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In an elementary chemical reaction, single molecu…

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Problem 39 Hard Difficulty

In Exercise 9.1.15 we formulated a model for learning in the form of the differential equation
$ \frac {dP}{dt} = k(M - P) $
where $ P(t) $ measures the performance of someone learning a skill after a training time $ t, M $ is the maximum level of performance, and $ k $ is a positive constant. Solve this differential equation to find an expression for $ P(t). $ What is the limit of this expression?


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 9

Differential Equations

Section 3

Separable Equations

Related Topics

Differential Equations

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Anaitl M.

March 10, 2021

A sphere with radius 1 m has temperature 11°C. It lies inside a concentric sphere with radius 2 m and temperature 19°C. The temperature at a distance r from the common center of the spheres satisfies the differential equation below. If we let S = dT/dr, t

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Video Thumbnail

13:37

Differential Equations - Overview

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

Video Thumbnail

33:32

Differential Equations - Example 1

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
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Problem 27
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Problem 37
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Problem 48
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Problem 53
Problem 54

Video Transcript

we have the differential equation. The P on the tee he cost too key. Have minus p. It was separate variables. You have he p on a minus bi he caused to Kate. Time to DT integrating both sides. He's S d P on M minus p. He closed two key integral of tea. It is not negative in the natural law. A minus p you close to you, t plus C, which is a constant of integration when team courses there, he becomes the courses there. So it implies that my next natural look and becomes you close to see hands. I questioned them. Become negative. Natural law. M minus key. You cost you Katie minus a journal off. What? Right to buy Negative one gives me natural long off M minus key. You cost her actual off. M minus. Katie. Who? Pee line dance. Lean off. Am I this p call em? You close to Katie? So m minus p I m. You cost. He exponent, you go by the negative. Katie, It's negative, Katie. So I m dynasty in question. M times he explain our negative, Katie. So I make Peter subject P becomes cost. I m minus E em. Want to plant? Yes, but negative. Katie Cook tonight. Isn it solution to the French? I questions Now, if I limit this as TR pushes infinity gonna have limit us. T approaches Infinity. Oh, m minus and e minus. Katie today, which gives me m minus. Um, I want to play by zero. That is nasty. Approaches infinity. Yes, but minus K t approaches. Oh, sure. Limit to Europe, which is infinity off p as a function of time. It comes in Khost. Thank you.

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Top Calculus 2 / BC Educators
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Lectures

Video Thumbnail

13:37

Differential Equations - Overview

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

Video Thumbnail

33:32

Differential Equations - Example 1

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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