Maths Quest Maths B Year 12 for QueenslandChapter 6 The calculus of periodic functions WorkSHEET 6.21
WorkSHEET 6.2The calculus of periodic functions
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Maths Quest Maths B Year 12 for QueenslandChapter 6 The calculus of periodic functions WorkSHEET 6.21
1 / Find the equation of the tangent to the curve y=sin x at the point (, 0). /2 / Find the equation of the tangent to the curve y= cos 2x at the point . /
3 / Find the equation of the tangent and the normal to the curve y = 3 sin 2x at the point where x = . / y= 3 sin 2x
= 6 cos 2xat x =
= 6 cos
= 3
Equation of tangent
y= 3(x)
2y 3= 6(x)
2y 3= 6x
6x + 2y + 3= 0
Equation of normal
y= (x)
6y 9= 2(x)
6y 9= 2x +
2x + 6y 9= 0
4 / Find the points on the curve y=2sin3x where the tangent is parallel to the xaxis. / y=2sin3x
= 6 cos 3x
Tangent is parallel to the x-axis where = 0
6 cos 3x= 0
cos 3x= 0
3x= , , , , ….
x= , , , , ….
5 / The depth of water (in metres) at an inlet is given by the equation
D = 8 + 2 cos
Where t is the number of hours since the last high tide.
(a)Find the depth of water at high tide.
(b)Find when low tide will occur (i.e. when the water depth is a minimum). / (a) At t= 0
D= 8 + 2 cos
= 8 + 2 cos 0
= 10 m
(b) = sin = 0 when
sin = 0
= 0, , 2
t= 0, 4, 8,….
At t = 0 is high tide
At t = 4 will be low tide
Low tide will occur 4 hours after the high tide.
6 / The velocity of a piston is given by the equation v = 1.5 sin 4t where t is the number of seconds since the engine was started.
Find the maximum and minimum velocity of the piston and when they first occur. / v = 1.5 sin 4t
= 6 cos 4t
Maximum and minimum occur when = 0
6 cos 4t= 0
cos 4t= 0
4t= , , …
t= , ,
At t = v= 1.5 sin 4()
= 1.5 sin
= 1.5
At t = v = 1.5 sin 4()
= 1.5 sin
= 1.5
Maximum velocity = 1.5 m/s after secs
Minimum velocity = 1.5 m/s after secs
7 / A body executing simple harmonic motion with a position x(t) is given by the equation
x(t) = 4 cos 2t + 1
State
(a)the initial position of the body.
(b)The period of the motion.
(c)The frequency of the motion
(d)The amplitude of the motion.
(e)The x value for the centre of the motion. / (a)at t = 0
x(t)= 4 cos 2t + 1
x(0)= 4 cos 4(0) + 1
= 4 cos 0 + 1
= 4 + 1
= 5
(b)T =
=
=
(c)F =
=
a= 4
x= 1
8 / A body is executing simple harmonic motion has the displacement equation
x(t) = 2 sin 3t
(a)Find the equation for velocity.
(b)Find the equation for acceleration.
(c)Use a graphics calculator to show the three equations on the same axes. / (a)x(t) = 2 sin 3t
v(t) = 6 cos 3t
(b)a(t) = 18 sin 3t
(c)
9 / The displacement of a particle executing simple harmonic motion can be given by the equation
x = 4 sin + 8
Find:
(a)the initial position
(b)the position after 3 seconds
(c)the position after 4 seconds
(d)the average velocity in the forth second
(e)an expression for velocity
(f)the initial velocity
(g)the velocity at t = 4
(h)when the particle is stationary. / (a)at t = 0
x= 4 sin + 8
x= 4 sin 0 + 8
x= 8
(b)at t = 3
x= 4 sin + 8
x= 4 sin + 8
x= 8
(c)at t = 4
x= 4 sin + 8
x= 4 sin + 8
x= 2+ 8
(d)2 m/sec
(e)v = cos
(f) at t= 0
v= cos 0
v=
(g) at t= 4
v= cos
v=
=
(h) stationary when v= 0
cos = 0
cos = 0
= , , , …
t= 1.5, 4.5, 7.5 …
10 / The population of frogs in a small pond through year can be approximated by SHM.
The mean population of frogs is 500, the period of the function is 12 months and the amplitude of the function is 200.
Write an equation for P in terms of t, where t is in months. / Equation will be of the form:
P = A sin nt + D
T = =12
2= 12n
n=
A = 200
D = 500
P = 200 sin + 500