By Munkres J.R.

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This result has the following geometric application. 5 where the edges are represented by a, band c. n Fig. 5. Parallelepiped volume The height of the parallelepiped is given by a cos ¢ and the area of the base is given by besin8. (b x c) = volume of parallelepiped. 29 Vector Algebra Trying to calculate the volume using the expression abc sin (J cos t/J is difficult if the angles (J and t/J have to be calculated. If the vectors are given in terms of cartesian components it is more convenient to calculate a determinant.

Find the rate of change of P with respect to t if P e2t = 1 +t2 . 19. 64x2 + 200x + 350 . Find the value of x such that :::: = O. 20. The power P developed in a resistor R by a battery with emf E and in~ernal resistance r is e2 R P=(R+r)2' Find the rate of change of power with respect to R. 21. l use the product rule to derive the quotient rule. 6 Higher derivatives So far we have reviewed methods for finding the derivative ~ of a function y = f(x). More formally, this is the first derivative; but ~ is itself a function of x and can usually be differentiated again.

This notation allows us to treat complex numbers in the same way as vectors written in component form. The real part is plotted along the horizontal axis and the imaginary part on the vertical axis. 1. Note the similarity to plotting the position vector 3i + 2j. B A Re D Fig. 1. 1 also shows: 1. 0 B represents the complex number -1 + 2i 2. OC represents the complex number 0·+ 3i 3. 0 D represents the complex number 3 - 2i 36 Complex Numbers Notice that OD is the complex conjugate of OA and is shown by reflection in the horizontal axis.