There is some overlap among these topics, so I recommend reading the whole page. Ultimately, what are the sources of errors and of misunderstanding? What kinds of biases and erroneous preconceptions do we have? Two of my favorite historic discoveries are Einstein's discovery of relativity and Cantor's discoveries of some of the most basic rules of infinities.

Thus a function is analytic in an open disc centered at b if and only if its Taylor series converges to the value of the function at each point of the disc. If f x is equal to its Taylor series for all x in the complex plane, it is called entire.

The polynomials, exponential function ex, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square rootthe logarithmthe trigonometric function tangent, and its inverse, arctan.

For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.

Uses of the Taylor series for analytic functions include: The partial sums the Taylor polynomials of the series can be used as approximations of the function.

These approximations are good if sufficiently many terms are included.

Differentiation and integration of power series can be performed term by term and is hence particularly easy. An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane.

This makes the machinery of complex analysis available. The truncated series can be used to compute function values numerically, often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm.

This result is of fundamental importance in such fields as harmonic analysis. Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.

Approximation error and convergence[ edit ] Main article: The pink curve is a polynomial of degree seven:How do you rewrite the expression as a single logarithm and simplify #ln(cos^2t)+ln(1+tan^2t)#? Exponential and Logistic Functions PreCalculus 3 - 2 Do you think it is reasonable for a population to grow exponentially indefinitely?

Logistic Growth Functions functions that model situations where exponential growth is limited.

An equation of the form _____ or _____. Learn about the properties of logarithms and how to use them to rewrite logarithmic expressions.

For example, expand log₂(3a). Intro to logarithm properties (1 of 2) Intro to logarithm properties (2 of 2) compressing a sum of two or more logarithms means writing it as a single logarithm.

The derivative of an exponential function. Illustration of how the derivative of the exponential function is a multiple of the function, where that multiple is the derivative at .

Functional language with intensional polymorphism and first-class staging. - mrakgr/The-Spiral-Language.

3) Ln (x - 4) + Ln x = Ln 21 Solution: Notice, this time we have a log on both sides. If we write the left side as a single log, we can use the rule that if the logs are equal, the quantity inside must be equal.

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Intro to logarithm properties (article) | Khan Academy