Implicit Differentiation, Related Rates Problem 1. In this problem, we consider the Lemniscate curve (1) (a2+y2)2 =3(z2 - ) 2 -2 Figure 1: Bernoulli's Lemniscate The goal of this problem is to find the four points of the Lemniscate curve with horizontal tangent lines. 2 -2 a. Use implicit differentiation to find the expression for dx b. To look for those points, set up the equation 0 (you should find an equation of the type x2y2 with a horizontal tangent line belong also to a circle centered at the origin since they have the same distance to the origin); C, with C> 0,... Yes! because the four points + c. In order to find the y coordinates of the points with a horizontal tangent line, replace y in equation (1) by the value C found in part b. then solve the equation for y (Hint: to eliminate 2, use the equation x2y C again) 1 d. Find the I and y coordinates of the four points of the Lemniscate curve with a horizontal tangent line. 1

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If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

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The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.