· Gradients of tangents to curves
The derivative of a function for some particular value
is also the gradient of the graph of the function at that point.
Example
A curve has equation
Find the gradient of the tangent at the point where x = 4.
· Finding the equation of a tangent
Example
The point A( -1,7) lies on the curve with equation
Find the equation of the tangent to the curve at point A.
· Increasing / Decreasing functions
The graph below is decreasing when x is less than zero:-
· The value of the function is decreasing as x is increasing
· The gradient is negative
and increasing when x is greater than zero:-
· The value of the function is increasing as x is increasing
· The gradient is positive
When x = 0, the gradient is zero and the graph changes from
a negative gradient to a positive gradient.
This turning point is called a stationary point.
The stationary point can be a :-
Maximum
Minimum
Rising point of inflection
Falling point of inflection
· Finding Stationary Points
To find the stationary points, set the first derivative
of the function to zero, then factorise and solve.
Example
Find the stationary points of the graph
To
find out if the stationary point is a
maximum,
minimum or point of inflection,
construct a nature table:-
·
Put in the values of x for the stationary points.
·
Copy these values, with a small minus and plus
sign.
·
Copy the first part of the factorised form of
the derivative.
·
Repeat for
subsequent parts of the factorised form of
the derivative.
·
Write down the whole factorised form of the
derivative.
Starting with the first
value,
·
Replace x with the value. Do the calculation.
·
Write in the sign of the answer that you get.
Repeat
for a number slightly smaller than x
Put
your answer in the small minus box.
Repeat
for a number slightly larger than x
Put
your answer in the small plus box.
Multiply
the signs together and put in the
factorised
form of derivative box.
Now
draw in the tangent shape
Repeat
for other stationary points
· Second Derivative
(Advanced Higher)
The
second derivative f’’(x) measures the gradient
with
respect to x.
When
this is positive, the curve is concave up it is a
minimum turning point.
When
f’’(x) is negative, the curve is concave down
it is a
maximum turning point.
When
f’’(x) is zero, there may be a point of inflexion.
Draw
a nature table to confirm.
In
a closed interval, the maximum and minimum values of the function
occur
either at a stationary point or an end
point.
Example
Find the maximum and minimum values of
For the given interval, the maximum value is 8
and the minimum value is -2.125
A continuous function defined in a closed interval
must have both a global maximum and minimum.
They are found by examining the local extrema
and end points.
Example
