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 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 asset value of a corporation is expected to change according to the formula: V = - 4 x^{6} - 5 x^{5} + 250 x^{4} + 40\,000
Find the asset growth rate, V'.
Find the value of V' when x = 7.
A materials laboratory is developing a new type of sponge. They find that the mass of liquid, in grams, a cube shaped sponge can hold is related to the side-length, in \text{cm}, of the cube by the following equation:
M = x^{3} + 18 x^{2} + 81 x
Find M'
State the unit that can be use to measure M'.
How much is the mass of liquid a sponge can absorb increasing by when the volume of the sponge is 29.791 \text{ cm}^{3}?
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.
A company's market research department finds that the revenue produced by pricing an item at p dollars is given by the equation: R = - 3 p^{2} + 45 p
Find \dfrac{d R}{d p}.
Calculate \dfrac{d R}{d p} when p = 4.
Calculate \dfrac{d R}{d p} when p = 8.
Should the market research team recommend a price of \$4 or \$8. Explain your answer.
A species of endangered marsupial is reintroduced to an area of bushland. The scientists involved monitor the population of the marsupial and the area inhabited by the population. They find that the relationship between the population and the area (in metres squared) can be modelled by the following equation:
A \left( n \right) = \dfrac{n^{3}}{2} - 410 n + 38
Determine the expression that describes change in the area of the range depending on the population.
How much is the area of the range increasing when the population is 110?
How much is the area of the range increasing when the population is 370?
Is the amount of space each new individual needs the same? For what reason?
Environmental scientists are monitoring the development of a blue-green algae bloom in a rural dam. They observe that the surface area (in \text{m}^2) of the bloom each day is given by the equation:A = \dfrac{t^{2}}{2} + \dfrac{9 t}{10}
How much is the surface area increasing each day?
Find the daily increase at day 20.
The scientists are allowed to use chemical treatment when the increase in surface area per day exceeds 38.9 \text{ m}^{2}/day. When is the earliest they expect to be able to start treatment?
The volume of a reservoir over time changes according to the equation V = 1000 \left(25 - t\right)^{3}, where V is in litres and t is in seconds.
Find an expression for rate of change of the volume at time t?
How much water is flowing out at time t = 20?
For what values of t is \dfrac{d V}{d t} equal to - 300\,000?
The average price in dollars of a particular share, P, on the Stock Exchange is modelled 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.
The hourly energy consumption in \text{kWh} of a city suburb can be modelled using the equation:
y = - \dfrac{1}{8} x^{2} + 50 x^{\frac{1}{2}} + 40 x^{\frac{1}{3}} + 19
Find \dfrac{d y}{d x}.
Find the value of \dfrac{d y}{d x} when x = 11.4. Round your answer to the nearest whole number.
State whether the rate of energy consumption increase or decrease at the time when x = 11.4.
The tide level at a beach is observed to rise and fall according to the following equation:
L = \dfrac{1}{1000} \left(t^{4} - 8 t^{3} + 425 t^{2} - 1300 t\right)
Find L'
Find the value of L' when t = 27.
State whether the tide is going in or out when t = 27.
A probe moving through the solar system uses a solar panel to charge its batteries. The amount of millivolts the panel generates depends on the distance in millions of kilometres x from the sun. This relationship can be described by the equation:
f \left( x \right) = 4000 x^{ - 1 } + 8000 x^{ - 2 } + 400
Find f' \left( x \right).
Describe what f' \left( x \right) represents physically.
Find f' \left( 790 \right). Round your answer to four decimal places
In an experiment involving mice, the daily population of the mice can be approximated by using the following equation:
P = 64 x^{\frac{1}{2}} + 80 x^{\frac{1}{3}} + 8
Find an expression for the daily growth rate of the population, P'.
Find the daily increase of mice for the following days. Round your answers down to the previous whole number.
After 10 days
After 40 days
After 80 days
Hence, describe the population growth.
The following equation can be used to determine the heart rate of a person whose heart pumps 6000 \text{ mL} of blood per minute:
R \left( v \right) = 6000 v^{ - 1 }
Find the rate of change of heart rate with respect to v, the output per beat.
What units will R' \left( v \right) be measured in?
Find the heart rate at v = 80 mL per beat.
Find the average rate of change between v = 80 mL per beat and 100 mL per beat.
Find the rate of change at v = 80 mL per beat.
A colony of bacteria is growing in a petri dish and, after t hours, covers an area in \text{ cm}^2 given by the formula:
A \left( t \right) = \dfrac{2 }{25}t^{2} + \dfrac{8}{5}t + 92
Find A' \left( t \right)
State the unit that can be use to measure A' \left( t \right).
Find the average growth rate of the bacteria between t = 6 and 9 hours.
Find the value of A' \left( t \right) when t = 6 hours.
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 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 grams of the supplement does the effectiveness of the drug start to decrease?
The height of a certain tree can be modelled by H \left( t \right) = 18 - \dfrac{87}{2 t + 5}where t is the time in years after the tree was planted from a nursery seedling and H is the height of the tree in metres.
