Consider the graph of y = \sin x:
Describe the motion of the graph of y = \sin x.
State whether each of the following describes the nature of the gradient of the curve:
Between points where the gradient is 0, the gradient is always negative.
The gradient of the curve is 0 once every 2 \pi radians.
Between points where the gradient is 0, the gradient is always positive.
The gradient of the curve is 0 once every \pi radians.
Between points where the gradient is 0, the gradient is positive and negative alternately.
List the intervals in the above graph in which the gradient of y = \sin x is positive.
List the intervals in the above graph in which the gradient of y = \sin x is negative.
Given that the gradient at x=0 is 1, sketch a graph of the gradient function y' for -2\pi \leq x \leq 2 \pi.
What is the equation of the gradient function y' graphed in the previous part?
Consider the graph of y = \cos x:
Describe the motion of the graph of y = \cos x.
State whether each of the following describes the nature of the gradient of the curve:
The gradient of the curve is only 0 once every 2 \pi radians.
Between points where the gradient is 0, the gradient is positive and negative alternately.
Between points where the gradient is 0, the gradient is always positive.
Between points where the gradient is 0, the gradient is always negative.
The gradient of the curve is 0 once every \pi radians.
List the intervals in the above graph in which the gradient of y = \cos x is positive.
List the intervals in the above graph in which the gradient of y = \cos x is negative.
Given that the gradient at x=\dfrac{\pi}{2} is - 1, sketch a graph of the gradient function y' for -2\pi \leq x \leq 2 \pi.
What is the equation of the gradient function y' graphed in the previous part?
Consider the function y = \sin a x, where a is a constant.
Let u = a x. Rewrite the function in terms of u.
Determine \dfrac{d u}{d x}.
Hence, determine \dfrac{d y}{d x} in terms of x.
Consider the function f \left( x \right) = \sin 3 x. State f' \left( x \right).
Hence, evaluate f' \left( \dfrac{\pi}{6} \right).
Differentiate the following functions:
Differentiate the following functions:
Consider the function y = \cos \left( 2 x\right) + x.
Differentiate y = \cos \left( 2 x\right) + x.
Solve for the first positive value of x at which \dfrac{d y}{d x} = \sqrt{3} + 1. Give your answer in radians.
Consider the graph of y = \cos x which is the gradient function of y = \sin x. A number of points have been labelled on the graph:
Name the point on the gradient function that corresponds to the following locations on the graph of y = \sin x :
Where y = \sin x is increasing most rapidly.
Where y = \sin x is decreasing most rapidly.
Where y = \sin x is stationary.
Consider the graph of y = - \sin x which is the gradient function of y = \cos x. A number of points have been labelled on the graph.
Name the point on the gradient function that corresponds to the following locations on the graph of y = \cos x :
Where y = \cos x is increasing most rapidly.
Where y = \cos x is decreasing most rapidly.
Where y = \cos x is stationary.
There is an expansion system in mathematics that allows a function to be written in terms of powers of x. The value of \sin x and \cos x, for any value of x, can be given by the expansions below:
\sin x = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \dfrac{x^{7}}{7!} + \dfrac{x^{9}}{9!} - \ldots
\cos x = 1 - \dfrac{x^{2}}{2!} + \dfrac{x^{4}}{4!} - \dfrac{x^{6}}{6!} + \dfrac{x^{8}}{8!} - \ldots
Use the expansions to find:
\dfrac{d}{dx} \left(\sin x\right) in terms of x.
\dfrac{d}{dx} \left(\cos x\right) in terms of x.
Hence express the derivatives of \sin x and \cos x in simplest form.
Consider the graph of y = \tan x:
Is the graph of y = \tan x constantly increasing or decreasing?
State whether each of the following describes the nature of the gradient of the curve:
The gradient is always negative.
The gradient to the curve is never 0.
The gradient function has the same period as the curve itself.
The gradient increases more and more rapidly as the curve approaches the asymptotes.
The gradient is always positive.
The gradient to the curve is 0 every \pi radians.
On the following graph, the tangent lines at the x-intercepts of the curve have been graphed for 0 \leq x \leq 2 \pi:
Using the graph, write down the gradient to the curve at \\ x = 0, \pi, 2 \pi, 3 \pi, \ldots
The gradient function of y = \tan x is y'. Which of the following is the correct graph of y'?
What is the equation of the gradient function y'?
Consider \tan x = \dfrac{\sin x}{\cos x} and differentiate y = \tan x using the quotient rule.
Differentiate:
Consider the function y = \tan \left(\dfrac{x}{2}\right) + 2.
Differentiate y = \tan \left(\dfrac{x}{2}\right) + 2.
Solve for the first positive value of x at which \dfrac{d y}{d x} = 2.
Differentiate the following:
Find y' for the function defined by x y = \sin 5 x.
For each of the following curves and given points:
Find an expression for \dfrac{dy}{dx}.
Find the exact value of the gradient of the curve at the given point.
y = \cos 3 x at x = \dfrac{\pi}{18}.
y = \cos 4 x at x =- \dfrac{\pi}{3}.
y = \sin 4 x at x = \dfrac{\pi}{16}.
Consider the function y = \sin 2 x + 4.
Differentiate y = \sin 2 x + 4.
Solve for the first positive value of x at which \dfrac{d y}{d x} = 1. Give your answer in radians.
State whether the following functions have the same gradient at x = \dfrac{\pi}{6}:
Consider the function y = \cos 6 x. Find the equation of the tangent to the graph at x = \dfrac{\pi}{18}.
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.
Consider the function f \left( x \right) = \cos 2 x - \sqrt{3} \sin 2 x and the first point to the right of the origin at which the curve intersects the x-axis.
Find \theta, the angle that the tangent makes with the positive x-axis at this point. Round your answer to the nearest degree.
Consider the function y = 4 \tan x.
Solve for all the values of x, where 0 \leq x \leq 2 \pi, for which the gradient of the tangent to the curve is 8.
For the function y = \sin x:
What is the equation of the tangent to the graph at each maximum?
What is the equation of the tangent to the graph at each minimum?
Find the equation of the tangent to the graph at x = \pi.
For the function y = 2 \cos x + 1:
What is the equation of the tangent to the graph at x = 0?
What is the equation of the tangent to the graph at each minimum?
Find the equation of the tangent to the graph at x = \dfrac{\pi}{2}.
Consider the function y = \tan \left( 2 x\right) + 3. Find the equation of the tangent to the graph at \\x = \dfrac{\pi}{2}.
Differentiate the following: