Math 31A 2009 10 20 MATH 31A DISCUSSION JED YANG 1 Differentiation 1 1 Basics Given a function f x The slope of the tangent line at x c is f c 1 1 1 Higher Derivatives Recursively define higher derivatives f n f n 1 1 1 2 Chain Rule If f and g are differentiable then f g x f g x is differentiable and f g x f g x g x 1 2 Exercise 3 6 51 Show that a nonzero polynomial function y f x cannot satisfy the equation y y Use this to prove that neither sin x nor cos x is a polynomial Proof Let y an xn an 1 xn 1 a1 x a0 with an 6 0 If the degree n 2 then y 0 so y 6 y Otherwise y nan xn 1 n 1 an 1xn 2 2a2 x a1 and y n 1 nan xn 2 2a2 Since y lacks a monomial xn we cannot have y y 1 3 Exercise 3 7 92 A Discontinuous Derivative Use the limit definition to show that g 0 exists but g 0 6 limx 0 g x where 2 x sin x1 if x 6 0 g x 0 if x 0 Proof Recall g x limh 0 g x h g x So by definition g 0 limh 0 g h g 0 h h limh 0 h sin h1 Using Squeeze Theorem and h h sin h1 h we get that g 0 0 Away from x 0 we can use the formula and get g x 2x sin x1 x2 cos x1 1 x12 2x sin x1 cos x1 Now limx 0 g x does not exist since limx 0 2x sin x1 0 by Squeeze Theorem but limx 0 cos x1 does not exist 1 4 Exercise 3 8 35 Implicit Differentiation If the derivative dx dy exists at a point and dx dy 0 then the tangent line is vertical Calculate dx dy for the equation y 4 1 y 2 x2 and find the points on the graph where the tangent line is vertical Solution By implicit differentiation we get 4y 3 2y 2x dx dy Setting dx dy 0 1 3 we get 4y 2y so y 0 2 If y 0 then x 1 if y 12 then x2 43 so we get 6 points Math 31A Yang 2 1 5 Exercise 3 9 44 Related Rates A wheel of radius r is centred at the origin As it rotates the rod of length L attached at the point P drives a piston back and forth in a straight line Let x be the distance from the origin to the point Q at the end of the rod a Use the Pythagorean Theorem to show that L2 x r cos 2 r2 sin2 b Differentiate part a with resepct to t to prove that dx d d 2 x r cos 2r2 sin cos r sin 0 dt dt dt c Calculate the speed of the piston when 2 assuming that r 10 cm L 30 cm and the wheel rotates at 4 revolutions per minute Solution Parts a and b are straightforward 4 revolutions per minute means d 2 2 2 dt 4 2 per minute From part a we get 30 x 10 so x 20 2 dx Plugging in we get 2 20 2 0 dx dt 10 8 0 0 So dt 80 cm per minute 1 6 Exercise 3 8 55 Lemniscate Curve The lemniscate curve x2 y 2 2 4 x2 y 2 was discovered by Jacob Bernoulli in 1694 who noted that it is shaped like a figure 8 or knot or the bow of a ribbon Find the coordinates of the four points at which the tangent line is horizontal Solution By implicit differentiation and chain rule we get 2 x2 y 2 2x 2yy 4 2x 2yy If y 0 we get 2 x2 y 2 2x 4 2x yielding x2 y 2 2 Substituting in to the lemniscate curve we get x2 y 2 1 So x2 3 2 and y 2 1 2