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STEP 2000, Math I, #8: Show that
and hence evaluate the integral.
Evaluate the following integrals:
(i)
(ii)
STEP 2009, Math II, #2: The curve C has equation
where
.
(i) Find the coordinates of the stationary points on C.
(ii) Use the approximation and (both valid for small values of ) to show that
for small values of
.
(iii) Sketch C.
(iv) By approximating C by means of straight lines joining consecutive stationary points, show that the area between C and the x-axis between the kth and the (k+1)th maxima is approximately
STEP 2001, Math II, #5: The curve passes through the origin in the xy-plane and its gradient is given by
Show that has a minimum point at the origin and a maximum point at
.
Find the coordinates of the other stationary point. Give a rough sketch of .
The curve passes through the origin and its gradient is given by
Show that
has a minimum point at the origin and a maximum point at , where . (You need not find .)
STEP 2006, Math I, #7: (i) Sketch on the same axes the functions
and , for .
Deduce that the equation
has exactly two roots in the interval .
Show that
where
is the larger of the roots referred to above.
(ii) Show that the region bounded by the positive x-axis, the y-axis and the curve
has area
.
STEP 2004, Math II, #5: Evaluate
and
The function
satisfies the equation
(*)
Show that
(**)
where
and
Use the expression (**) to find
and by substituting for and in (*) and equating coefficients of and .
2003, Math II, #7: Show that,
, then
You may assume that
, then
Deduce that
where
is any integer greater than 1.
STEP 2004, Math III, #7: For , let
By considering the greatest value taken by for show that .
Show also that
Deduce that
Prove that
and hence show that
STEP 2005, Math II, #3: give a sketch, for , of the curve
and show that .
Show that:
(i)
(ii)
STEP 2011, Math I, #2: The number
is defined by
Show that
and evaluate
in terms of
and .
Evaluate also, in terms of
and as appropriate
(i)
(ii)
2009, Math I, #7: Show that, for any integer
,
(i) Expand . Hence show that
.
(ii) Evaluate
.
STEP 2011, Math II, #6: For any given function , let
(*)
where
is a positive integer. Show that, if satisfies for some constant , then (*) can be integrated to obtain an
expression for
in terms of , , and .
(i) Verify your result in the case . Hence fin
(ii) Find
Using the Chart
STEP 1998, Math II, #4: The integral is defined by
where
is a positive integer. Evaluate and hence evaluate leaving your answer in the form of a sum.
STEP 1998, Math III, #7: Sketch the graph of for , Taking , find the smallest positive integer , , such that .
Now let
where
is a positive constant. Show that has a single turning point in . By considering the behaviour for small and for large ,
sketch
for .
STEP 2012, Math I, #5: Show that
and that
Hence evaluate
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