The graph of a sinusoidal function is as shown.
a) Identify the x-values in the interval 0 \leq x \leq 2\pi
where the slope is
i) zero
ii) a local maximum
iii) a local minimum
b) Sketch a graph of the instantaneous rate of change of this function with respect to x
.
a) Sketch a graph of the function y= -3\sin x
b) Sketch a graph of the instantaneous rate of change of this function with respect to x
.
Find the derivative of each function with respect to x
.
a) y=\cos x
Find the derivative of each function with respect to x
.
b) f(x) = -2\sin x
Find the derivative of each function with respect to x
.
c) y = \cos x - \sin x
Find the derivative of each function with respect to x
.
d) f(x) = 3\sin x - \pi\cos x
Determine the slope of the function y = 4 \sin x
at x= \dfrac{\pi}{3}
a) Find the equation of the line that is tangent to the curve y = 2 \sin \theta + 4 \cos \theta
at \theta = \dfrac{\pi}{4}
.
b) Find the equation of the line that is tangent to to the curve y = 2 \cos \theta - \dfrac{1}{2}\sin \theta
at \theta = \dfrac{3\pi}{2}
c) Graph each function and it's tangent lines from parts a) and b).
Differentiate.
y = -\cos^2x
Differentiate.
y = \sin 2 \theta - 2\cos 2\theta
Differentiate.
f(\theta) = -\dfrac{\pi}{2} \sin (2\theta - \pi)
Differentiate.
f(x) = \sin (\sin x)
Differentiate.
f(x) = \cos (\cos x)
Differentiate.
f(\theta) = \cos 7\theta - \cos (5\theta)
Determine the derivative of each function.
y = 3x\sin x
Determine the derivative of each function.
f(t) = 2t^2\cos 2t
Determine the derivative of each function.
y = \pi t\sin (\pi t - 6)
Determine the derivative of each function.
y = \cos (\sin \theta) + \sin (cos \theta)
Determine the derivative of each function.
f(x) = \cos^2 (\sin x)
Determine the derivative of each function.
f(\theta) = \cos 7\theta - \cos ^2 5\theta
a) Find and equation of a line that is tangent to the curve f(x) = 2\cos 3x
and whose slope is a maximum.
b) Is this the only possible solution? Explain. If there are other possible solutions, how many solutions are there?
The voltage of the power supply in an AC-DC couple circuit is given by the function V(t) = 130 \sin 5t + 18
, where t is time, in seconds, and V
is the voltage in volts, at time t
.
a) Find the maximum and minimum voltages and the times at which they occur.
b) Determine the period, T
, in seconds, the frequency, f
, in hertz, and the amplitude, A
, in volts, for this signal.
A block is positioned on a frictionless ramp as shown.
The gravitational force component acting on the block directed along the ramp as a function of the angle of inclination, \theta
, is F= mg\sin \theta
, where m
is the mass of the block and g
is the acceleration of gravity.
a) For what angle of inclination, 0 \leq \theta \leq \dfrac{\pi}{2}
, will this force be
i) a maximum?
ii) a minimum?
b) Explain how you found your answers in part a). Explain why these answers make sense.
Newton's second law of the motion was originally written as F= \dfrac{dp}{dt}
, where
F
is the force of acting on the object, in newtons
p
is the momentum of the object, in kilogram metres per second, given by the equation p =mv
(where m
is the mass of the object, in kilograms, and v
is the velocity of the object in metres per second).
t
is time, in seconds.
a) Assuming an object's mass is constant, use this definition to show that Newton's second law can also be written as F = ma
, where a
is the acceleration of the object.
b) Suppose an object is oscillating such that its velocity, in metre per second, at time t
is given by the function v(t) = 2\cos 3t
, where t
is time, in seconds. Find the times when the force acting on the object is zero.
c) What is the speed of the object at these times?