# Mathematics

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## Riemann Sum and Area Approximation

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very
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## Antiderivatives and Indefinite Integrals

Given a function f, the indefinite integral of f, denoted ∫f(x)dx, is the most general antiderivative of f. If F is an antiderivative of f, then ∫f(x)dx = F(x) +
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## Tangent Line Approximation and Differentials

In general, for a differentiable function f, the equation of the tangent line to f at x = a can be used to approximate f(x) for x near a. Therefore,
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## Optimization Problems

Optimization problems in calculus involve finding the maximum or minimum value of a function while satisfying certain constraints. Let’s explore an example to illustrate this concept....
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## Curves of f , f’ , f” and Curve Sketching

Curve Sketching. To produce an accurate sketch a given function f, consider the following steps. Find the domain of f. Generally, we assume that the...
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## The Second Derivative Test

The second derivative test comes in handy when we have twice differentiable functions (meaning, we can take the derivative $f (x)$ twice in a row). We can confirm the relative
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## The First Derivative Test and the Extreme Values of Functions

The First Derivative Test is a powerful tool in calculus that helps us identify local maximum and minimum points of a function. Here’s how it works...
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## Position, Velocity, and Acceleration

Acceleration is a second derivative of the position. Given a(t) a (t), the acceleration as a function of t t, we can use antidifferentiation to obtain the velocity v(t) v
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## Related Rates

This is the core of our solution: by relating the quantities (i.e. A and r) we were able to relate their rates (i.e. A ′ and r ′) through differentiation.
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## Approximating a Derivative

In general, for a differentiable function f, the equation of the tangent line to f at x = a can be used to approximate f(x) for x near a. Therefore,