For each of the following derivatives, find an equation for the primitive function. Use C as the constant of integration:
Find the equation of a curve p given that \dfrac{d p}{d t} = 6 t - 5, and when t = 3, \dfrac{d p}{d t} = 13 and p = 15.
Find the equations of the curve y given that:
\dfrac{d y}{d x} = 4 x + 7 and the curve passes through the point \left(3, 41\right).
\dfrac{d y}{d x} = 9 x^{2} - 10 x + 2 and the curve passes through the point \left(2, 13\right).
\dfrac{d y}{d x} = 10 x^{4} + 20 x^{3} + 6 x^{2} + 6 x + 9 and the curve passes through the point \left( - 3 , - 133 \right).
\dfrac{d y}{d x} = 9 x^{\frac{2}{3}} and the curve passes through the point \left(8, \dfrac{889}{5}\right).
Find an equation for y in terms of x given that y'' = 0. Use c and d to represent the constants.
Find the equation of the curve that has a gradient of 15 x^{2} + 7 and passes through the point \left(2, 59\right).
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) = 6 t^{2} - 14 t + 3, where v \left( t \right) is the velocity in metres per second and t is the time in seconds.
The displacement after 2 seconds is 4 \text{ m} to the left of the origin.
Calculate the initial velocity of the particle.
Find the function x \left( t \right) for the position of the particle. Use C as the constant of integration.
Calculate the displacement of the particle after 4 seconds.
Assuming the object continues to move in the same direction, find the total distance travelled between 2 and 4 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.
Find 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 = 2 t - 15 where t \geq 0. The object is initially moving to the right at 56 \text{ m/s}.
Find the velocity v of the particle at time t.
Find the times, t, 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 \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.
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.
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 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.
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.