Find the height of the tree when it was planted in metres.
Find the tree’s average rate of growth in the first 3 years in metres per year.
Round your answer to two decimal places.
Find H' \left( t \right).
At what rate was the tree growing when t = 5? Round your answer to two decimal places.
A cat is given an antibiotic tablet, and the amount of the drug D (mg) still present in its bloodstream after t hours is given by D \left( t \right) = \dfrac{3 t}{5 + 2 t^{2}}.
Determine the amount of drug present in the blood stream initially.
Determine the function that gives us the rate of change of the drug in the bloodstream over time.
Calculate the rate of change of the amount of the drug in the bloodstream at t = 2 hours. Express the rate as an exact value.
Calculate the average rate of change of the drug in the bloodstream over the first 2 hours.
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 variable 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.
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.
The voltage in a circuit is given by V \left( t \right) = 240 \sin \left( 200 \pi t\right) volts, where t is time in seconds.
Find V' \left( t \right).
At what exact rate does the voltage change when t = 20?
At what exact rate does the voltage change at t = 30?
A radioactive isotope decays continuously and can be modelled by A = A_0 e^{ k t}, where A is the kilograms of isotope remaining after t years.
10 kilograms is produced in an industrial process and the isotope decays at a continuous rate of 30\% per year.
State the value of A_0.
Find the value of k.
Hence find the equation for A, the amount remaining after t years.
Find an equation for the instantaneous rate of change of A after t years.
Hence find the value of m if \dfrac{d A}{d t} = m A.
How much of the isotope remains after 4 years? Round your answer to five decimal places.
A radioactive isotope decays continuously such that \dfrac{d A}{d t} = k A.
If the rate of decay is 40\%, find the value of k.
Show that A = C e^{ - 0.4 t } satisfies the equation \dfrac{d A}{d t} = - 0.4 A, where C is some positive constant.
After 20 years, there is 2 kg of the isotope remaining. Find the value of C, the initial amount (in kg) of the isotope present. Round your answer to one decimal place.
A radioactive substance decays at a continuous rate of 5\%. Initially there were 500 grams of this substance.
Let A be the amount of the substance remaining after t years.
Find the amount of the substance left after 50 years. Round your answer to one decimal place.
Find the instantaneous rate of change after 50 years. Round your answer to one decimal place.
Under certain climatic conditions the number P of blue-green algae is modelled by the equation P = B e^{ 0.0008 t} where t is measured in days from when measurement began, and B is constant.
Show that the number of algae increases at a rate proportional to the number present.
At what rate is the number of algae increasing when there are 300\,000 algae present?
Find t, the number of days it takes the initial number of algae to double. Round your answer to two decimal places.
The population of a country town close to a major mine site is growing continuously at 8\% per year. At the beginning of 2015, the population is 400\,000.
The population t years after 2015 can be modelled by P = P_0 e^{ k t}.
State the value of k.
State the value of P_0.
Find the equation for the instantaneous rate of growth of the population in terms of t.
Find the number of years, t, after which the population will double. Round your answer to one decimal place.
Find the instantaneous rate of growth when the population has doubled.
The temperature of a cup of coffee placed in a room can be modelled by T \left( x \right) = 22 + 68 e^{ - 0.07 x }where x is the time in minutes and T is the temperature in degrees Celsius.
What was the initial temperature of the coffee?
What is the approximate room temperature?
What is the average rate of temperature change of the coffee in the first 2 minutes?
Find T' \left( x \right).
Find the instantaneous rate of change in temperature at x = 2. Round your answer to two decimal places.
A radioactive isotope decays continuously such that \dfrac{d A}{d t} = k A, where A is the amount of isotope remaining after t years.
If the rate of decay is 40\%, find the value of k.
Initially, 20 kg of the isotope were produced in an industrial process. Write an equation for A, the weight (in kg) of the isotope, at time t.
Find the time, t years, when the instantaneous rate of decay is 1.3 kg/year. Round your answer to one decimal place.
At the site of a nuclear accident, the amounts of two radioactive substances are monitored. The amount of radioactive substance A remaining is modelled by A = 67 e^{ - 0.02 t } and the amount of radioactive substance B remaining is modelled by B = 38 e^{ - 0.002 t }, where t is time in years, and A and B are amounts in micrograms.
Which of these two substances is decaying faster?
How much of substance A remains after 4 years? Round your answer to the nearest tenth of a microgram.
How much of substance B has decayed after 4 years? Round your answer to the nearest tenth of a microgram.
Find the number of years t at which the amounts of A and B remaining are equal. Round your answer to one decimal place.
Researchers have created a model to project the country’s population for the next 10 years, where P is the population (in thousands), t years from now. The model is defined by the function: P \left( t \right) = \dfrac{57\,460 e^{\frac{t}{7}}}{t + 13}
State the current population of the country.
According to the model, state the current rate of growth of the population, to the nearest thousand.
Find the rate of population growth 7 months from now, to the nearest thousand.
Find the rate of population growth 10 years from now, to the nearest thousand.