"Explore
and Visualize" Please respond to the following:
•At this week's
applet: Curves, the applet graphs the function, the
derivative and the second derivative in the window. At the bottom, you
can enter any function and see it graphed. Additionally, the first
derivative is given at the bottom. You can manually change the x value or
use the slider bar to the right to move along the function. The left
graph is the function. The red line is the tangent line at the selected
value of x. The middle graph is the derivative. The green line is
the tanget line at the selected value of x. The right graph is the second
derivative.
Read
the explanation on the page and play with the applet. You might also
select a function from one of this week's section to enter and play with as
well. Come back and share what you learned from this applet. Be sure to
respond to at least two classmates.
One
Paragraph:
Start
Here:
Here
are the two classmates to respond to them all you can use is one or less of a
paragraph.
1.
*?Concavity describes the way that a curve bends. A functionfisconcave
upon
an open interval iff' is increasing
andconcave downiff' is decreasing.
This also means that it is concave up if the second derivativef'' is positive and
is concave down if the second derivative is negative. Aninflection
pointis
where the function has a tangent and the concavity changes
*The
second derivative can also be useful in determining whether a critical point is
a maximum or a minimum.
*Critical points occur when the first derivative
is zero or undefined
Respond here less than a paragraph unifolks:
2.I learned a number of things with the curve
applet.
First,
a function is increasing when f(x) increases as x increases. A function is
decreasing when f(x) decreases as x increases.
Second,
critical points occur when the first derivative is zero or is undefined.
Third,
critical points can be relative extrema. A critical point is a local minimum at
x = c if f (c) <= f(c-1) and f(c+1). A critical point is a local maximum at
x = c if f (c) >= f(c-1) and f(c+1).
Fourth,
concavity describes the way that a curve bends. A function f(x) is concave up
like a bowl if f '(x) is increasing. A function f(x) is concave down like a
dome if f '(x) is decreasing.
Fifth,
inflections points occur when concavity changes from up to down or vice versa.
Respond
less than a paragraph unifolks: