Examples of mathematical writing
... Prime numbers are essential to crytography, Euclid’s famous theorem has held generations of mathematicians spellbound in it’s inescapable beauty. Theorem. (Euclid, 400) There are infinitely many prime numbers, where a prime is a number only divisible by itself and 1. (Throughout this project, number ...
... Prime numbers are essential to crytography, Euclid’s famous theorem has held generations of mathematicians spellbound in it’s inescapable beauty. Theorem. (Euclid, 400) There are infinitely many prime numbers, where a prime is a number only divisible by itself and 1. (Throughout this project, number ...
NESTED INTERVALS
... Sufficiency. Suppose jupuqj < for all p; q > N and any > 0. Then all the numbers u N; uN1;. . . lie in a finite interval, i.e., the set is bounded and infinite. Hence, by the Bolzano–Weierstrass theorem there is at least one limit point, say a. If a is the only limit point, we have the desired proof and ...
... Sufficiency. Suppose jupuqj < for all p; q > N and any > 0. Then all the numbers u N; uN1;. . . lie in a finite interval, i.e., the set is bounded and infinite. Hence, by the Bolzano–Weierstrass theorem there is at least one limit point, say a. If a is the only limit point, we have the desired proof and ...
Calculus - maccalc
... 6. To what new value should f(1) be changed to make f continuous at x = 1? ______ 7. What is the domain of this function? (Interval Notation) ________________________ 8. What is the range of this function? (Interval Notation) ______________________ ...
... 6. To what new value should f(1) be changed to make f continuous at x = 1? ______ 7. What is the domain of this function? (Interval Notation) ________________________ 8. What is the range of this function? (Interval Notation) ______________________ ...
Big Picture Day: Slopes of Tangent Lines and Derivative Techniques
... Big Picture Day: Slopes of Tangent Lines and Derivative Techniques This activity should help you understand the connections between graphs of a function, its derivative, and the meaning of several things we have done in class. It will also help you see the power behind the graphing calculator. 1) Pu ...
... Big Picture Day: Slopes of Tangent Lines and Derivative Techniques This activity should help you understand the connections between graphs of a function, its derivative, and the meaning of several things we have done in class. It will also help you see the power behind the graphing calculator. 1) Pu ...
Math 3005 – Section 5.4 Homework
... f (2/3) and f (1), the Intermediate Value Theorem of Calculus guarantees that there is at least one value c ∈ (2/3, 1) such that f (c) = 0; that is, there is at least one real solution to x3 + x2 − 1 = 0 between x = 2/3 and x = 1. We will now show that there is only one such number c. Assume, to the ...
... f (2/3) and f (1), the Intermediate Value Theorem of Calculus guarantees that there is at least one value c ∈ (2/3, 1) such that f (c) = 0; that is, there is at least one real solution to x3 + x2 − 1 = 0 between x = 2/3 and x = 1. We will now show that there is only one such number c. Assume, to the ...
Test #3 Topics
... The area of a region bounded by the graph of a function over some interval (more simply, the area beneath a curve) is approximated by summing the areas of rectangles whose heights are given by the values of the function (that is, by evaluating Riemann sums). In this section, regions are assumed to ...
... The area of a region bounded by the graph of a function over some interval (more simply, the area beneath a curve) is approximated by summing the areas of rectangles whose heights are given by the values of the function (that is, by evaluating Riemann sums). In this section, regions are assumed to ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.