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MATH260—Week 4 Lab Name: Part I: The Trig DerivativesAlthough the derivative of each trig function can be found by using trig identities and the formulaf ( x+ h)−f ( x )lim, it is far simpler to memorize them because they will be used in many of thehh→0techniques that follow in this class as well as in Calculus II.Directions: Look at the examples and answer questions 1–4 below.Function Derivative f(x) = sin (2x) f’(x) = 2 cos (2x) f(x) = cos (5x2) f’(x) = -10x sin (5x2) f(x) = tan (x2 + 3x) f’(x) = (2x + 3) sec2 (x2 + 3x) f(x) = cot (5x2) f’(x) = -10x csc2 (5x2) f(x) = csc (2x) f’(x) = -2csc (2x) cot (2x) f(x) = sec (x2 + 3x) f’(x) = (2x + 3) sec (x2 + 3x) tan (x2 + 3x) 1.) What two things do all of the derivatives of the trig functions above have in common? 2.) What do all the derivatives of the functions that start with a c have in common? 3.) How are the derivatives of the cotangent and tangent different than the derivatives of thecosine and sine? 4) Write the formulas for the derivatives of the secant and the cosecant. Let u be theargument for the trig function.Function Derivative f(u) = csc (u)f(u) = sec (u)What do these formulas have in common? For problems 5 through 10, find the derivatives. Show all work.5) f(x) = 2 sin (3x2 – 2) 6) f(x) = - cos(x2 – 5x) 7) f(x) = ½ tan(x4) 8) f(x) = 6 cot (x3 – 2x2) 9) f(x) = -2 sec (x2) 10) f(x) = 4csc(2x3 – 1) Combining Rules: Trig Derivatives + the Product Rule11) f(x) = cos(2x) sin(2x) Find the derivative. Show all work. Combining Rules: Trig Derivatives + the Quotient Rulesec(5 x )12) f (x)=Find the derivative. Show all work and factor/simplify your answer.x2 Combining Rules: Trig Derivatives + the Chain Rule. Recall: sin2(x) = (sin(x))213) Find the derivative: f(x) = tan3(3x) Simplify the final answer to cosines and sines. When trying to find some of the more complicated trig derivatives, the basic reciprocal andPythagorean identities can help. Reciprocal relationships for the 6 trig functions:1secθ1sinθ =cscθ1sin xtan x==cot x cos xcosθ= 1cosθ1cscθ=sinθ1cos xcot x==tan x sin xsecθ= Sometimes, if you first simplify a trigonometric expression, finding the derivative becomes simpler.14) Simplify, then differentiate: f (x)= csc(3 x)cot (3 x) . Show all work. 15) Simplify, then differentiate: f(x) = tan(2x) cot(2x) 16) Simplify, then differentiate: f(x) = tan(5x + 3) csc(5x + 3) 17) In studying water waves on Lake Michigan, the movement of the waves was determined tobe s(t) = .7sin(2t) + .4 cos(t). Find the velocity and acceleration at t = 1 sec., s in meters.Use radian mode. Part II: Inverse Trigonometric Functions: arcsine or sin-1(u) and arccosine or cos-1(u)f(x) = arcsin(2x)2f ' (x)=√ 1−4 x2 f(x) = 3sin-1(x2) f(x) = arccos(5x)−5f ' (x)=√ 1−25 x 2 f(x) = cos-1(x3)−3 x 2f ' →( x )=√1−x 6 f ' (x)= 6x√ 1−x 4 18) What is the formula for the derivative of sin-1(u)? What about for cos-1(u)?How are the two formulas different? 19) Find the derivative: f(x) = 2arcsin (3x) 20) Find the derivative: f(x) = -arccos(5x2) arcTangent or tan-1(u) and arcCotangent or cot-1(u)f(x) = arctan(3x)3f ' (x)=21+9 x f(x) = 7tan-1(x2) f(x) = arccot(5x)−6f ' ( x)=21+36 x f(x) = cot-1(x3)−2 x 2f ' (x)=1+4 x 6 f ' (x)= 14 x41+ x 21) What are the formulas for the derivative of arctangent and arccotangent? Let u be theargument for the trig function. How are these formulas different from the other derivativeformulas for inverse trig functions? 22) Find the derivative: f(x) = ½ arctan(x4 + 3x). Simplify your answer. 23) Find the derivative: f(x) = 6 arccot (7x3). Simplify your answer. 24) Below is the graph of one period of f(x) = sin(x) and its derivative f ’(x) = cos x.Answer the questions a-e below.Solid linedotted line 1 π2 π 3π2 2π -1 a) What does the graph of f ’(x) on the intervalinterval? (0, π )2 say about the slope of the sin(x) on that b) What does the graph of f ’(x) at x = π2 say about the slope of sin(x) at that point? c) What does the graph of f ’(x) on the interval ( π2 , 32π ) say about the slope of sin(x) on that interval? d) What is the slope of f(x) at x = 3π2 ? e) What is the equation of the line that is tangent to f(x) at x = π2 and at x = 3π?2
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