A curve passes through point and has a gradient of .
Find the gradient of the curve at point .
Find the equation of the tangent to the curve at point .
Give your answer in the form .
Determine the equation of the curve .
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A curve passes through point and has a gradient of .
Find the gradient of the curve at point .
Find the equation of the tangent to the curve at point .
Give your answer in the form .
Determine the equation of the curve .
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A point lies on the curve that has a gradient of .
Find the gradient of the curve at point .
Find the equation of the tangent to the curve at point .
Give your answer in the form .
Determine the equation of the curve .
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The following table shows the and coordinates of five points that lie on a curve .
0 | 0.25 | 0.5 | 0.75 | 1 | |
1 | 2.25 | 4 | 6.25 | 9 |
Estimate the area under the curve over the interval .
The equation of the curve was found to be .
Find the exact value of the area under the curve over the interval .
Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.
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The following diagram shows an arch that is tall and wide. The arch crosses the -axis at the origin, , and at point , and its vertex is at point . The arch may be represented by a curve with an equation of the form , where all units are measured in metres.
Find
the coordinates of
the coordinates of
Find the cross-sectional area under the arch.
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The diagram below shows a part of the curve . Points and represent the -intercepts, point V represents the vertex of the curve, and the shaded region represents the area between the curve and the -axis.
Find the values of and .
Find the coordinates of points and .
Find the area of region .
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The following diagram shows part of the graph of . The shaded region is bounded by the -axis, the -axis and the graph of .
Write down an integral for the area of region
Find the area of region .
The three points and define the vertices of a triangle.
Find the value of , the -coordinate of , given that the area of the triangle is equal to the area of region .
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A rice farm sells kg of rice every week.
It is known that where is the weekly profit, in dollars ($), from the sale of kg of rice.
Find the amount of rice, in kg, that should be sold each week to maximise the profit.
The profit from selling kg of rice is $480.
Find .
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A paint company sells hundred of litres of paint every week.
It is known that where is the weekly profit, in euros (€), from the sale of hundred litres of paint.
Find the number of litres that should be sold each week to maximise the profit.
The profit from selling litres of paint is €.
Find .
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A river has a cross-sectional area shown by the shaded region of the diagram below, where the and values are in metres. The riverbed (the curved part of the region shown) has an equation of the form . Point is the origin, and points and are the vertices of a rectangle. Point , the deepest point of the riverbed, is situated on the -axis.
Find
the coordinates of
Determine the value of .
Find the cross-sectional area of the riverbed.
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A trough has a cross-sectional area shown by the shaded region of the diagram below, where the and values are in centimetres. The curved bottom of the trough has an equation in the form . Point is the origin, and points are the vertices of a rectangle. Point , the deepest point of the trough, is situated on the -axis.
Determine the value of .
Find the cross-sectional area of the trough.
The length of the trough is 1.2 m.
Find the volume of the trough. Give your answer in cm3
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A function is defined by the equation .
Sketch the graph of in the interval .
Use your sketch from part (a), along with relevant area formulae, to work out the value of the integral
You should not use your GDC to find the value of the integral.
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The derivative of the function is given by
and the curve passes through the point .
Find an expression for .
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A curve has the gradient function , where is a constant. The diagram below shows part of the curve, with the and intercepts labelled and where represents the vertex of the curve.
Find
the value of
the equation of the curve
Find the area between the curve and the -axis.
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A section of the curve with equation is shown below:
The shaded region in the diagram is bounded by the curve, the -axis and the line .
The shaded region in the diagram is bounded on three sides by the curve, the -axis and the -axis. The boundary on the fourth side is a straight line parallel to the -axis, and that line, the curve and the line all intersect at a single point.
Find the area of region . Give your answer as a fraction.
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A company is designing a plastic piece for a new game. The piece is to be in the form of a prism, with a cross-sectional area as indicated by the shaded region in the following diagram:
Region is bounded, as shown, by the positive - and -axes and the curve with equation . All units are in centimetres.
Using technology, or otherwise, find the coordinates of the points of intersection of the curve with the - and -axes.
The volume of the puzzle piece is to be 30 .
Find the length of the puzzle piece, giving your answer correct to 3 significant figures.
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The following diagram shows part of the graph of , . The shaded region is bounded by the -axis, the -axis and the graph of .
ABCD is a parallelogram with vertices , , and , as shown in the diagram below. The area of ABCD is equal to the area of region above.
By first finding the value of , the -coordinate of point , determine the coordinates of point . The coordinates should be given as exact fractions.
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A curve has the equation . Consider the area enclosed by the curve and the positive -axis.
Sketch the curve, shading the area indicated above.
Using the trapezoidal rule with 5 strips, determine an approximation for the shaded area.
Explain, using your sketch from part (a), why it is not possible to determine immediately whether your approximation will be an underestimate or an overestimate.
