Fundamentals And Functions Of Calculus! Quiz

Explanation
Rolle's Theorem can be thought of as a special case of the Mean Value Theorem because it is a specific condition of the Mean Value Theorem. Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function's values at the endpoints of the interval are equal, then there exists at least one point in the interval where the derivative of the function is zero. This is a specific case of the Mean Value Theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the derivative of the function is equal to the average rate of change of the function over the interval.

Explanation
The polynomial x to the fifth +12x-7 is a fifth-degree polynomial. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots, including both real and imaginary roots. Since the polynomial in question has a degree of 5, it can have a maximum of 5 roots. However, it is not specified whether the roots need to be real or complex. In this case, the answer is 1, which implies that the polynomial has only one real root.

Explanation
The integral from 0 to Pi over 2 of (Sine of x divided by the sum of Sine of x and Cosine of x) is Pi over 4.

Explanation
The number of roots a polynomial of degree n will have is exactly n. This is because the Fundamental Theorem of Algebra states that a polynomial of degree n will have exactly n complex roots, counting multiple roots the number of times they occur. Therefore, we can conclude that the correct answer is "Exactly n".

Explanation
The statement is false. Two infinitely differentiable functions can have the same value and derivatives at a point, but still be different functions. This is because the functions can have different behavior outside of that point. The value and derivatives at a single point do not provide enough information to determine the entire functions.

Explanation
The correct answer is A is true, but not B. This means that if a function is differentiable at 0, it is continuous there (statement A is true), but its derivative may not necessarily be continuous at 0 (statement B is false). Differentiability implies continuity, but continuity does not necessarily imply differentiability.

Explanation
Calculus involves many other components when defining it, but in general it is the study of change. The study of operations and applications is algebra. The study of shape is geometry. The study of collections of data is statistics.

Explanation
The Fundamental Theorem of Calculus relates the two branches by stating they are inverse operations of each other.

Explanation
Today, they both share credit for developing differential and integral calculus. It has long been disputed as to who actually developed it first. Leibniz developed infinitesimal calculus independently and published his work first, but Newton began work on it before and took many years to publish his work.

Explanation
The answer is both because there are two ways in defining the derivative of a function: physically and geometrically.