The graph below shows the education level of youths over the past 100 years. Describe the rate of change of education level over this period of time.
The size of classes enrolling, E, in Physics at a local university is decreasing, and the rate at which this is happening is decreasing. Sketch a graph to show this trend.
As an ice cream melts, the rate at which it melts increases. Sketch a graph showing the amount of solid ice cream, S, over time described by this information.
The electrical resistance, R, of a component at temperature, t, is given by: R = 9 + \dfrac{t}{17} + \dfrac{t^{2}}{108}
Find \dfrac{d R}{d t}, the instantaneous rate of increase of resistance with respect to temperature.
The temperature, T, in degrees Celsius, of a body at time t minutes is modelled by: T = 37 + 1.4 t - 0.02 t^{2}
Find the initial temperature of the body.
Find the average change in temperature in the first 5 minutes.
Find an equation for \dfrac{d T}{d t}, the instantaneous rate of change of the temperature with respect to time.
Find the instantaneous rate of change of the temperature when t = 5.
Find the time, t, when the instantaneous rate of change of temperature equal to zero.
The average price in dollars of a particular share, P, on the Stock Exchange is modeled by: P = \dfrac{1}{3} x^{3} - 8 x^{2} + 63 x + 54Where x is the time in years since it was first listed.
Find the price of the share when it was first listed on the Stock Exchange.
Find the average rate of change of price for 4 \leq x \leq 6, correct to two decimal places.
Find an equation for \dfrac{d P}{d x}, the instantaneous rate of change of price with respect to time.
Find the instantaneous rate of change of price with respect to time when x = 4.
Find the time(s), x, when the instantaneous rate of change of price is equal to zero.
A company manufactures and sells x items of a given product. The total cost, C, in dollars of producing x items given by: C = 7 x + 120 \sqrt{x} + 5000
Find the cost of producing 9 items.
Find the average cost per item when x = 900.
Find an equation for \dfrac{d C}{d x}, the instantaneous rate of change of cost with respect to items produced.
Find the instantaneous rate of change of cost with respect to items produced when 900 items are produced.
The area, in square millimetres, of a colony of bacteria growing on petri dish after t hours (for 0 \leq t \leq 24) is given by:A\left(t\right)=2t^2+10
Find the average growth rate of the colony between t=1 hours and t=3 hours.
Write an expression for the average rate of change between t and t+h.
Hence, using limits find the instantaneous rate of change of the size of the colony for any time 0 \leq t \leq 24.
State the instantaneous rate of change of the bacteria at t=3 hours.
The volume of gas, V, is related to the pressure, P, by the equation P V = k, where k is a constant. Find \dfrac{d V}{d P}, the rate of increase of volume with respect to pressure.
The effectiveness of a supplement is given by \dfrac{d E}{d g}, where E is the body’s reaction to the supplement and g is the number of grams of the supplement administered to the subject. For a particular supplement the body's reaction can be modelled by:E = \dfrac{g^{2} \left(400 - g\right)}{2}
Find the equation for the effectiveness of this particular supplement \dfrac{d E}{d g}.
Find the effectiveness of the drug when g = 30.
After how many units of the supplement does the effectiveness of the drug start to decrease?
A tank initially holds 3600 litres of water. The water drains from the bottom of the tank. The tank takes 60 minutes to empty. The volume V \left( t \right), in litres, of water remaining at time t minutes is given by: V \left( t \right) = 3600 \left(1 - \frac{t}{60}\right)^{2} \text{ where } 0 \leq t \leq 60
Find the volume, in litres, that remains after 10 minutes.
Find the rate of change of the volume inside the tank after 20 minutes.
Find the time when the volume be drained from the tank at the fastest rate.
As the sand in a hourglass is poured, the radius, r, of the cone formed by the sand expands according to the rule r = \dfrac{3 t}{5}, where t is the time in seconds.
Given that the sand falls such that the height of the cone is the same as the radius at all times, find an equation for the volume, V, of the cone of sand with respect to time, t.
Find an equation for \dfrac{d V}{d t}, the rate of change of the volume of the cone of sand with respect to time.
