11.18 Applications of the differentiation of trigonometric functions
Find the gradient of the normal to the curve y = x^{2} + \cos \left( \pi x\right), at the point where x = 1.
For the following functions, find the equation of the tangent at the given point:
y = - 5 x \sin x at \left(\pi, 0\right).
y = \cos 6 x at the point where x = \dfrac{\pi}{18}.
Consider the function f \left( x \right) = \cos 2 x - \sqrt{3} \sin 2 x.
Find the first positive value of x at which the curve intersects the x-axis.
Find the gradient of the tangent at this point.
Hence determine the positive angle, \theta, that the tangent makes with the positive \\x-axis at this point. Round your answer to the nearest degree.
Consider the function f \left( x \right) = - 4 \cos 3 x.
Find \theta, the angle that the tangent at x = \dfrac{\pi}{12} makes with the positive x-axis. Round your answer to the nearest degree.
For each of the following functions, find:
The derivative function.
The gradient of the normal at the point \left(\dfrac{\pi}{4}, 1\right).
The equation of the normal at the point \left(\dfrac{\pi}{4}, 1\right).
For the function y = \dfrac{\sin x}{x}, find:
The derivative function.
The gradient of the normal at the point \left( \dfrac{\pi}{2}, \dfrac{2}{\pi} \right).
The equation of the normal at the point \left( \dfrac{\pi}{2}, \dfrac{2}{\pi} \right).
Consider the function y = \dfrac{\sin x}{1 + \cos x}.
Find \dfrac{dy}{dx}.
Determine the number of turning points function y has.
Find the x-coordinate of each point of inflection on the curve y = 3 \cos \left( 2 x + \dfrac{\pi}{4}\right) for the interval 0\leq x\leq 2\pi.
Consider the function f \left( x \right) = - 4 \sin \left(x + \dfrac{\pi}{6}\right) on the interval 0 \leq x \leq 2 \pi.
Find the coordinates of any turning points.
Find an expression for f'' \left( x \right).
Hence classify the turning points.
For each of the following functions on the interval 0 \leq x \leq 2\pi:
Find the axes intercepts.
Find an expression for f' \left( x \right).
Determine the coordinates of any stationary points.
Find an expression for f'' \left( x \right).
Hence classify the stationary points.
Sketch the graph of f \left( x \right).
Determine the possible values of k, given that y = \cos k x satisfies the following equation for all values of x: y'' + 25 y = 0
The displacement of an object is a measure of how far it is from a fixed origin point at time, \\t seconds. Displacement to the left of the origin point is negative and displacement to the right of the origin is positive. At time t seconds, velocity is given by v \left( t \right) = x'(t), and acceleration is given by a \left( t \right) = x''(t).
If the displacement of an object x about a fixed origin is given by: x \left( t \right) = - 5 \cos 2 t
Describe the initial position of the object, from the fixed origin point.
Write an expression for v \left( t \right).
Show that a \left( t \right) = - 4 x \left( t \right).
Find the acceleration of the object, when t = \dfrac{\pi}{2}.
A particle moves in a straight line and its displacement, x \text{ cm}, from a fixed origin point after t seconds is given by the function:
x \left( t \right) = \sin t - \sin t \cos t - 5 tState the initial displacement of the particle.
The velocity is the rate at which displacement changes over time. Find an equation for v \left( t \right), in terms of \cos t.
Hence find the initial velocity, v \left( 0 \right).
The arm of a pendulum swings between its two extreme points A to the left and B to the right. Its horizontal displacement x \text{ cm} from the centre of the swing, at time t seconds after it starts swinging, is given by:
x \left( t \right) = 16 \sin 3 \pi tFind the intial position of the pendulum.
Find the maximum distance of the pendulum from the central position of its swing.
Find an expression for the velocity function, v \left( t \right)= x'\left( t \right).
Find the first two times at which the pendulum comes to rest, v \left( t \right)= 0 .
Find the displacement of the pendulum when it first comes to rest.
Find the displacement of the pendulum when it comes to rest for the second time.
Hence determine the distance between the two points A and B.
The arm of a pendulum swings between its two extreme points A to the left and B to the right. Its horizontal displacement x \text{ cm} from the centre of the swing at time t seconds after it starts swinging is given by:
x \left( t \right) = 19 \sin 4 \pi tFind an expression for the velocity function, v \left( t \right)=x'\left( t \right).
Find the maximum velocity of the pendulum.
Find the first two times, after it starts swinging, at which the pendulum reaches its maximum velocity, v' \left( t \right) = 0 and x' \left( t \right) < 0 .
Describe the position of the pendulum when it reaches its maximum velocity.