12.03 Finding f(x) from f'(x)
For each of the following find the equation of y:
Find the equation of the curve that has a gradient of 15 x^{2} + 7 and passes through the point \left(2, 59\right).
Find the equation of y in terms of x given that \dfrac{d y}{d x} = 12 \left(x - 1\right)^{3} and y = 247 when x = - 2.
Find the equation of the curve that has a gradient of \dfrac{d y}{d x} = \left(x - 6\right)^{2} and the point \left(3, - 4 \right) lies on the curve.
The velocity v \left( t \right), in \text{m/s}, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t, \text{ where } t \geq 0The object is initially at the origin.
Find the displacement x \left( t \right)= \int v(t) \, dt of the particle at time t.
Find the time at which x(t) = 54 \text{ m}.
The velocity v, in metres, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 6 t + 10, \text{ where } t \geq 0
Find the displacement s \left( t \right) of the particle at time t, given that the object starts its movement at 8 \text{ m} to the right of the origin.
Find the displacement of the object after 5 seconds.
The velocity v \left( t \right), in metres, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t^{2} + 30 t + 9, \text{ where } t \geq 0The object starts its movement at 6 \text{ m} to the left of the origin.
Find the displacement, s \left( t \right), of the particle at time t.
Find the displacement of the object after 5 seconds.
The velocity of a particle moving in rectilinear motion is given by v \left( t \right) = 4 t - 4 where v is the velocity in metres per second from the origin and t is the time in seconds. The particle is instantaneously stationary when it is 1 \text{ m} right of the origin.
Find the time when the particle is stationary.
Determine the function x \left( t \right) for the position of the particle.
Find the value of x(t) at:
t=0
t=1
t=2
The velocity v \left( t \right) of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t^{2} - 48 t, \text{ where } t \geq 0The object starts its movement 5 \text{ m} to the right of the origin.
Find the displacement, x \left( t \right), of the particle at time t.
Find the times, t, when the object is at rest.
Find the displacement at which the object is stationary other than its initial position.
The velocity v \left( t \right), in \text{ m/s}, of an object along a straight line after t seconds is modelled by v \left( t \right) = 12 \sqrt{t}The object is initially 5 \text{ m} to the right of the origin.
Find the function x \left( t \right) for the position of the particle.
Hence, calculate the position of the object after 9 seconds.
The acceleration a, in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by a \left( t \right) = 6 t - 27, \text{ where } t \geq 0After 10 seconds, the object is moving at 90 \text{ m/s} in the positive direction.
State the velocity, v, of the particle at time t.
Find all the times at which the particle is at rest.
The acceleration a, in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by a = 2, \text{ where } t \geq 0The object is initially 8 \text{ m} to the right of the origin and moving to the left at 5 \text{ m/s}.
Find the velocity, v, of the particle at time t.
Find the displacement, x, of the particle at time t.
Find the position of the object at 7 seconds.
Find the time at which the particle is moving at a speed of 3 \text{ m/s} to the right.
In a closed habitat, the population of kangaroos P \left( t \right) is known to increase according to the function P' \left( t \right) = \dfrac{t}{2} + 9 where t is measured in months since counting began.
Calculate the total change in the population of kangaroos in the first 4 months since counting began.
Find the number of months it will take from when counting began for the population of kangaroos to increase by 88.
The rate of emission \dfrac{d E}{d t} of CFCs in Australia, measured in tonnes per year, was given by \dfrac{d E}{d t} = 100 + \left(\dfrac{60}{1 + t}\right)^{2}where t is the time in years after 1 September 1988.
Find the rate of emission on 1 September 1988.
Find the rate of emission on 1 September 1997.
Find the value that \dfrac{d E}{d t} approaches as the years pass by.
Calculate the total amount of CFCs emitted in Australia during the years 1988 to 1997.
The total revenue, R (in thousands of dollars), from producing and selling a new product, t weeks after its launch, is given by \dfrac{d R}{d t} = 401 + \dfrac{500}{\left(t + 1\right)^{3}}
Given that the initial revenue at the time of launch was zero, state the revenue function.
Find the average revenue earned over the first 5 weeks.
Calculate the revenue earned in the 6th week.
An ice cube with a side length of 25 \text{ cm} is removed from the freezer and starts to melt at a rate of 25 \text{ cm}^{3}/\text{min}. Let V be its volume t minutes after it is removed from the freezer.
State the equation for the rate of change of volume.
State the equation for the volume, V, of the cube as a function of time t.
Find the value of t when the ice cube has melted completely.
M \left( x \right) is the cost of producing x units of a certain product, where M \rq \left( x \right) = 9 x^{2} - 70 x + 500.
Determine the cost function M \left( x \right) in terms of x.
Hence, determine the extra cost incurred by producing 60 units rather than 20 units.
Water flows into, then out of, a container at a rate \dfrac{d V}{d t} litres per minute given by \dfrac{d V}{d t} = t \left(10 - t\right) where the number of minutes, t \geq 0.
Sketch the graph of \dfrac{d V}{d t}.
Hence, find the maximum flow rate.
Find an expression for the volume of water, V litres, in the container at time t minutes, assuming that the container is initially empty.
Find the total time taken for the container to fill and then empty.
The mass, m grams, of a raindrop falling for t seconds is increasing at a rate given by \dfrac{d m}{d t} = \dfrac{1}{120} \left(\sqrt{t} + \dfrac{t^{2}}{12}\right) \text{ g/s}
Given that the initial mass of the raindrop is zero, determine the mass m of the raindrop after t seconds.
Determine the exact mass of the raindrop after 16 seconds have passed.
Another raindrop starts as a gas particle with a mass of 0.004 \text{ g}. How much heavier will it be after 16 seconds than a raindrop that is initially weightless?
Wheat is poured from a silo into a truck at a rate of \dfrac{d M}{d t} , where \dfrac{d M}{d t} = 81 t - t^{3} \text{ kg/s} and t is the time in seconds after the wheat begins to flow.
Find an expression for the mass M \text{ kg} of wheat in the truck after t seconds, if initially there was 1 tonne of wheat in the truck.
Calculate the total mass of wheat in the truck after 8 seconds.
Find the largest value of t for which the expression for \dfrac{d M}{d t} is physically possible.
A pen moves along the x-axis ruling a line. The graph of the velocity of the tip of the pen as a function of time is shown below. The velocity, in \text{cm/s}, is given by the equation v = 4 t^{3} - 36 t^{2} + 72 twhere t is the time in seconds. The tip is initially 5 \text{ cm} to the right of the origin.
Find an expression for x, the position of the tip of the pen, as a function of time.
What feature will the graph of x as a function of t have at the point where t = 3?
What happens to the pen at t = 3?
The pen uses 0.02 \text{ mg} of ink per centimetre travelled. How much ink is used between t = 0 and t = 4? Round your answer to two decimal places.