Show that the derivative function of the curve given by
is given by
Find the equation of the normal to the curve given in part (a) at the point where , giving your answer in the form where and c are integers to be found.
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Show that the derivative function of the curve given by
is given by
Find the equation of the normal to the curve given in part (a) at the point where , giving your answer in the form where and c are integers to be found.
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Given that , find the general solution to the differential equation
Find the general solution to the differential equation
giving your answer in the form .
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Write in the form , where and are integers to be found.
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The functions and are given as follows
Expand , in ascending powers of up to and including the term in .
Expand , in ascending powers of up to and including the term in .
Find the expansion of in ascending powers of , up to and including the term in .
Find the values of for which your expansion in part (c) is valid.
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The graph of the curve C shown below is defined by the parametric equations
Find an expression for in terms of .
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A curve C has parametric equations
Find a Cartesian equation for the curve C in the form .
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Use calculus and the substitution to find the exact value of
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Find the coordinates of the point on the line that is closest to the point and hence determine the minimum distance from point to the line.
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Use integration by parts to find, in terms of e, the exact value of
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The diagram below shows the graph of the curve with equation
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A large weather balloon is being inflated at a rate that is inversely proportional to the square of its volume.
Defining variables for the volume of the balloon (m3) and time (seconds) write down a differential equation to describe the relationship between volume and time as the weather balloon is inflated.
Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m3, find the particular solution to your differential equation.
Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m3. If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?
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Prove by contradiction that if is odd, then must be odd.
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