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Curve sketching techniques
What can you get from the function?
domain
points
vertical and horizontal asymptotes
zeros
intercepts
What can you get from the first derivative?
stationary points
increasing/decreasing intervals
maximum or minimum
What can you get from the second derivative?
concavity
STEP 2004, Math III, #2: The equation of a curve is
where
(i) Write down the equations of the vertical and oblique asymptotes to the curve and show that the oblique asymptote is a
tangent to the curve.
(ii) Show that the equation has a double root.
(iii) Sketch the curve.
STEP 2001, Math I, #3: Sketch, without calculating the stationary points, the graph of the function
given by
,
where
. By considering the quadratic equation [Hint: think about discriminants], or otherwise, show that
.
By considering
, or otherwise, show that is a sufficient condition but not a necessary condition for
the inequality
to hold.
STEP 2003, Math 3, #3: If
is a positive integer, show that for any real .
The function is defined by
Find and simplify an expression for .
In the case
, sketch the curves and .
STEP 2010, Math I, #2: The curve
where
and are constants, has two stationary points. Show that
or [Hint: think about discriminants].
(i) Show that, in the case
and , there is one stationary point on either side of the curve's vertical asymptote, and
sketch the curve.
(ii) Sketch the curve in the case
and .
STEP 2002, Math III, #3: Let
where
and . Sketch the graph of . Hence show that the equation , where , has no solution
when
. Find conditions on in terms of and for the equation to have exactly one or exactly two solutions.
Solve the equations (i)
and (ii) .
STEP 2006, Math III, #1: Sketch the curve with cartesian equation
and give the equations of the asymptotes and the tangent to the curve at the origin. Hence determine the number of real roots
of the following equations:
(i)
(ii)
(iii)
STEP 1999, Math III, #2: (i) Let
. Show that and sketch the graph of . Hence or otherwise,
show that the equation
,
where
is a constant, has exactly one real root if and no real roots if .
(ii) Determine the number of real roots of the equation
in the case (a)
and (b) .
STEP 2008, Math I, #4: A function is said to be convex in the interval
if for all in this interval.
(i) Sketch on the same axes the graphs of and in the interval .
The function is defined for by
.
Determine the intervals in which is convex.
(ii) The function
is defined for by
If
and , show that is convex in the interval , and give one other interval in which
is convex.
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