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Calculus is a big step up from Algebra and TutorVista guides the students in understanding it. Precalculus which is the base required to start off with Calculus will be tutored on a one on one basis with emphasis on Algebra, Trigonometry, Functions and Geometry. Give a strong foundation and move on to Calculus, starting from defining Limits and L’Hospital’s rule to Derivatives which will introduce the student to definition of a derivative to Product and Chain rules and to Integrals and Anti-Derivatives
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Functions Limits and Continuity |
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Left Hand Limit: Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit. |
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Right Hand Limit: Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit. |
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We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal. |
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1. We say that f(x) is continuous if f(x) is continuous at every point in its domain. |
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2. If f and g are two continuous functions then f + g, f - g, fg are continuous functions. |
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3. Every polynomial function is continuous. |
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4. Every rational function is continuous at each point of its domain. |
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5. Composition of two continuous functions is continuous. |
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Differentiation |
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The derivative, measures the rate at which the dependent variable changes with respect to the independent variable. It is one of the most important ideas in Calculus. The differentiation of functions are widely used in science, economics, medicine and computer science. |
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Differential Equations |
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Differential Equation: A differential equation is a relation between the independent, dependent variables and their differential coefficients. |
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Order of Differential Equation: The order of differential equation is defined to be the order of the highest order derivative of the dependent variable occurring in the differential equation. |
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Degree of a Differential Equation: The degree of a differential equation is the highest power of the highest order derivative after making the equation free from radicals and fractional indices as far as the derivatives are concerned. |
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Indefinite Integrals |
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The expression ∫ f(x) dx |
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is read "the indefinite integral of f(x) with respect to x," and stands for the set of all antiderivatives of f. |
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Thus, ∫ f(x) dx is a collection of functions; it is not a single function, nor a number. The function f that is being integrated is called the integrand, and the variable x is called the variable of integration. (The expression dx is short for "with respect to x."). |
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Definite Integrals |
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Let f (x) be a single valued continuous function defined in the interval [a,b] where b > 0 and let the interval [a,b] be divided into n equal parts each of length h, so that nh = b - a; then we define |
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The method of evaluating  by using the above definition is called integration from first principles. |
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Application of Derivatives |
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Differential calculus can be considered as mathematics of motion, growth and change where there is a motion, growth, change. Whenever there is variable forces producing acceleration, differential calculus is the right mathematics to apply. Application of derivatives are used to represent and interpret the rate at which quantities change with respect to another variable. Most of the changes are considered in terms of independent variable time. But there is no restriction that the changes are considered with respect to time only. |
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Exponential and Logarithmic Series |
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The sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...¥ is called the exponential number. |
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If x is any complex number then the series  is called the exponential series. It can be proved mathematically that this exponential series has a sum and we denote it by ex. |
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 If x is a real number such that |x|<1, then the series is
called the logarithmic series. It can be proved mathematically that this logarithmic series has the sum equal to log(1 + x).
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