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The Shape of a Graph, Part II : In previous we saw how we could use the first derivative of a function to obtain some information regarding the graph of a function. In this section we will look at the information which the second derivative of a function can give us a regarding the graph of a function.
Some definitions
Concavity: The main concept which we'll be discussing in this section is concavity. Concavity is easiest to see with a graph .
Concave up
A function is concave up if it "opens" up and
Concave down
The function is concave down if it "opens" down.
Notice that concavity has not anything to do with increasing or decreasing. Any function can be concave up and either increasing or decreasing. Likewise, a function can be concave down and either increasing or decreasing.
It's possibly not the best way to described concavity by saying which way it "opens" since it is a somewhat nebulous definition. Following is the mathematical definition of concavity.
Examples on Log rules: Example: Calculate (1/3)log 10 2. Solution: log b n√A = log b A 1/n = (1/n)log b A (1/3)log 10 2 = log 10 3 √2 = log 10 1.
Simple derivatives Example Differentiate following. (5x 3 - 7 x + 1) 5 ,[ f ( x )] 5 ,[ y ( x )] 5 Solution: Here , with the first function we're being asked to
Method to determine solution is absolute minimum/maximum value Let's spend a little time discussing some methods for determining if our solution is in fact the absolute minimum
1. His favorite current carrot drink contain 40%. but h eneeds add to 80 quarts wife brought his perfect drink mix.
Even and Odd Functions : This is the final topic that we have to discuss in this chapter. Firstly, an even function is any function which satisfies,
a) Write a summary on Tower of Hanoi Problem. How can it be solved using recursion ? b) Amit goes to a grocery shop and purchases grocery for Rs. 23.
On a picnic outing, 2 two-person teams are playing hide-and-seek. There are four hiding locations (A, B, C, and D), and the two members of the hiding team can hide separately in a
MATH
Q. What is Box-and-Whisker Plot? Ans. Line graphs or stem-and-leaf plots become difficult to manage when there is a large amount of data. Box-and-whisker plots help summa
F(x)=2x+3
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