Find
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The curve has equation
The curve
passes through the point
has a turning point at
Given that
where is a constant,
show that .
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Figure 1 shows a sketch of a curve with equation
where
is a cubic expression in
.
The curve
passes through the origin
has a maximum turning point at
has a minimum turning point at
Write down the set of values of for which
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Find an expression for when
.
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Find the gradient of at the points where
(I) ,
(ii) .
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(i) Find the gradient of the tangent at the point (2 , 3) on the graph of .
(ii) Hence find the equation of the tangent at the point (2 , 3).
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(i) Find an expression for when
.
(ii) Solve the equation .
(iii) Hence, or otherwise, find the values of x for which is a decreasing function.
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The curve C has equation
Find expressions for and
.
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(i) Evaluate and
when
.
(ii) What does your answer to part (b) tell you about curve C at the point where ?
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For the graph with equation , find the gradient of the tangent at the point where
.
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(i) Find the gradient of the normal at the point where .
(ii) Hence find the equation of the normal at the point where .
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Find the values of for which
is an increasing function.
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Find the x-coordinates of the stationary points on the curve with equation
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Show that the point (2 , 1) is a (local) maximum point on the curve with equation
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A curve has equation
Find writing your answer in simplest form.
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Hence find the range of values of for which
is decreasing.
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A curve has equation
Find
(i)
(ii)
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Verify that has a stationary point when
.
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Determine the nature of this stationary point, giving a reason for your answer.
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Figure 1 shows part of the curve with equation
The point lies on the curve.
Find the gradient of the tangent to the curve at .
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The point with
coordinate
also lies on the curve.
Find the gradient of the line , giving your answer in terms of
in simplest form.
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Explain briefly the relationship between part (b) and the answer to part (a).
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In this question your must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
The curve has equation
The point lies on
and has
coordinate
The line is tangent to
at
.
Show that has equation
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Find the values of x for which is an increasing function.
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Show that the function is increasing for all
.
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The curve C has equation.
Show that the point P(2, 9) lies on C.
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Show that the value of at P is 16.
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Find an equation of the tangent to C at P.
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The curve C has equation . The point P
lies on C.
Find an expression for .
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Show that an equation of the normal to C at point P is .
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This normal cuts the x-axis at the point Q.
Find the length of PQ, giving your answer as an exact value.
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Given that , find
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A curve has the equation .
Find expressions forand
.
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Determine the coordinates of the local minimum of the curve.
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The diagram below shows part of the curve with equation . The curve touches the x-axis at A and cuts the x-axis at C. The points A and B are stationary points on the curve.
Using calculus, and showing all your working, find the coordinates of A and B.
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Show that (-1, 0) is a point on the curve and explain why those must be the coordinates of point C.
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A company manufactures food tins in the shape of cylinders which must have a constant volume of 150π cm3. To lessen material costs the company would like to minimise the surface area of the tins.
By first expressing the height h of the tin in terms of its radius r, show that the surface area of the cylinder is given by .
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Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.
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The curve has equation
Find
(i)
(ii)
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(i) Verify that has a stationary point at
(ii) Show that this stationary point is a point of inflection, giving reasons for your answer.
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A company makes drinks containers out of metal.
The containers are modelled as closed cylinders with base radius cm and height
cm and the capacity of each container is
cm3
The metal used
for the circular base and the curved sides costs pence/cm2
for the circular top costs pence/cm2
Both metals used are of negligible thickness.
Show that the total cost, pence, of the metal for one container is given by
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Use calculus to find the value of for which
is a minimum, giving your answer to 3 significant figures.
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Using prove that the cost is minimised for the value of
found in part (b).
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Hence find the minimum value of , giving your answer to the nearest integer.
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A company makes toys for children.
Figure 5 shows the design for a solid toy that looks like a piece of cheese.
The toy is modelled so that
face is a sector of a circle with radius
cm and centre
angle radians
faces and
are congruent
edges and
are perpendicular to faces
and
edges and
have length
cm
Given that the volume of the toy is show that the surface area of the toy,
, is given by
making your method clear.
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Using algebraic differentiation, find the value of for which
has a stationary point.
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Prove, by further differentiation, that this value of gives the minimum surface area of the toy.
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Find the values of x for which is a decreasing function.
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Show that the function is decreasing for all
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The curve C has equation . The point P(2, 2) lies on C.
Find an equation of the tangent to C at P.
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The curve C has equation The point P
lies on C.
The normal to C at P intersects the x-axis at the point Q.
Find the coordinates of Q.
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Given that , find
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A curve has the equation
The point P is the stationary point of the curve.
Find the coordinates of P and determine its nature.
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The diagram below shows a part of the curve with equation , where
Point A is the maximum point of the curve.
Find .
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Use your answer to part (a) to find the coordinates of point A.
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A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:
The base of each triangle is 2x metres, and the equal sides are each y metres in length.
Although x and y can vary, the total amount of fencing to be used is fixed at P metres.
Explain why .
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Show that
where A is the total area of the garden bed.
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Using your answer to (b) find, in terms of P, the maximum possible area of the garden bed.
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Describe the shape of the bed when the area has its maximum value.
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[A sphere of radius has volume
and surface area
]
A manufacturer produces a storage tank.
The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9.
The walls of the tank are assumed to have negligible thickness.
The cylinder has radius metres and height
metres and the hemisphere has radius
metres.
The volume of the tank is 6 m3.
Show that, according to the model, the surface area of the tank, in m2, is given by
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The manufacturer needs to minimise the surface area of the tank.
Use calculus to find the radius of the tank for which the surface area is a minimum.
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Calculate the minimum surface area of the tank, giving your answer to the nearest integer.
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Find the values of x for which is a decreasing function, where
.
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Show that the function, is increasing for all x in its domain.
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A curve has equation .
A is the point on the curve with x coordinate 0, and B is the point on the curve with x coordinate 6.
C is the point of intersection of the tangents to the curve at A and B.
Find the coordinates of point C.
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Calculate the area of triangle ABC.
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A curve is described by the equation , where
P is the point on the curve such that the normal to the curve at P also passes through the origin.
Find the coordinates of point P. Give your answer in the form , where a and b are rational numbers to be found.
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Write down the equation of the normal to the curve at P.
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Show that an equation of the tangent to the curve at P is
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A curve is described by the equation , where
Find and
.
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P is the stationary point on the curve.
Find the coordinates of P and determine its nature.
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The diagram below shows the part of the curve with equation for which
. The marked point P
lies on the curve. O is the origin.
Show that
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Find the minimum distance from O to the curve, using calculus to prove that your answer is indeed a minimum.
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The top of a patio table is to be made in the shape of a sector of a circle with radius r and central angle , where .
Although r and may be varied, it is necessary that the table have a fixed area of A m2.
Explain why .
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Show that the perimeter, P, of the table top is given by the formula
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Show that the minimum possible value for P is equal to the perimeter of a square with area A. Be sure to prove that your value is a minimum.
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