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Basic Formulas
There are hundreds of integral formulas. However, the vast majority of integrals you will encounter can be evaluated using just a few of them. The following formulas should be committed to memory because they will arise frequently in applications.
The Power Rule
The Log Rule
Exponential Rules
Trigonometric Rules
Inverse Trigonometric Rules
STEP 2000, Math I, #3: For any number
, the largest integer less than or equal to is denoted . For example, and .
Sketch the graph of for and evaluate
Sketch the graph of for , where is an integer, and show that
STEP 2004, Math I, #2: The square bracket notation
means the greatest integer less than or equal to . For example, , and .
(i) Sketch the graph of and show that
when
is a positive integer.
(ii) Show that
when
is a positive integer.
(iii) Determine an expression for
when
is positive but not an integer.
2010, Math II, #8: The curves and are defined by
and
respectively. Sketch roughly and on the same diagram.
Let
denote the x-coordinate of the th point of contact between the two curves, where , and let denote the area of the region enclosed by the two curves between
and . Show that
and hence find
STEP 2001, Math I, #5: Show that (for
)
(i)
(ii)
Noting that the right hand side of (ii) is the derivative of the right hand side of (i) conjecture that value of
(You need not verify your conjecture.)
STEP 2002, Math I, #4: Give a sketch of the curve
for .
Find the equation of the line that intersects the curve at
and is tangent to the curve at some point with . Prove that there are no further intersections between the line and the curve. Draw the line on your sketch.
By considering the area under the curve for
, show that .
Show also, by considering the volume formed by rotating the curve about the y-axis, that .
[Note: .]
STEP 2003, Math II, #6: The function is defined by
,
where the domain is
, the set of all real numbers. The function , with domain , so for example
. In separate diagrams, sketch graphs of , , and .
The function
is defined by
where the domain is
. Show that if is even,
STEP 2002, Math I, #7: Let
and
where . By considering and , show that .
Find also
(i)
(ii)
STEP 2011, Math I, #5: Given that
, show with the help of a sketch that the equation
has a unique solution in the range
.
Let
Show that
where
is the unique solution of (*).
Show that
, regarded as a function of , has a unique stationary value and that this stationary value is a minimum.
Deduce that the smallest value of
is
STEP 2011, Math III, #4: The following result applies to any function
which is continuous, has positive gradient and satisfies :
(*)
where denotes the inverse function of , and and .
(i) By considering the graph of , explain briefly why the inequality (*) holds.
In the case
and , state a condition on and under which equality holds.
(ii) By taking in (*) , where , show that if
then
verify that equality holds under the condition you stated above.
(iii) Show that, for and
Deduce that, for ,
STEP 2012, Math III, #3: (i) Sketch the curve
for and add to your diagram the tangent to the curve at the origin and
the chord joining the origin to the point
, where .
By considering areas, show that
(ii) By considering the curve , where , show that
[Hint: You may wish to write
as .]
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