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
I have plotted graphs from both sets of calculated gradients however I will concentrate on the graph plotted from the results show above as
This graph shows the result that I expect to get, I expect to see a
gradient of this graph = - (Ea / RT) which can be used to calculate
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.
X value, the height is X2 and the width is 1/2 the X value. This shows
Find the values of x2 – x1 and x4 – x3 and name them respectively SL and SR .
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)
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.
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
with an argument y which represent the state of the system at times i, i 2
... or odd, and positive or negative before you can determine your answer. Third, you have to see if your graph is above or below the x-axis between your x-intercepts and plug a value between these intercepts into your function. Last but not least, you plot your graph.
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: