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1 Lesson Plan #28 Class: AP Calculus Date: Wednesday November 17th, 2010 Topic: Derivatives of Sine and Cosine functions functions? Aim: How do we differentiate trigonometric Objectives: 1) Students will be able to differentiate trigonometric functions HW# 28: Page 140 #’s 1-9 (odd) Squeeze Theorem: The squeeze theorem is formally stated as follows. Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have: and also suppose that: Then Illustration of the Squeeze Theorem On your graphing calculator, set the mode to radian mode. In y1 enter cosx, in y2 enter sin x , in y3 enter 1 x sin x 1 x sin x 1 Since there is an interval for which cos x x and lim cos x 1 Verify that there is an interval for which cos x x0 and lim 1 1 x 0 then by the squeeze theorem lim x0 sin 5 x x 0 x 1) Find the limit lim sin x 1 x sin x x0 5 x 2) Find the limit lim 1 cos x 0 x0 x Comment: By the Squeeze Theorem it can also be shown that lim 2 3(1 cos x) x0 x 3) Find the limit lim sin( x ) x0 x 4) Find the limit lim 5) Find the limit lim x0 sin x x 2 3x Note: Also do above question as x 4) If 3x f ( x) x3 2 for 0 x 2 , evaluate lim f ( x ) x1 Do Now: In Precalculus we learned how to use the Sum and Difference formulas for trigonometric functions. For example, recall sin( x y ) sin x cos y cos x sin y o o o Use this formula to find the exact value of sin 75 , assuming you know the exact values of sin 30 , cos 30 , sin 45 , cos 45o 2) On your graphing calculator, set you graphing window to an appropriate size and graph the function f ( x ) What y value is the graph approaching as x 0? sin x x What is another way to rewrite this question? What is f (0) ? 3) On your graphing calculator, set you graphing window to an appropriate size and graph the function f ( x) What y value is the graph approaching as x 0? What is another way to rewrite this question? What is f (0) ? Procedure: Write the Aim and Do Now Get students working! Take attendance Give back work Go over the HW Collect HW Go over the Do Now Recall the long way to find the derivative. Let’s use this long way to find the derivative of Definition of the Derivative of a Function: The derivative of a f at f ' ( x) lim x 0 f ( x x) f ( x) x x is given by provided the limit exists sin x 1 cos x x d sin( x x) sin x [sin x] lim x 0 dx x Use the formula for the sin of the sum of angles to expand the first part of the limit d sinxcosx cosxsinx sin x [sin x] lim x0 dx x Rewrite the right side of the equation so that the middle term is first, the last term is in the middle and the middle term is last d cos x sin x sin x sin x cos x [sin x] lim x 0 dx x Factor sin x from the last two terms d cos x sin x (sin x)(1 cos x) [sin x] lim x 0 dx x Put the denominator under each term of the numerator d sin x 1 cos x [sin x] lim (cos x) (sin x) x 0 dx x x Evaluate limit d [sin x] (cos x)(1) (sin x)(0) dx cos x So we have the derivative of sin x. d [sin x] cos x dx By other proofs we can get the derivatives of the other five trigonometric functions. Below are the derivatives of the six trig functions. d [sin x] cos x dx d [cos x] sin x dx d [tan x] sec 2 x dx d [cot x] csc 2 x dx d [sec x] sec x tan x dx d [csc x] csc x cot x dx Assignment: I. Find the derivative of each of the following 1) y 2 sin x tan x 3 4 2) y x 2 sec x 3) y 2 x cot x 4) y x cos x 5) y 4 x 2 sin x 6) y x sin x cos x 7) y sec x csc x III. Find the slope of the line tangent to y 2 cot x at x 1 radian x 5 Sample Test Questions: 1) If f ( x ) 2) Let sin x , find f ' ( x ) x2 g ( x) cos x 1 . The maximum value attained by g on the closed interval A) -1 B) 0 C) 2 3) The period of f ( x) sin A) 1 3 B) 2 3 0,2 is for x equal to D) 2 2 x is 3 3 C) 2 E) D) 3 E) 6 4) Which of the following functions is not odd? A) f ( x) sin x D) f ( x) B) f ( x) sin 2 x x x 1 E) f ( x) 2 3 2 B) C) f ( x) x 3 1 2x 5) The smallest positive x for which the function A) C) 3 2 x f ( x) sin 1 is maximum is 3 D) 3 E) 6 dy 5 if y x tan x dx 4 5 2 4 2 A) 5 x tan x B) x sec x C) 5 x sec x 4 2 4 5 2 D) 5 x sec x E) 5 x tan x x sec x 6) Find , is 2 2 7) The equation of the tangent to the curve y x sin x at the point A) y x B) y 2 C) y x D) y x 2 E) y x 8) Evaluate lim sin x x A) -1 B) C) oscillates between -1 and 1 D) 0 E) Does not exist