Find the exact area bounded by the following:
Find the exact area of the shaded region bounded by the curve y = 3 \cos x and the x-axis, between x=\dfrac{\pi}{2} and x=\dfrac{3\pi}{2}:
Find the exact area of the shaded region enclosed by the curve y = \cos 3 x and the x-axis, between x = 0 and x=\dfrac{\pi}{3}:
Find the exact area of the region enclosed by the curve y = \sin 2 x, the x-axis and the lines x = \dfrac{\pi}{4} and x = \dfrac{3 \pi}{4}:
Find the exact area of the region enclosed by the curve y = 3 \sin x and the x-axis, from x = 0 to x = 2 \pi:
Consider the function y = - \tan x shown:
Show that \dfrac{d}{dx}\left(\ln\left(\cos x \right) \right) = - \tan x
Find the exact area bounded by the curve y = -\tan x, the coordinate axes and the line x = \dfrac{\pi}{4}
Consider the curve y = \sin 3 x on the interval 0 \leq x \leq 2 \pi.
Sketch the graph of the function.
Calculate the exact area of the region bound by the curve y = \sin 3 x, the x-axis, x = 0 and x = \dfrac{\pi}{4}.
Consider the curve y = \cos \left(\dfrac{x}{4}\right).
Find the area bound by the curve and the coordinate axes on the interval 0 \leq x \leq \dfrac{\pi}{2} .
Hence find the area bound by the curve and the x-axis between x = - 4 \pi and x = 4 \pi.
Consider the curve y = 3 \sin 2 x on the interval 0 \leq x \leq 2 \pi.
Sketch the graph of the function.
Find the exact area bound by the curve y = 3 \sin 2 x and the x-axis between:
x = 0 and x = \dfrac{\pi}{2}.
x = \dfrac{\pi}{4} and x = \dfrac{2 \pi}{3}.
Consider the curve y = 4 \sin \left(\dfrac{\pi x}{5}\right).
Determine the period of the curve.
Find the exact area bound by the curve and the x-axis between x = \dfrac{5}{4} and x = \dfrac{5}{2}.
Consider the function y = \tan x on the interval 0 \leq x \lt \dfrac{\pi}{2}. Find the exact area bound by the curve y = 3 \tan x, the coordinate axes and x = \dfrac{\pi}{6}.
Consider the functions y=\sin x and y = \cos x on the interval 0\leq x\leq 2\pi.
Find the points of intersection of the two curves.
Calculate the exact area of the region bounded by the two curves.
Explain why the integral \int_{ - \frac{\pi}{12} }^{\frac{\pi}{12}} \sin 4 x \, dx has a value of 0.
The average rate of change of function f \left( x \right) over the domain a \leq x \leq b is given by:
\dfrac{ \int_{a}^{ b} f' \left( x \right) \, dx}{b - a}Calculate the average rate of change of the function f \left( x \right) over the domain \dfrac{\pi}{6} \leq x \leq \dfrac{\pi}{3} if:
The acceleration, a \left( t \right) , of a particle moving in a straight line, where t is time in seconds and a \left( t \right) is in \text{m/s}^{2} is given by:
a \left( t \right) = 64 \sin 4 tWrite an expression for v \left( t \right), the velocity in \text{m/s} at time t, if the particle is initially traveling at 2 \text{ m/s}.
Hence, calculate the speed of the particle at t = \dfrac{\pi}{3}.
Find an expression for x \left( t \right), the displacement in metres at time t, if the particle is initially located 3 \text{ m} from the origin.
Hence, calculate the displacement at t = \dfrac{\pi}{3}.
The acceleration, a \left( t \right) , of a particle moving in a straight line, where t is time in seconds and a \left( t \right) is in \text{m/s}^{2} is given by:
a \left( t \right) = 50 \cos 5 tWrite an expression for v \left( t \right), the velocity in \text{m/s} at time t, if the particle is initially traveling at 6 \text{ m/s}.
Hence, calculate the speed of the particle at t = \dfrac{\pi}{6}.
Find an expression for x \left( t \right), the displacement in metres at time t, if the particle is initially located at 3 \text{ m} from the origin.
Hence, calculate the displacement of the particle at t = \dfrac{\pi}{6}.
The acceleration, a \left( t \right) , of a particle moving in a straight line, where t is time in seconds and a \left( t \right) is in \text{m/s}^{2} is given by:
a \left( t \right) = 6 \sin \left(\dfrac{\pi t}{3}\right)Calculate the change in velocity between t = 1 and t = 7.
Given that the speed function is the absolute value of the velocity function, calculate the total change in speed in the initial 4 seconds.
The acceleration, a \left( t \right) , of a particle moving in a straight line, where t is time in seconds and a \left( t \right) is in \text{m/s}^{2} is given by:
a \left( t \right) = 4 \cos \left(\dfrac{\pi t}{4}\right)Calculate the change in velocity between t = 1 and t = 3.
Calculate the total change in speed in the first 4 seconds.
The velocity of a particle moving in a straight line, where t is time in seconds and v \left( t \right) is in metres per second, is given by:
v \left( t \right) = 6 \sin 3 tFind the acceleration of the particle at t = \dfrac{\pi}{3}.
Find the displacement of the particle at t = \dfrac{\pi}{9}, given that when t = 0, the displacement is 2 \text{ m}.
The velocity of a particle moving in a straight line, where t is time in seconds and v \left( t \right) is in metres per second, is given by:
v \left( t \right) = 6 \cos 3 tFind the acceleration of the particle at t = \dfrac{\pi}{9}.
Find the displacement of the particle at t = \dfrac{\pi}{3}, given that the inital displacement is 3 \text{ m}.
The velocity of a particle moving in a straight line, where t is time in seconds and v \left( t \right) is in metres per second, is given by:
v \left( t \right) = 4 \sin \left(\dfrac{\pi t}{3}\right)Calculate the change in displacement in the first 8 seconds.
Given that the distance function is the absolute value of the displacement function, calculate the change in distance in the first 4 seconds.
The velocity of a particle moving in a straight line, where t is time in seconds and v \left( t \right) is in metres per second, is given by:
v \left( t \right) = 6 \cos \left(\dfrac{\pi t}{4}\right)Calculate the change in displacement in the first 9 seconds.
Calculate the change in distance in the first 8 seconds.
For each of the following marginal profit functions, which give the profit, in dollars, of producing and selling x items of a product. Calculate:
The extra profit associated with producing and selling the 101st item.
The net change in profit from producing and selling the first 100 items.
The average profit from producing and selling the first 100 items.
To determine the number of staff required to serve customers at a new cafe, a consultancy company monitored the number of customers over a ten hour period.
The rate of change of the number of customers seen during t hours of service was seen to be modelled by: \dfrac{d C}{d t} = 50 \pi \cos \left(\dfrac{\pi \left(t - 3\right)}{20}\right)
Approximately how many customers visited the cafe in the first hour?
Approximately how many customers visited the cafe in the last two hours?
Calculate the average number of customers per hour over the entire ten hour day.