The equation of a curve isÂ
FindÂ
The gradient of the tangent to the curve at point  is .
Find
    Â
Give your answer in the form .
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The equation of a curve isÂ
FindÂ
The gradient of the tangent to the curve at point  is .
Find
    Â
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Consider the function .
Find .
Find the gradient of the graph of at .
Find the coordinates of the points at which the normal to the graph of has a gradient of .
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The equation of a curve is .
Find the equation of the tangent to the curve atÂ
Give your answer in the form .
Find the coordinates of the points on the curve where the gradient is .
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Consider the function .
Calculate
A line,  is tangent to the graph of  at the point .
Find the equation of . Give your answer in the form .
The graph of  and  have a second intersection at point .
Use your graphic display calculator to find the coordinates of .
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Consider the function .
Find .
The equation of the tangent line to the graph  at  is .
Calculate the value of .
Calculate the value of and write down the function .
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The equation of the curve  is . A section of the curve  is shown on the diagram below.
Find .
Points and represent the local maximums on the diagram above.
Write down the coordinates of
There are two points, and , along the curve  at which the gradient of the normal to the curve  is equal to .
Calculate the -coordinates of points and .
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The daily cost function of a company producing pairs of running shoes is modelled by the cubic function
where  is the number of pairs of running shoes produced and  the cost in USD.
Write down the daily cost to the company if no pairs of running shoes are produced.
The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.
Find an expression for the marginal cost, , of producing pairs of running shoes.
Find the marginal cost of producing
pairs of running shoes
The optimum level of production is when marginal revenue, , equals marginal cost, . The marginal revenue, , is equal to 4.5.
Find the optimum level of production.
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A cyclist riding over a hill can be modelled by the function
where  is the cyclist’s altitude above mean sea level, in metres, and is the elapsed time, in seconds.
Calculate the cyclist’s altitude after a minute.
Find .
Calculate the cyclist’s maximum altitude and the time it takes to reach this altitude.
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A company produces and sells cricket bats. The company’s daily cost, , in hundreds of Australian dollars , changes based on the number of cricket bats they produce per day. The daily cost function of the company can be modelled by
where  hundred cricket bats is the number of cricket bats produced on a particular day.
Find the cost to the company for any day zero cricket bats are produced.
The company’s daily revenue, of , from selling  hundred cricket bats is given by the function .
Given that profit revenue cost, determine a function for the profit,  in hundreds of AUD from selling  hundred cricket bats.
Find .
The derivative of  gives the marginal profit. The production of bats will reach its profit maximising level when the marginal profit equals zero and  is positive.
Find the profit maximising production level and the expected profit.
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Dora decides to build a cardboard container for when she goes strawberry picking from a rectangular piece of cardboard, . She cuts squares with side length  cm from each corner as shown in the diagram below.
Show that the volume, , of the container is given by
Find .
Find
the value of that maximises the volume of the container
the maximum volume of the container. Give your answer in the form , where and .
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The equation of a curve is  for .
Find .
The gradient of the tangent to the curve at point  is .
Find the coordinates of point .
Find the equation of the normal to the curve at point Give your answer in the form  .
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The volume of a sphere of radius  is given by the formula  .
Find .
Find the rate of change of the volume with respect to the radius when .
Give your answer in terms of .
Show that  is an increasing function for all relevant values of .
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A curve has the equation
Points  and  are the two points on the curve where the gradient is equal to 1, and the  -coordinate of  is less than zero.
Find the coordinates of points and .
Find the equations of
Point  is the point of intersection of the two lines found in part (b).
Find the coordinates of point .
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The gradient of the tangent to the curve with equation  at the point  is 14.
Find the values of  and .
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Patroclus, a would-be Olympic javelin thrower, throws a javelin during a training session. The height of the javelin’s point can be modelled by the equation
Â
where  is the time, in seconds, that has passed since the javelin was released, and  is the height of the javelin above the ground, in metres.
Find .
Find the greatest vertical distance that the javelin’s point travels above the height from which it was released.
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Check, Mate! is a company that produces luxury chess sets for discerning chess set connoisseurs. The company’s profits , in thousands of UK pounds (£1000), can be modelled by the function
where  is the number of chess sets (in hundreds) sold per year. Because of manufacturing constraints, the maximum number of chess sets that the company can sell in a year is 2500.
Calculate
In each case include the units, and explain the meaning of the value you find.
State the values of for which the instantaneous rate of change of is negative. Explain the meaning of this result.
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The diagram below shows a part of the graph of the function , where
Calculate the instantaneous rate of change of when .
