星辉国际教育 STEP班
第一次作业
1.
. It has a maximum point at (�, �) and a
A curve has equation
minimum point at (�, �) where � > 0 and � > 0 . Let R be the region enclosed by
the curve, the line
and the line
.
(i)
Express � and � in terms of � and �.
(ii)
Sketch the curve. Mark on your sketch the point of inflection and shade the region
R. Describe the symmetry of the curve.
(iii) Show that
(iv) Show that the area of R is
.
.
2.
The curve C has equation
where the square root is positive. Show that, if
point.
Sketch C when (i)
and (ii)
.
, then C has exactly one stationary
3.
Sketch the graph of the function h, where
Hence, or otherwise, find all pairs of distinct positive integers m and n which satisfy the
equation
�� = � � .
4.
Sketch the curves given by
where � and � are non-negative, in the cases:
(i) 2� < � 3 , (ii) 2� = � 3 ≠ 0, (iii) � 3 < 2� < 2�, (iv) � = � ≠ 0,
(v) � > � > 0, (vi) � = 0, � ≠ 0, (vii) � = � = 0.
Sketch also the curves given by
in the cases (i), (v) and (vii).
5.
Sketch the curve with cartesian equation
And give the equations of the asymptotes and of the tangent to the curve at the origin.
Hence determine the number of real roots of the following equations:
(i)
(ii)
(iii)
6.
It is given that the two curves
and
Where
(i)
, touch exactly once.
In each of the following four cases, sketch the two curves on a single diagram, noting the
coordinates of any intersections with the axes:
(a) k < 0 ;
(b) 0 < k < 16, k/m < 2 ;
(c) k > 16, k/m > 2 ;
(ii)
(d) k > 16, k/m < 2 .
Now set m = 12.
Show that the
Let
-coordinate of any point at which the two curves meet satisfies
be the value of
at the point where the curves touch. Show that
And hence find the three possible values of .
Derive also the equation
Which of the four sketches in part (i) arise?
satisfies
7.
(i)
Sketch the curve
, where
( ≠ ±1),
and
(ii)
The function
Where
cases
is a constant.
and
is defined by
are constants, and
and
. Sketch the curves
, finding the values of
( ≠ ±1,
≠ ± 1),
in the two
at the stationary points.