Calculus Concepts and Function Analysis

<p>1/2 (A)</p> Signup and view all the answers

<p>Convergent (C)</p> Signup and view all the answers

<p>Critical value (B)</p> Signup and view all the answers

<p>$rac{a e^{mx}}{m} + c$ (A)</p> Signup and view all the answers

<p>Lower half plane (D)</p> Signup and view all the answers

The range of a function f(x) is the set of all possible output values of the function. For the absolute value function, f(x) = |x|, the output is always non-negative, regardless of the input.

A function f(x) is continuous at x = c if the following three conditions are met:

1. The limit of f(x) as x approaches c exists.
2. The function f(x) is defined at x = c.
3. The limit of f(x) as x approaches c is equal to the value of the function at x = c. In other words, the graph of the function should not have any breaks or jumps at x = c for the function to be continuous.

A function is even if it is symmetric about the y-axis. This means that f(x) = f(-x) for all x. For example, the function f(x) = |x| is even because |x| = |-x| for all x.

The limit of (1 - cos x)/x^2 as x approaches 0 is 1/2. This can be found using L'Hopital's rule, or by using a Taylor series expansion of cos x.

The Taylor series expansion of a function can be used to approximate the value of the function at a point. This expansion is only valid if the series converges, meaning that the sum of the infinite terms is a finite number.

A critical value of a function is a point where the derivative of the function is either equal to zero or does not exist. These points are important because they can represent potential maxima, minima, or inflection points of the function.

The integral of the function y = a*emx, with respect to x is (a/m) * e^(mx) + c, where c is the constant of integration.

If the integral of f(x) is F(x) + c, then the integral of f(ax + b) is (1/a) * F(ax + b) + c. This is a consequence of the chain rule for differentiation and the fact that integration is the inverse of differentiation.