Hence, calculate the instantaneous rate of change of the volume when t = 4. Give an exact answer.
A spherical balloon, initially flat and without any air, is being inflated such that the radius of the balloon, r \text{ cm}, after t seconds of inflation is given by r = 4.5 t^{\frac{1}{3}}.
Find an equation for \dfrac{d r}{d t}, the instantaneous rate of change of the radius with respect to time.
As t increases, does r increase more slowly, more quickly or at a constant rate?
Find an equation for the volume, V, of the balloon in terms of t.
Find an equation for \dfrac{d V}{d t}, the instantaneous rate of change of volume with respect to time.
Hence, find the instantaneous rate of change of the volume when t = 8.
Describe the rate of change of the volume of the balloon over time.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 6 t^{2}
State the velocity v \left( t \right) of the object at time t.
State whether the following represent the velocity of the object after 4 seconds:
x' \left( 4 \right)
v' \left( 4 \right)
x \left( 4 \right)
v \left( 4 \right)
Hence, find the velocity of the object after 4 seconds.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 3 t^{3} - 4 t^{2}
State the velocity v \left( t \right) of the object at time t.
Hence, find the velocity of the object after 2 seconds.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 3 t^{2} + 5 t + 2
Find v \left( t \right), the velocity function.
Find the velocity of the object after 4 seconds.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 18 \sqrt{t}
Find v \left( t \right), the velocity function.
Find the velocity of the object after 9 seconds.
The displacement (in metres) of a particle after t seconds is given by: s \left( t \right) = \dfrac{5 t - 2}{3 t + 1}
Find v \left( t \right), the velocity function.
Find a \left( t \right), the acceleration function.
The velocity (in metres per second) of a body moving in rectilinear motion after t seconds is modelled by: v \left( t \right) = 5 t^{2} - 23 t + 24
Find the time(s), t, when the body is instantaneously at rest.
Find a \left( t \right), the acceleration function.
Calculate the acceleration at time t = 3.
The velocity (in metres per second) of a body moving in rectilinear motion after t seconds is modelled by: v \left( t \right) = t^{2} - 11 t + 24
Find for the time(s), t, when the body is instantaneously at rest.
Find a \left( t \right), the acceleration function.
Calculate the acceleration at time t = 2.
A particle moves in a straight line. Its velocity (in metres per second), t seconds after passing the origin is given by: v = 2 t^{2} - 10 t Find the velocity when the acceleration of the particle is zero.
The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by: x \left( t \right) = - 2 t \left(t + 10\right)
Find the initial displacement of the particle.
Find v \left( t \right), the velocity function.
Find the initial velocity of the particle.
Find the initial speed of the particle.
The displacement (in metres) of a particle moving in rectilinear motion after t seconds is modelled by: x \left( t \right) = 2 t^{3} - 7 t^{2} + 3 t - 4
Find the initial displacement of the particle.
Find v \left( t \right), the velocity function.
Find the initial velocity of the particle.
Find the speed of the particle when t = 2.
The displacement (in metres) of a body from an origin O at time t seconds is modelled by: x \left( t \right) = t^{2} - 7 t + 5
Find the velocity function, v \left( t \right).
Find the initial velocity of the body.
Find the acceleration function, a \left( t \right).
Find the acceleration of the body at t = 8.
Find the value(s) of t for which the body has a velocity of 3 \text{ m/s}.
Find the value(s) of t for which the body has a speed of 3 \text{ m/s}.
The displacement (in metres) of a body moving along a straight line after t seconds is modelled by: x \left( t \right) = - t^{3} + a t^{2} + b t + 7 The initial velocity of the body is 8 \text{ m/s}. The body is momentarily at rest at t = 2 seconds.
Find the values of a and b.
A ball is projected vertically and the height of the ball, y (in metres), after x seconds (for 0 \leq x \leq 8) is modelled by:y=40x-5x^2
Find the average rate of change in the height of the ball between x=2 and x=4.
What does your answer to part (a) represent?
From first principles find the instantaneous rate of change of the height of the ball for any time 0 \leq x \leq 8.
State the instantaneous rate of change of of the height of the ball at x=2 seconds.
What does your answer to part (d) represent?