Investigation : average gradient - parabolic function
Fill in the table by calculating the average gradient over the indicated
intervals for the function
$f\left(x\right)=2x-2$ :
${x}_{1}$
${x}_{2}$
${y}_{1}$
${y}_{2}$
$\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
A-B
B-C
C-D
D-E
E-F
F-G
What do you notice about the average gradient over each interval? What can you
say about the average gradients between A and D compared to the averagegradients between D and G?
The average gradient of a parabolic function depends on the interval and is the gradient of a straight line that passes through the points on the interval.
For example, in
[link] the various points have been joined by straight-lines. The average gradients between the joined points are then the gradients of the straight lines that pass through the points.
Method: average gradient
Given the equation of a curve and two points (
${x}_{1}$ ,
${x}_{2}$ ):
Write the equation of the curve in the form
$y=...$ .
Calculate
${y}_{1}$ by substituting
${x}_{1}$ into the equation for the curve.
Calculate
${y}_{2}$ by substituting
${x}_{2}$ into the equation for the curve.
Calculate the average gradient using:
$$\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$$
Find the average gradient of the curve
$y=5{x}^{2}-4$ between the points
$x=-3$ and
$x=3$
Label the points as follows:
$${x}_{1}=-3$$
$${x}_{2}=3$$
to make it easier to calculate the gradient.
We use the equation for the curve to calculate the
$y$ -value at
${x}_{1}$ and
${x}_{2}$ .
The average gradient between
$x=-3$ and
$x=3$ on the curve
$y=5{x}^{2}-4$ is 0.
Summary
Definition of average gradient
Average gradient of straight line
Average gradient of parabola
End of chapter exercises
An object moves according to the function
$d=2{t}^{2}+1$ , where
$d$ is the distance in metres and
$t$ the time in seconds. Calculate the average speed of the object between 2 and 3 seconds. The speed is the gradient of the function
$d$
Given:
$f\left(x\right)={x}^{3}-6x$ .
Determine the average gradient between the points where
$x=1$ and
$x=4$ .
Questions & Answers
where we get a research paper on Nano chemistry....?
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?