Calculate the average rate of change of between andÂ
Explain what would happen if you continued to calculate the average rates of change in part (b), moving the second value closer and closer to 2 each time.
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A manufacturing company is producing tins that must have a capacity of 470 cm3. The tins are in the shape of a cylinder with a height of  cm and a base radius of  cm.
Show that the surface area of the cylinder in cm2, including the two circular ends, may be written as
Sketch the graph of .
The company would like to minimise the amount of metal used to make the tins.
A commercially available tin of chopped tomatoes on sale in the UK has a capacity of 470 Â and a base radius of 3.7 cm.
Determine the percentage difference between the surface area of that tin of chopped tomatoes and the minimum possible surface area for a tin with the same capacity.
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Two numbers,  and , are such that  and the difference between the two numbers is .
Find the minimum possible value of the product , and the values of  and  that correspond to that minimum value.
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A curve is given by the equation
Determine the coordinates of the points on the curve where the gradient is 2. You must show all your working, and give your answers as exact fractions.
Find the range of values for for which the curve is increasing.
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An engineer is designing a right cone that is to be produced on a 3D printer. The cone has a base radius of  cm and a height of  cm, and while the radius may vary freely the height must always be 7 cm more than the radius.
Write down, in terms of only, the formula for the volume of the cone.
Find the exact value of the radius at the point where the instantaneous rate of change of the volume with respect to the radius is .Â
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A curve has the equation
Points  and  are the two points on the curve where the gradient is equal to 3, and the  -coordinate of  is less than zero.
Find the coordinates of points and .
Find the equations of
Point  is the point of intersection of the two lines found in part (b).
Find the coordinates of point . Give your answers as exact fractions.
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A curve has equation .
The gradient of the tangent to the curve at the point  is 25.
The gradient of the tangent to the curve at the point  is .
Find the values of , , Â and .
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A newly-commissioned attack submarine is performing a series of manoeuvres to test its propulsion and steering systems. The vertical position of the submarine relative to sea level (where sea level is represented by ) is given by the equation
where is the time, in minutes, that has passed since the submarine began its manoeuvres, and is the vertical position of the submarine in metres.
Find the stationary points for .
For each of the stationary points found in part (a), determine whether the point is a maximum point or a minimum point. Justify your answer in each case.
Explain why, in order to find the maximum and minimum depths reached by the submarine in the interval , it is not sufficient merely to consider the stationary points found in part (a).
Find the greatest vertical distances that the submarine travels in the interval   above and below the depth from which it started its manoeuvres.
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Muggins! is a company that produces luxury cribbage boards for discerning collectors of pub game paraphernalia. For sales of between 0 and 100 cribbage boards in a month, the company’s profits , in thousands of UK pounds (£1000), can be modelled by the function
where  is the number of cribbage boards (in hundreds) sold during the month. For sales of between 100 and 1000 cribbage boards in a month, the corresponding formula isÂ
Because of manufacturing constraints, the maximum number of cribbage boards that the company can sell in a month is 1000.
On the same set of axes, sketch the two profit functions. Each function should only be sketched over the interval of  values for which it is valid.
Show that the combined profit function sketched in part (b) is an increasing function for all valid values greater than zero.
Considering only values of for which , find the value of  for which the instantaneous rate of change of  is a minimum. Give the value of the corresponding instantaneous rate of change, and explain the meaning of that value in context.
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The diagram below shows a part of the graph of the function , where
Calculate the average rate of change of between and
Explain what would happen to the values of the average rates of change in part (b) if you continued to calculate them, moving the second value closer and closer to 3 each time.
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An artist is producing large pieces of sculpture for an art installation. Each piece is in the form of a cylinder with base radius metres, on top of which is a hemisphere with the same radius as the cylinder’s base radius. The hemisphere is fitted exactly to the top of the cylinder, so that the circular bottom of the hemisphere lines up exactly with the circular top of the cylinder.Â
Every side of each piece of sculpture must be painted, so the artist is eager to find a design for his sculptures such that, for any given volume of a piece of sculpture, the total surface area will be the minimum possible.
Show that for a piece of sculpture with volume , the minimum surface area occurs when
Find the minimum possible surface area for a piece of sculpture with volume . Give your answer as an exact value.
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Two numbers, and , are such that  and the difference between the two numbers is , where  is a positive constant.
Find the minimum possible value of the sum , and the values of  and  that correspond to that minimum value. Your answers should be given in terms of .
Justify that your answer in part (a) is a minimum value.
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