Derivatives
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Half a century before direct experimental observation became possible, most structures of organic molecules were assigned by inspired guessing based on plausibility. But Wilhelm Körner developed a strictly logical system for proving the structure of benzene and its derivatives based on isomer counting and chemical transformation. His proof that the six hydrogen positions in benzene are equivalent is the outstanding example of this...more
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Intuition on what happens to the slope/derivative and second derivatives at local maxima and minima.
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Professor Diamond continues her discussion of the nervous system, finishing her discussion of the derivatives of the neural tube. She begins by discussing the lamina terminalis and the four ventricles, relating each to the source of their derivation and the areas of the brain in which they are found. Next, she continues her discussion of diencephalon from the last lecture and describes the roles of the thalamus and hypothalamus. After...more
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Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".
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Determining the derivatives of simple polynomials.
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At the top and bottom of a curve (Max and Min), the slope is zero. The second derivative shows whether the curve is bending down or up. Here is a real-world example of a minimum problem: What route from home to work takes the shortest time?
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Professor Diamond continues her discussion of the nervous system beginning with a discussion of myelin-forming oligodendrocytes and Schwann cells, saltatory conduction from the nodes of ranvier, and the similarity of the function of microglia to monocytes. She moves on to describe the development of the neural tube by drawing a cross-section of the neural tube and depicting the changes it undergoes, forming the ventricles of the brain,...more
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Calculus finds the relationship between the distance traveled and the speed — easy for constant speed, not so easy for changing speed. Professor Strang is finding the rate of change, the slope of a curve, and the derivative of a function.
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This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.
