"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 function
f
is
concave up
on an open interval if
f
' is increasing and
concave down
if
f
' is decreasing. This also means that it is concave up if the second derivative
f
'' is positive and is concave down if the second derivative is negative. An
inflection point
is 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: