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The figure shows a beam of length $ L $ embedded in concrete walls. If a constant load $ W $ is distributed evenly along its length, the beam takes the shape of the deflection curve$$ y = -\dfrac{W}{24EI} x^4 + \dfrac{WL}{12EI} x^3 - \dfrac{WL^2}{24EI} x^2 $$where $ E $ and $ I $ are positive constants. ($ E $ is Young's modulus of elasticity and $ I $ is the moment of inertia of a cross-section of the beam.) Sketch the graph of the deflection curve.

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Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 5

Summary of Curve Sketching

Derivatives

Differentiation

Volume

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

04:35

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

06:14

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

04:22

The figure shows a beam of…

0:00

01:35

01:08

A beam of length $L$ is em…

So first, what we want to do is, um, sketch this curve so we see that we'll have why equals w divided by e i Times X to the fourth. And that can be a minus. And we also have minus 24 minus 24 ei times x to the fourth, actually plus w l divided by 12 ei times X to the third or X Cube. And then lastly, we'll have minus W l squared. Yeah, divided by 24 e I X squared. You know, creation. Using these properties, we can. Now I'm just select constants for our values, and we end up getting this graph. Now we can change these values. But what we ultimately see is that we'll have this somewhat parabolic shape that's always concave down S O. This is the general shape of our graph, Andi. That is the deflection curve that we can sketch based on the values and plotting points

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