Gradient Function
For this investigation, I have to find the relationship between a
point of any non-linear graph and the gradient of the tangent, which
is the gradient function. First of all, I have to define the word,
'Gradient'. Gradient means the slope of a line or a tangent at any
point on a curve. A tangent is basically a line, curve, or surface
that touches another curve but does not cross or intersect it. To find
a gradient, observe the graph below:
[IMAGE][IMAGE]
All you have to do to find the gradient is to divide the change in X
with the change in Y. In this case, on the graph above, AB and you
would have gotten the
BC
gradient for that particular point of the graph.
I am going start by finding the gradient function of y=x², y=2x², and
then y=ax². I will move on finding the gradient function of y=x³,
y=2x³, and finally y=ax³. I will then find the similarities and
generalise y=axâ¿ where 'a' and 'n' are constants, and then investigate
the Gradient function for any curves of my choice.
I will first find the gradient of tangents on the graph y=x² by
drawing the graph (page 3), and then find the gradient for a number of
selected points on the graph:
Point
X
Change in Y
Change in X
Gradient
a
-3
6
-1
-6
b
-2
4
-1
-4
c
-1
2
-1
-2
d
1
2
1
2
e
2
4
1
4
f
3
6
1
6
As you can see, the gradient is always twice the value of its original
X value Where y=x². So the gradient function has to be f `(x)=2x for
If I were using a cut out of length 1cm, the equation for this would
Living in a divided society based upon the religions of the Puritans and the Quakers, Evan Feversham sought out his own religious faith through his daily interactions with both religious groups.
The gradient of the graph tells us whether the different rate curves have the same relation, meaning if they have a similar rate of reaction. Reactions can take place in a variety of customs; they can bee steep or steady. The steeper the slope, the faster the reaction takes place. The steadier the slope, the slower the reaction takes place. Aim:
X value, the height is X2 and the width is 1/2 the X value. This shows
... the value of D will be 0, as it is similar to the cubic graph.
Directional derivative: Directional derivative represents the instantaneous rate of modification of the function. It generalizes the view of a partial derivative.
(% change in the quantity demanded of good B)/(% change in price of good A)
I created my graph by entering the original function P(t) = 145e-0.092 in the first box. In the second box, I entered y = 70. My graph is now created. Where the red line crosses the blue, I plotted a dot where it was closer to 8 minutes. I also plotted a dot on the blue line closest to 70 beats. As one can see, the answer on the graph is the same as my calculations
When “a” is increased in the equation for the curve, the entire curve increases in size, giving it a larger area. The value for “x” is greatly increased on the right side positive y-axis, while the value for “x” on the left side negative y-axis becomes gradually more negative at a much lower rate then that of the right side positive y-axis.
slope. I think that out of all the variables, this is the one which is
with an argument y which represent the state of the system at times i, i 2
Next you would create mathematical equations that link to what you are trying to solve. If you are looking at the rate of change in more than one variable you will end up with some differential equations that need to be derived.
However, the most straightforward method is to assume a linear model, that is to set b to one and then use regression analysis to estimate the slope i-e a and possibly introduce an intercept so that the model becomes: