Business Calculus I
... positive it slopes up when you graph it. If the tangent line is sloping up, then the function is increasing. Likewise, when the first derivative is negative, the slopes of the tangent lines are negative. If the slope of a line is negative the line slopes down when you graph it. If the tangent line i ...
... positive it slopes up when you graph it. If the tangent line is sloping up, then the function is increasing. Likewise, when the first derivative is negative, the slopes of the tangent lines are negative. If the slope of a line is negative the line slopes down when you graph it. If the tangent line i ...
Math 500 – Intermediate Analysis Homework 8 – Solutions
... Solution: No, the convergence is not uniform. Indeed, notice that although each function fn is continuous on [0, 1], the pointwise limit f is not continuous on [0, 1]. Thus, the convergence can not be uniform by Theorem 24.3. 24.12 Let (fn ) be a sequence of functions defined on a set S ⊂ R. Then (f ...
... Solution: No, the convergence is not uniform. Indeed, notice that although each function fn is continuous on [0, 1], the pointwise limit f is not continuous on [0, 1]. Thus, the convergence can not be uniform by Theorem 24.3. 24.12 Let (fn ) be a sequence of functions defined on a set S ⊂ R. Then (f ...
Syllabus: MAT 242 CALCULUS II
... Syllabus: MAT 242 CALCULUS II Goals: To learn the fundamental theorem of calculus, some techniques for applying the concepts embodied in it, and to transfer knowledge gained to formulating and solving problems. The language of mathematics is powerful; a major thrust of this course is to help you bui ...
... Syllabus: MAT 242 CALCULUS II Goals: To learn the fundamental theorem of calculus, some techniques for applying the concepts embodied in it, and to transfer knowledge gained to formulating and solving problems. The language of mathematics is powerful; a major thrust of this course is to help you bui ...
Piecewise Defined Functions
... The function g is a piecewise defined function. It is defined using three functions that we’re more comfortable with: x2 − 1, x − 1, and the constant function 3. Each of these three functions is paired with an interval that appears on the right side of the same line as the function: x ≤ 0, and 0 ≤ x ...
... The function g is a piecewise defined function. It is defined using three functions that we’re more comfortable with: x2 − 1, x − 1, and the constant function 3. Each of these three functions is paired with an interval that appears on the right side of the same line as the function: x ≤ 0, and 0 ≤ x ...
An investigation in the Hailstone function
... Hailstone function. As this factor is smaller than one we may conclude on statistical grounds that the number will converge to one. Please note that the constant +1 is ignored. For large numbers this constant is not significant. For the algorithm of course it is! We define the sequence length as N a ...
... Hailstone function. As this factor is smaller than one we may conclude on statistical grounds that the number will converge to one. Please note that the constant +1 is ignored. For large numbers this constant is not significant. For the algorithm of course it is! We define the sequence length as N a ...
Lecture 4: Cauchy sequences, Bolzano
... certain conditions under which we are guaranteed that limits of sequences converge. Definition We say that a sequence of real numbers {an } is a Cauchy sequence provided that for every > 0, there is a natural number N so that when n, m ≥ N , we have that |an − am | ≤ . Example 1 Let x be a real n ...
... certain conditions under which we are guaranteed that limits of sequences converge. Definition We say that a sequence of real numbers {an } is a Cauchy sequence provided that for every > 0, there is a natural number N so that when n, m ≥ N , we have that |an − am | ≤ . Example 1 Let x be a real n ...
Does
... Let x1, x2, x3,…, xn denote a set of n numbers. x1 is the first number in the set. xi is represents the ith number in the set. The summation sign, the symbol,∑, which is the ...
... Let x1, x2, x3,…, xn denote a set of n numbers. x1 is the first number in the set. xi is represents the ith number in the set. The summation sign, the symbol,∑, which is the ...
Pythagorean Triples and Fermat`s Last Theorem
... The proof for exponent 5 is shared by two very eminent mathematicians, the young, 20 years old, Peter Gustav Lejeune Dirichlet(1805-1859) and the aged, 73 years old, AdrienMarie Legendre(1752-1833). They proved the result for n = 5 in 1825 using one of the first general results on the general an + b ...
... The proof for exponent 5 is shared by two very eminent mathematicians, the young, 20 years old, Peter Gustav Lejeune Dirichlet(1805-1859) and the aged, 73 years old, AdrienMarie Legendre(1752-1833). They proved the result for n = 5 in 1825 using one of the first general results on the general an + b ...
Math Review
... Then show that some known property would be false as well. Example: “There is an infinite number of prime numbers” Proof: • Assume the theorem is false (so there are only finite prime) • Let P1, P2, ..., Pk be all the primes in increasing order. • Let N = P1P2 Pk + 1,N is > Pk , so it is not ...
... Then show that some known property would be false as well. Example: “There is an infinite number of prime numbers” Proof: • Assume the theorem is false (so there are only finite prime) • Let P1, P2, ..., Pk be all the primes in increasing order. • Let N = P1P2 Pk + 1,N is > Pk , so it is not ...
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.