is the set of all real numbers denoted by R. To obtain its range, one should consider the domain −∞ < x < ∞ using the axioms of inequality in −∞ < x < ∞ . 3. , 2 D = b − 4 ac ≥ 0 for real x. 4. In case the domain of a function y = f (x) is a finite set D = {a1, a2, a3, …, an}, then its range is obtained by forming the set whose members are the values of [f (x)] x = a1, a2, a3, …, an.

999 ∈ [–1, 0) = −1 ≤ x < 0 ∴ f (–1 + h) = (1 + x)x =–1 + h = 1 + h – 1 = h Domain of a Function Sometimes a function of an independent variable x is described by a formula or an equation or an expression in x and the domain of a function is not explicitly stated. In such circumstances, the domain of a function is understood to be the largest possible set of real numbers such that for each real number (of the largest possible set), the rule (or, the function) gives a real number or for each of which the formula is meaningful or defined.

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