Calculus Continued
Tangents and Normals
Example
2
Find the equations of the tangent and normal to the graph of y  5x  x
at the point where x  1
Example
Find the equation of the tangent and the normal to the curve
y  x3  4x 2  8x  2 at the point A where x  2
Stationary Points
Stationary Points on the graph of a function
y  f x  are points at which the
gradient is zero. Hence to obtain coordinates of stationary points on the graph of
y  f x 
1. Solve
f ' x   0
2. Substitute in
-gives the x coordinates then
y  f x  -gives the y coordinates
Stationary points will be one of the following types:
Minimum point
Maximum point
Points of inflection
Example
Find the stationary points to the graph of y  x 2  4 x  1
Hence sketch the graph of y  x 2  4 x  1
.
For type?
We can determine type of any stationary point by looking at the change in
its gradient as we go ‘through’ the stationary point.
+
–
Minimum
+
–
Maximum
+
–
–
+
Inflections
Example
Obtain the stationary point and determine type of the graph of
y  5  6x  x2
Example
Obtain the stationary point and determine type of the graph of
y  12x  x3
Example
Find the maximum and minimum values of y when y  x3  6x 2  9 x  1
Hence sketch the graph of y  x3  6x 2  9 x  1