💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

Get the answer to your homework problem.

Try Numerade Free for 7 Days

In an episode of The Simpsons television show, Homer reads from a newspaper and announces "Here's good news! According to this eye-catching article, SAT scores are declining at a slower rate." Interpret Homer's statement in terms of a function and its first and second derivatives.

$f^{\prime}(x) < 0$ and $f^{\prime \prime}(x) > 0$

01:33

Fahad P.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Derivatives

Differentiation

Volume

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

02:34

On May $9,2007,$ CBS Eveni…

0:00

The president announces th…

06:01

From the mid-l960s to the …

01:28

SAT Scores. The graph of f…

00:16

Use the following graph. I…

00:09

04:36

In Exercises $19-24,$ find…

00:10

02:19

Find the directions in whi…

So we're told in this question that s A T scores are decreasing at a slower rate. So what that means is that we're decreasing but we're actually not as decreasing as much as we were before. So if we were to look at a graph that's decreasing at a slower rate at some point would be decreasing, decreasing decreasing and then we're decreasing less and less and less like this or going towards a positive slope, but we're still decreasing. So this is what a graph would look like, where we could be saying, somewhere over here we're decreasing less and less and less, and then we start increasing. So what this tells us about our first derivative is that we know that our first derivative f prime of X is going to be negative since we're decreasing whenever we're decreasing, our first derivative is going to be negative. So we know that it's going to be less than zero. Our second derivative is actually going to be positive. And the reason for this is because we're concave up at this point where we're decreasing less and less. If we were decreasing more and more, we would be concave down. We can also think of this as if we were looking at a graph of our first derivative, we're decreasing less. So we start at some negative value and we're getting closer and closer to zero, which means that we have a positive slope for our first derivative, which means that the derivative of that of our first derivative has to be positive at that point. So those are two different ways to know that we're going to be concave up at that point in our second derivative is going to be positive.

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

On May $9,2007,$ CBS Evening News had a 4.3 point rating. (Ratings measure t…

The president announces that the national deficit is increasing, but at a de…

From the mid-l960s to the early 1990 s, there was a slow but steady decline<…

SAT Scores. The graph of function $m$ in the next column gives the average s…

Use the following graph. In the graph, 0 represents the average critical rea…

In Exercises $19-24,$ find the directions in which the functions increase an…

Find the directions in which the functions increase and decrease most rapidl…