Lecture Notes for Section 6.1
... u x f u x dx F u x c . Thus, when we see an integrand that is the product of a composition of functions and the derivative of the inner function, we’ll know that its antiderivative is simply the antiderivative of the outer function of the composition (still composed with t ...
... u x f u x dx F u x c . Thus, when we see an integrand that is the product of a composition of functions and the derivative of the inner function, we’ll know that its antiderivative is simply the antiderivative of the outer function of the composition (still composed with t ...
SOLUTIONS TO PROBLEM SET 4 1. Without loss of generality
... vanishes at ∞. Then, g : R 7→ R defined by g(x) = f (|x|) belongs to C0 (R). Note that Qt f (x) = Pt g(x), x ≥ 0. Thus, the desired Feller property follows from the Feller property of Brownian motion. One can also have a direct proof of this fact by using the explicit form of q using the arguments l ...
... vanishes at ∞. Then, g : R 7→ R defined by g(x) = f (|x|) belongs to C0 (R). Note that Qt f (x) = Pt g(x), x ≥ 0. Thus, the desired Feller property follows from the Feller property of Brownian motion. One can also have a direct proof of this fact by using the explicit form of q using the arguments l ...
Lecture 17 - University of Chicago Math
... The definition of indefinite integral is unrelated to the area cut out by f (which is why this notation and name is unfortunate when you first learn it). ...
... The definition of indefinite integral is unrelated to the area cut out by f (which is why this notation and name is unfortunate when you first learn it). ...
Notes - Ryan, Susan
... If the average score on an exam is 84, then 84 lies somewhere between the highest and lowest values, but no student had to receive the score of 84. But a function will always take on its mean value somewhere in the interval. ...
... If the average score on an exam is 84, then 84 lies somewhere between the highest and lowest values, but no student had to receive the score of 84. But a function will always take on its mean value somewhere in the interval. ...
The Fundamental Theorem of Calculus
... Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations. ...
... Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations. ...
The Fundamental Theorems of Calculus
... First Fundamental Theorem of Calculus • Given f is continuous on interval [a, b] F is any function that satisfies F’(x) = f(x) ...
... First Fundamental Theorem of Calculus • Given f is continuous on interval [a, b] F is any function that satisfies F’(x) = f(x) ...
Fundamental Theorem of Calculus, Riemann Sums, Substitution
... Notice that the integral involves one of the terms above. Substitute the appropriate u. Make sure to change the dx to a du (with relevant factor). Simplify the integral using the appropriate trig identity. Rewrite the new integral in terms of the original non-Ѳ variable (draw a reference right-trian ...
... Notice that the integral involves one of the terms above. Substitute the appropriate u. Make sure to change the dx to a du (with relevant factor). Simplify the integral using the appropriate trig identity. Rewrite the new integral in terms of the original non-Ѳ variable (draw a reference right-trian ...
lesson 29 the first fundamental theorem of calculus
... discovered some members of that family by evaluating the definite integral 2t dt and ...
... discovered some members of that family by evaluating the definite integral 2t dt and ...
Solutions for Exam 4
... may choose any six to do. Please write DON’T GRADE on the one that you don’t want me to grade. In writing your solution to each problem, include sufficient detail and use correct notation. (For instance, don’t forget to write “=” when you mean to say that two things are equal.) Your method of solving ...
... may choose any six to do. Please write DON’T GRADE on the one that you don’t want me to grade. In writing your solution to each problem, include sufficient detail and use correct notation. (For instance, don’t forget to write “=” when you mean to say that two things are equal.) Your method of solving ...
The Fundamental Theorem of Calculus [1]
... when f is continuous. Roughly speaking, 2.3 says that if we first integrate f and then differentiate the result, we get back to the original function f . This shows that an antiderivative can be reversed by a differentiation, and it also guarantees the existence, continuity, differentiability of antider ...
... when f is continuous. Roughly speaking, 2.3 says that if we first integrate f and then differentiate the result, we get back to the original function f . This shows that an antiderivative can be reversed by a differentiation, and it also guarantees the existence, continuity, differentiability of antider ...
[Write on board:
... You may have seen this in past calculus classes (and it’s closer to Riemann’s original definition): We estimate the integral of f on [a,b] via the sum R(f,P) = 1kn f(ck) (xk – xk–1) ...
... You may have seen this in past calculus classes (and it’s closer to Riemann’s original definition): We estimate the integral of f on [a,b] via the sum R(f,P) = 1kn f(ck) (xk – xk–1) ...
Math 223 - Vector Calculus (Fall 2016) Homework 4
... 2. (a) We have grad f = 3i + 4j . The value of the line integral will be maximised when the line goes in the same direction as grad f . Let r (t) = (2 + 3t)i + (1 + 4t)j be a parameterisation for the line starting at (2, 1) and going in the same direction as grad f . We want the distance from (2, 1) ...
... 2. (a) We have grad f = 3i + 4j . The value of the line integral will be maximised when the line goes in the same direction as grad f . Let r (t) = (2 + 3t)i + (1 + 4t)j be a parameterisation for the line starting at (2, 1) and going in the same direction as grad f . We want the distance from (2, 1) ...
Section 6.6
... shaped (conical) container of radius 10 ft and height 30 ft. Suppose that this container is filled with water to a depth of 15 ft. How much work is required to pump all of the water out through the hole in the top of the container? We will divide the water into thin layers, approximate the work re ...
... shaped (conical) container of radius 10 ft and height 30 ft. Suppose that this container is filled with water to a depth of 15 ft. How much work is required to pump all of the water out through the hole in the top of the container? We will divide the water into thin layers, approximate the work re ...
Definite Integrals - West Virginia University
... Really, the definite integral computes the area under the curve by adding up the area of an ‘infinite’ number of rectangles ...
... Really, the definite integral computes the area under the curve by adding up the area of an ‘infinite’ number of rectangles ...
Unit 5
... B. Find antiderivatives, apply basic integration formulas, and find distance and velocity functions from initial conditions. C. Understand and apply the concepts and properties of definite integrals including the Fundamental Theorem . D. Solve simple first order differential equations, recognize and ...
... B. Find antiderivatives, apply basic integration formulas, and find distance and velocity functions from initial conditions. C. Understand and apply the concepts and properties of definite integrals including the Fundamental Theorem . D. Solve simple first order differential equations, recognize and ...
MATH 1325 – BUSINESS CALCULUS Section 11.4/11.5 The
... Ex: A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the daily marginal cost function associated with producing these toaster oven is given by C0 (x) = 0.0003x2 − 0.12x + 20 where C0 (x) is measured in dollars per unit and x denotes the number of uni ...
... Ex: A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the daily marginal cost function associated with producing these toaster oven is given by C0 (x) = 0.0003x2 − 0.12x + 20 where C0 (x) is measured in dollars per unit and x denotes the number of uni ...