2/3 Unit Math Homework for Year 12

Transcription

1 Yimin Math Centre 2/3 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 12 Trigonometry The Derivative of Trigonometric Functions Using Graphs of Trigonometric Functions Applications of the Derivative Practical Exam Questions Miscellaneous Exercise This edition was printed on July 18, Camera ready copy was prepared with the L A TEX2e typesetting system. 2/3 Unit Math Homework for Year 12

2 Year 12 Topic 12 Homework Page 1 of Trigonometry The Derivative of Trigonometric Functions Definition: d d (sin x) = cos x d (cos x) = sin x d d (tan x) = sec2 x sin(ax + b) = a cos(ax + b) cos(ax + b) = a sin(ax + b) d tan(ax + b) = a sec2 (ax + b) Example Find the derivative of : 1. y = tan(5x 2) Solution: Let u = 5x 2, so du = 5, dy = dy du du = sec 2 u 5 = 5 sec 2 (5x 2) dy and y = tan u, so = du sec2 u 2. y = ex cos x Solution: Let u = e x, so du = ex, and v = cos x, so dv = sin x du dv dy v. u. = v 2 = cos x.ex e x. sin x cos 2 x = ex (cos x+sin x) cos 2 x 3. y = ln(tan x) Solution: Let u = tan x, so du = sec2 x and y = ln u, so dy = 1 u dy = dy du = 1 du u sec2 x = 1 tan x sec2 x = cos x 1 sin x cos 2 x 1 = cos x sin x = sec x cosec x

3 Year 12 Topic 12 Homework Page 2 of 11 Example y = sin(e x ) Solution: Let u = e x, so du = ex, and y = sin u, so dy = cos u dy = dy du du = cos u.e x = e x cos(e x ) 2. y = cos 3 4x Solution: Put y = cos 3 4x = (cos 4x) 3, let u = cos 4x, so du By the chain rule dy = dy du du = 3u 2 4 sin 4x and y = u 3, so dy du = 3u2 = 12 sin 4x(cos 4x) 2 = 12 sin 4x cos 2 4x = 4 sin 4x 3. y = 5 cos x 12 sin x, show that d2 y 2 + y = 0 Solution: dy d 2 y 2 = 5 sin x 12 cos x = 5 cos x 12. sin x = 5 cos x + 12 sin x So d2 y + y = ( 5 cos x + 12 sin x) + (5 cos x 12 sin x) = 0 4. Find the second derivative of sin 2 x Solution: Put y = sin 2 x = (sin x) 2 = u 2, so dy dy = dy du du = 2 sin x cos x by the product rule: d2 y 2 du = 2 sin x.( sin x) + cos x.2 cos x = 2(cos 2 x sin 2 x) = 2u and u = sin x, du = cos x

4 Year 12 Topic 12 Homework Page 3 of 11 Exercise Differentiate the following: 1. y = sin 3 2x 2. y = cos(e x ) 3. y = cos(2x 3) 4. y = x 3 cos 2x 5. y = tan x x

5 Year 12 Topic 12 Homework Page 4 of Using Graphs of Trigonometric Functions Example Solve sin x = 2 2, 0 x 2π Solution: (1) Sketch the graphs of y = sin x and y = 2, and note where they intersect. 2 (2) Find the first value using your calculator or from the standard values. A is where x = π 4 or (45 ), sin x = 2 2, (3) By symmetry, B is where x = π π 4 = 3π 4, x = π 4, 3π 4 Example Solve cos 2x = 1 2, 0 x 2pi Solution: Sketch the graphs y = cos 2x and y = 1 2. y = cos 2x : n = 2, Period is 2π 2 = π There are four solutions, from the standard values, if cos 2x = 1 : 2x = π 2 3 x = π (solution A). 6 Using symmetry, B is where x = π π 6 = 5π 6 C is where x = π + π 6 = 7π 6 and D is where x = 2π π 6 = 11π 6

6 Year 12 Topic 12 Homework Page 5 of 11 Exercise Solve the following equations, where 0 x 2π 1. sin x = tan x = 1 3. sin 3x = 1 2

7 Year 12 Topic 12 Homework Page 6 of Applications of the Derivative Example Find the gradient of the tangent to the curve y = 2 sin x at the point where x = π 3 dy Solution: If y = 2 sin x, = 2 cos x at x = π, dy = 2 cos π 3 3 = = 1, the gradient of the curve is m = 1 Example Find in general form the equation of the tangent to the curve y = cos 3x at the point where x = π 6 Solution: If y = cos 3x, dy dc = 3 sin 3x. At x = π 6, dy = 3 sin(3 π 6 ) So m = 3. Also, at x = π 6, y = cos(3 π 6 ) = 0, So the point is the point ( π 6, 0) Now y y 1 = m(x x 1 ) y 0 = 3(x π 6 ) y = 3x + π 2. the equation in general form is 3x + y π 2 = 0 Exercise Find the equation of the tangent to the curve y = 2 cos x at the point where x = π 6

8 Year 12 Topic 12 Homework Page 7 of 11 Exercise Find the equation of the tangent to the following curves: 1. y = 3 sin x at the point where x = π 3 2. y = cos 2x at the point where x = π 4 3. y = cos πx 1 at the point where x = 1 2.

9 Year 12 Topic 12 Homework Page 8 of Practical Exam Questions Exercise Show that x = π is a solution of sin x = 1 tan x On the same set of axes, sketch the graphs of the functions y = sin x and y = 1 tan x for 2 π x π. 3. Hence find all solutions of sin x = 1 2 tan x for π 2 < x < π Use your graphs to solve sin x 1 2 tan x for π 2 < x < π Find the equation of the tangent to y = cos πx 1 at the point where x = 1.

10 Year 12 Topic 12 Homework Page 9 of 11 Exercise Solve 2 sin 2 x 3 = 1 for π x π. 2. Solve 2 sin x = 1 for 0 x 2π 3. Find the equation of the tangent to the curve y = cos 2x at the point whose x-coordinate is π 6.

11 Year 12 Topic 12 Homework Page 10 of Miscellaneous Exercise Exercise Solve the equation cos 2x = 3 2 where 0 x 2π. 2. Solve the equation sin 3x = 1, where 0 x 2π. 3. If f(x) = x cos x, find f ( π 3 ). 4. Differentiate y = e 2x sin 2x

12 Year 12 Topic 12 Homework Page 11 of 11 Exercise If y = sin x cos x, show that d2 y 2 = y. 2. If dy = cos 3x, and y = 1 when x = 0, Find y when x = π 4 3. Defferentiate y = e 3x cos 3x 4. Find the derivative of tan(ln x)