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Math 31 – Derivatives of Trigonometric Functions Review 1. Evaluate the following limits: a. lim x cot x c. omit e. g. Trig review b. x 0 d. x sin 2 2 lim x 0 sin x omit lim x 0 x 0 sin x 5x 1 cos x x 0 sin x lim f. h. lim 1 cos 2 x x 0 2 x cos x 1 lim tan x tan 2 x 1 2. Find dy for each of the following: dx a. y cos3 2 x 1 b. y sec 2 x 1 c. x y sin 2 x cos 2 d. y e. 1 y sin 2x 1 f. y cot 2 x 2 2 g. y tan 2 sin 3x h. y Trig review 3 x3 cos x cos x sec x tan x 2 3. Find an equation of the tangent for each of the following: 5 6 a. y 2 cos 2 x at x b. y 2sin x at x c. y csc x 2sin x at x Trig review 3 6 3 4. Determine the maximum and minimum points, the concavity and points of inflection for each of the following: a. y sin 2 x on the interval 0, 2 b. y sin 2 x Trig review x on the interval 0, 2 4 5. Find the derivatives of the following using implicit differentiation: a. x 2 sin y 2 y cos x 6 b. cos x y cos x y 1 c. x sin y cos 2 y cos y Trig review 5 6. A 6 m ladder is leaning against a wall and begins to slide. The foot of the ladder slides outward at a rate of 0.2 m/s. At what rate is the angle between the ladder and the wall changing when the top is 3 m from the ground? 7. The hypotenuse of a right triangle has length of 10 cm. Determine its maximum possible area. 8. A vehicle moves along a straight road at a speed of 4 m/s. A searchlight is located on the ground 20 m from the road and is focused on the vehicle. At what rate (in rad/sec) is the searchlight rotating when the vehicle is 15 m from the point on the road closest to the searchlight? 9. A lighthouse is situated 200 m from a straight shoreline. The light rotates clockwise at 2 revolutions per minute. At what speed is the beam of light moving along the shore when the angle between the beam and the shore is ? 6 Trig review 6 Answer Key 1. 2. 3. 4. a. 1 b. 1 5 c. omit d. 0 e. 0 f. 0 g. omit h. a. 6cos2 2 x 1 sin 2 x 1 1 2 b. 6sec 2 x 1 tan 2 x 1 2 x 1 c. 1 x x sin x 4cos x cos sin x sin 2 2 2 2 x2 sec x x tan x 3 OR x2 sec2 x 3cos x x sin x e. f. 4 x cot x 2 2 csc 2 x 2 2 g. 6 tan sin 3x sec2 sin 3x cos3x h. a. cos x 6 3x 6 y 5 3 15 0 b. 3x 3 y 3 3 0 c. 6 3x 2 y 3 2 0 a. 5 3 7 min , 1 , , 1 max ,1 , ,1 3 4 4 4 4 , inflection point 2 , 0 , , 0 , 2 , 0 a. b. 1 cos 2x 1 2 x 1 2 2 5 3 min , 0.06 , max , 0.28 , inflection point , 0.11 , , 0.68 12 12 4 4 2 2 x sin y y sin x y cos x 2 x 2 y cos y 2 sin x y sin x y y sin x y sin x y sin y x cos y 2sin 2 y sin y 0.067 rad/sec 25 cm 2 0.128 rad/sec 3200 m/min c. 6. 7. 8. 9. 3 d. b. 5. 3 Trig review y 7