Explanation Given information: The given function is f ( x ) = x e x Concept used: Concepts of derivatives: If f ′ is positive, f is increasing If f ′ is negative, f is decreasing If f ′ is zero, f is constant If f ″ is positive, f is concave up if f ″ is negative, then f is concave down. To find the local extrema use the following second derivative test: Suppose x = c is the location of a critical point of a function f with f ′ ( c ) = 0 ,and suppose both f and f ′ are differentiable and f ″ is continuous on an interval around x = c . If f ″ ( c ) is positive, then f has a local minimum at x = c If f ″ ( c ) is negative, then f has a local maximum at x = c If f ″ ( c ) = 0 , then this test says nothing about whether or not f has extremum at

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 3, Problem 21SC

To determine

To construct: the sign charts for f,f′ and f″ . To examine: the limits at any interesting values and at ±∞ . To interpret the information to sketch the graph of f .

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Chapter 3, Problem 20SC

Chapter 3, Problem 22SC

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Taylor Series Approximation Example- H.W More terms used implies better approximation f(x) 4 f(x) Zero order f(x + 1) = f(x;) First order f(x; + 1) = f(x;) + f'(x;)h 1.0 Second order 0.5 True f(x + 1) = f(x) + f'(x)h + ƒ"(x;) h2 2! f(x+1) 0 x; = 0 x+1 = 1 x h f(x)=0.1x4-0.15x³- 0.5x2 -0.25x + 1.2 51 Taylor Series Approximation H.w: Smaller step size implies smaller error Errors f(x) + f(x,) Zero order f(x,+ 1) = f(x) First order 1.0 0.5 Reduced step size Second order True f(x + 1) = f(x) + f'(x)h f(x; + 1) = f(x) + f'(x)h + "(xi) h2 f(x,+1) O x₁ = 0 x+1=1 Using Taylor Series Expansion estimate f(1.35) with x0 =0.75 with 5 iterations (or & s= 5%) for f(x)=0.1x 0.15x³-0.5x²- 0.25x + 1.2 52

Could you explain this using the formula I attached and polar coorindates

Could you explain this using the formula I attached and polar coordinates

Chapter 3 Solutions

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To construct: the sign charts for f , f ′ and f ″ . To examine: the limits at any interesting values and at ± ∞ . To interpret the information to sketch the graph of f .

Solution Summary: The author explains how to construct the sign charts for f,fprime, and to examine the limits at any interesting values.

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To construct: the sign charts for f,f′ and f″ . To examine: the limits at any interesting values and at ±∞ . To interpret the information to sketch the graph of f .

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Chapter 3 Solutions