Using integration, determine the exact value of the shaded area.
Find the percentage error of the approximation found in part (b), compared with the exact value.
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The shaded region in the following diagram is bounded by the -axis, the line and the curve .
Using technology, or otherwise, find the coordinates of
Show that the area of region is equal to exactly units2 . Be sure to show all of your working.
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For a particle travelling in a straight line, the velocity, m/s, of the particle at time seconds is given by the equation
Sketch the graph of in the interval .
The distance travelled between times and by a particle moving in a straight line may be found by finding the area beneath the particle’s velocity-time graph between those two times.
Find the distance travelled by the particle between the times and .
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After analysing several years of company data, a fast food company has determined that the rate of change of its sales figures can be modelled by the equation
where represents the number of meals sold in a week (in thousands of meals sold), and represents the amount spent on advertising during the preceding week (in thousands of euros).
It is known as well that 5988 meals are sold in a week where 2000 euros had been spent on advertising during the preceding week.
Find an expression for .
Find the maximum number of meals that the company can expect to sell in a week, and the amount of money that the company should spend on advertising during the preceding week to bring about that level of sales. Give your answers to the nearest meal sold and the nearest euro, respectively. Be sure to justify that the value you find is indeed a maximum.
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A function is a piecewise linear function defined by
Sketch the graph of in the interval .
Use your sketch from part (a), along with relevant area formulae, to work out the value of the integral
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The derivative of the function is given by
and the curve passes through the point .
Find an expression for .
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A curve has the gradient function . The diagram below shows part of the curve, with the - and -intercepts labelled.
Find
the value of
the equation of the curve
Find the area of the region enclosed by the curve and the -axis.
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Celebrity chef Pepper Bee has opened a new restaurant and is charging diners £630 for a piece of his signature ‘Croesus’ cake. The chef claims that the price reflects the high cost of the gold foil that is placed on top of each slice of cake, but a suspicious and disgruntled customer has decided to investigate this claim.
The shaded area in the diagram below shows the shape of the piece of gold foil that is placed on top of each slice of cake:
The shape is that of a rectangle, from which four identical curved sections have been removed. The rectangle is bounded by the positive - and -axes and the lines and . The shape of one of the curved sections in the diagram can be described by the curve with equation
All units are given in centimetres.
Given that gold foil costs per , work out the cost of the gold foil on a piece of Pepper Bee’s Croesus cake. Give your answer to 2 decimal places.
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A company is designing a piece for one of the plastic wargaming models they produce. The piece is to be in the form of a prism, with a cross-sectional area as indicated by the shaded region in the following diagram:
Region is bounded, as shown, by the positive -axis and the curve with equation . All units are in centimetres.
Given that the model piece will have a volume of 50.3 cm3, find the length of the piece.
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The following diagram shows part of the graph of , . The shaded region is bounded by the -axis, the -axis and the graph of .
Find the area of region
A trapezoid is shown below.
is perpendicular to and parallel to . . The coordinates of points , and are , and respectively, where is a constant.
Given that has the same area as the region R above, find the value of , the -coordinate of point .
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A curve has the equation .
Sketch the curve.
Using the trapezoidal rule with , determine an approximation for the integral
Give your answer as an exact value.
Explain, using your sketch from part (a), why your approximation will be an underestimate.
Explain how you might modify your method in part (b) in order to get a more accurate approximation.
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The shaded region in the following diagram is bounded by the two curves and .
The two curves intersect at points and as shown. and are the -coordinates of points and respectively.
By setting up and solving an appropriate quadratic equation, find the values of and
Find the area of region , giving your answer as an exact value.
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For a particle travelling in a straight line, the velocity, m/s, of the particle at time seconds is given by the equation
At time the particle reaches its maximum velocity, while at time the particle comes momentarily to rest.
Find the values of and , justifying your answers in each case.
The distance travelled between two times by a particle moving in a straight line may be found by finding the area beneath the particle’s velocity-time graph between those two times.
Find
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Donty is a would-be social media celebrity who is obsessed with the number of ‘likes’ his posts receive. He hires a statistician to study his social media accounts, and after analysing several years of data she determines that the rate of change of his number of ‘likes’ can be modelled by the equation
where represents the number of likes received on a given day (in thousands of likes), and represents the amount of new video content Donty uploaded on the preceding day (in hours). Because of technical limitations, Donty is unable to upload more than 12 hours of new video content on any given day.
It is known as well that 36075 ‘likes’ are received on a day after 5 hours of video content was uploaded the day before
Find the maximum and minimum number of ‘likes’ that Donty can expect to receive in a day, and the corresponding number of hours of new video content that Donty should upload on the preceding day to attain that maximum or minimum. Be sure to justify that the values you find are indeed the maximum and minimum possible.
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