Periodic functions and applications iii
This presentation is the property of its rightful owner.
Sponsored Links
1 / 35

Periodic Functions And Applications III PowerPoint PPT Presentation


  • 102 Views
  • Uploaded on
  • Presentation posted in: General

Periodic Functions And Applications III. Significance of the constants A,B,C and D on the graphs of y = A sin(Bx+C) + D, y = A cos(Bx+C) + D Application of periodic functions Solution of simple trig equations within a specified domain Derivatives of functions involving sin x and cos x

Download Presentation

Periodic Functions And Applications III

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Periodic functions and applications iii

Periodic Functions And Applications III

Significance of the constants A,B,C and D on the graphs of y = A sin(Bx+C) + D, y = A cos(Bx+C) + D

Application of periodic functions

Solution of simple trig equations within a specified domain

Derivatives of functions involving sin x and cos x

Applications of the derivatives of sin x and cos x in life-related situations


Periodic functions and applications iii

Periodic Functions And Applications III


Fm page 118 ex 5 8

REVISION

FM Page 118 Ex 5.8

New Q

Page 351 Ex 10.1

No. 1-12 (parts a & b only), leave out no.10


Solving trig onometric equations

Solving Trig(onometric) Equations

  • ModelFind all values of x (to the nearest minute) where 0< x <360 for which

  • (a) sin x = 0.5

  • (b) tan x = -1


Periodic functions and applications iii

sin is positive

 angle is in Q1 or Q2

(a) sin x = 0.5

x = 30 or x = 180 - 30

= 30 or 150

30

30

Value of sin x is 0.5

 30 off x-axis


Periodic functions and applications iii

tan is negative

 angle is in Q2 or Q4

(b) tan x = -1

x = 180 - 45 or x = 360 - 45

= 135 or 315

45

45°

Value of tan x is -1

 45 off x-axis


Fm page 119 ex 5 9 1 orally

New Q

Ex 10.3

Page 363 2,6

FM Page 119 Ex 5.91 (orally)


General solution of a trig function

General Solution of a Trig Function

So there appears to be more than one solution

cos θ = 0.643

θ = cos-1 (0.643)

θ ≈ 50°

But cos 310° = 0.643 also

So, how many solutions are there?


Periodic functions and applications iii

y=0.643

cos curve

cos θ = 0.634

θ = 50° orθ = 310°

or θ = 50° + 360°orθ = 310° + 360°

or θ = 50° + 2 x 360°orθ = 310° -360°

or θ = 50° + 3 x 360°orθ = 310° - 2 x 360°

or θ = 50° - 360°orθ = 310° - 3 x 360°

or θ = 50° - 2 x 360° etc

θ = 50° + 360° x n θ = 310° + 360° x n


The general solution for cos 0 643

θ = 50° + 360n

θ = 310° + 360°n

The General Solution forcos θ = 0.643

For all integer values of n


Periodic functions and applications iii

Model:

Find all values of x (to the nearest minute) for which

(a) sin x = 0.5


Periodic functions and applications iii

sin is positive

 angle is in Q1 or Q2

(a) sin x = 0.5

x = 30 or x = 180 - 30

= 30 or 150

 general solution is

x = 30 + n x 360 or x = 150 + n x 360

30

30

Value of sin x is 0.5

 30 off x-axis


Periodic functions and applications iii

ModelFind all values of x (to the nearest minute) where 0 ≤ x ≤ 360 for which

(a) sin2x = 0.25

(b) tan 3x = -1


Periodic functions and applications iii

sin is pos or neg

 angle is in Q1,Q2,Q3 or Q4

(a) sin2x = 0.25

sin x = ± 0.5

x = 30 or x = 150 or x = 210 or x = 330

30

30

30

30

Value of sin x is 0.5

 30 off x-axis


Periodic functions and applications iii

tan is negative

 angle is in Q2 or Q4

(b) tan 3x = -1

3x = 135° + 360n or 3x = 315 + 360n

x = 45 + 120n or x = 105 + 120n

 45, 165, 285, 105, 225, 345

45

45°

Value of tan x is -1

 45 off 3x-axis


Fm page 304 ex 14 16 1 2 3 with gc

New Q

Page 371 Ex 10.5

1, 2, 4,10, 14

FM Page 304 Ex 14.161 -23 with GC


Derivatives of functions involving sin x and cos x

Derivatives of functions involving sin x and cos x

  • Derivatives of functions involving sin x and cos x


Derivative of sin x and cos x

Derivative of sin x and cos x

y = sin x

 dy = cos x

dx

y = cos x

 dy = -sin x

dx


Model find the derivative of a sin 2x b sin 3 2x c sin 2 x cos3x d sin 3x

Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x)

