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(a) Draw the graph of a function which is increasing on (0,2) and such that it is concave upward on (0,1) and concave downward on (1,2) . (b) Does this function have an inflection point?

(a)(b) yes

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:12

help

01:21

Suppose that the second de…

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03:24

(a) Draw the graph of a co…

01:04

question 18. In part, A is asking you to draw a graph for a function which is increasing from zero to such that it is concave, up from 0 to 1 and concave down from 1 to 2. So drawing writing that interval here, this is our X axis. This is our Y axis, Uh, and then you have one and two. So, you know, it's increasing the entire time. Um and it's conclave up from 0 to 1. So increasing in con cape up would look something like that. And then once you get to one, you're going to be concave down. But you're still increasing, so connecting that it would look something like this. So you have that wave shape here for part B. They are asking, Ah, if this function has any inflection points Yes, it does, because an inflection point is where your con cavity is changing. So we're going from concave up two concave down, so inflection point would be at X equals one. And that is your answer to question 18

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