  • do some examples on Graphmatica


Model find the derivative of a sin 2x b sin 3 2x c sin 2 x cos3x d sin 3x1

Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x)

________________________________________

(a) y = sin 2x

= sin u where u = 2x

dy= cos u du= 2

du dx

dy=dy . du

dx du dx

= 2 cos u

= 2 cos 2x


Model find the derivative of a sin 2x b sin 3 2x c sin 2 x cos3x d sin 3x2

Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x)

________________________________________

(b) y = sin32x ( = (sin 2x)3 )

= u3 where u = sin 2x

dy= 3u2du= 2 cos 2x

du dx

dy=dy . du

dx du dx

= 3u2 . 2 cos 2x

= 6 sin22x cos 2x


Model find the derivative of a sin 2x b sin 3 2x c sin 2 x cos3x d sin 3x3

Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x)

________________________________________

(c) y = sin2x cos3x

= uv where u = sin2x and v = cos3x

du= 2 sinx cosx dv = -3 sin3x

dx dx

dy= u dv + v du

dx dx dx

= -3 sin3x sin2x + cos3x  2 sin x cos x

= -3 sin3x sin2x + 2 cos3x sin x cos x


Model find the derivative of a sin 2x b sin 3 2x c sin 2 x cos3x d sin 3x4

Model : Find the derivative of(a) sin 2x(b) sin32x(c) sin2x cos3x(d) sin(π-3x)

________________________________________

(d) y = sin (π-3x)

= sin 3x

dy= 3 cos 3x

dx


Newq p50 2 4 no 1 a b d f h j 3 8 all fm page 447 ex 19 5 1 4 5

NEWQ P50 2.4No. 1(a,b,d,f,h,j), 3 – 8 (all)FM Page 447 Ex 19.51,4,5


Model find the gradient of the curve y sin 2x at the point where x 3

Model : Find the gradient of the curve y = sin 2x at the point where x = π/3


Model find the gradient of the curve y sin 2x at the point where x 31

Model : Find the gradient of the curve y = sin 2x at the point where x = π/3


Trig functions and motion

Trig functions and motion

  • Consider the motion of an object on the end of a spring dropped from a height of 1m above the equilibrium point which takes 2π seconds to return to the starting point.

1m

0m

-1m

s = cos t

v = -sin t

a = -cos t


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(a) How far from the fixed point is the object at the start?(b) How long does it take for the object to return to its starting point?(c) Find the object’s velocity (i) at the start (ii) as it passes the fixed point (iii) after 2 sec(d) Find its acceleration as it passes the fixed point


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(a) How far from the fixed point is the object at the start?

  • At the start, t = 0

  • When t = 0, s = 4 cos(3x0)

  • = 4 cos0

  • = 4 x 1

  • = 4

  • i.e. at the start, object is 4 metres from the fixed point.


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds. (b)How long does it take for the object to return to its starting point?

Returns to starting point  s = 4

 4 cos3t = 4

 cos3t = 1

 3t = 2nπ

 t = 2nπ/3

 t = 2π/3, 4π/3, 6π/3, …

i.e. first returns to starting point after 2π/3 secs


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(c) Find the object’s velocity (i) at the start(ii) as it passes the fixed point (iii) after 2 sec

s = 4 cos3t

 v = -12 sin3t

(i) At the start

When t = 0, v = -12 sin 0

= 0


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(c) Find the object’s velocity (i) at the start (ii) as it passes the fixed point(iii) after 2 sec

  • (ii) at what time does it pass the fixed point

     s = 0

  •  4 cos 3t = 0

  • cos 3t = 0

  • 3t = π/2, …

  • t = π/6, …

  • When t = π/6, v = -12 sin 3π/6

  • = -12


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(c) Find the object’s velocity (i) at the start(ii) as it passes the fixed point (iii) after 2 sec

  • (iii) After 2 sec

  • v = -12 sin 3x2

  • = -12 sin 6

  • = 3.35


Periodic functions and applications iii

Model: The motion of an object oscillates such that its displacement, s, from a fixed point is given by s = 4cos3t where s is in meters and t is in seconds.(d) Find its acceleration as it passes the fixed point

  • s = 4 cos3t

  • v = -12 sin3t

  • a = -36 cos3t

    It passes the fixed point when t = π/6,

    a = -36 cos 3π/6

    = 0


Newq p59 ex 2 6 no 1 4 7

NEWQ P59 Ex 2.6No. 1 – 4, 7


  • Login