Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h} $

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

$f(x)=\sqrt{x}$$a=9$

01:57

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Campbell University

Baylor University

Idaho State University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

0:00

Each limit represents the …

01:12

02:09

02:36

06:27

02:05

01:27

03:12

01:03

$31-36$ Each limit represe…

01:35

00:55

$33-38$ Each limit represe…

01:04

01:17

The limit represents $f^{\…

if f of x is equal to the square root of X. And if A Is the number nine than using the definition to limit definition of the derivative, F prime at A would be Let me raise that. I got to put into words, limit F prime of A would be to limit as H approaches zero of the function evaluated at X plus h minus two function evaluated at X uh over each. Uh And actually instead of X since we're evaluating the derivative at A instead of X, we're going to put a. So F prime of A will be limited as H approaches zero of the function at a plus H minus two function of F at a over H. No, using uh this definition for the function F of X is the square root of X. And our number A is nine. I'll continue down here. Uh This will equal the limit As a church approaches zero. Now the function F of a plus H is going to be the square root of a plus H minus the function at a f of X is squared of X. So if a baby is a square root of X, all of this is divided by H. And we're taking the limit of this expression as H goes towards zero. That's how we're gonna find uh F prime of X. Keep in mind, let's scroll up just a minute. So you remember remember that in the very top of the problem here. Yeah. Uh F of X. I defined to be the function squared of X. And a the number A equals nine. So let's go ahead and plug into number nine. Since that is um our actual uh what we're looking for, So f prime of nine, The derivative of our function squared of x at a equals nine is going to be to limit As a church approaches zero. Now the square root of A plus H is the square root of nine plus H. Since a. is nine, subtract the square root of A. Since A is nine, the square root of nine is three. And this is being divided by each. And so you can see uh the derivative F prime at nine equals to limit of the square root of nine plus H minus three over H as H approaches zero. When remember what our function And what are # eight was? This is true when F of X is equal to square root of X. And our number A. Was nine.

View More Answers From This Book

Find Another Textbook

Numerade Educator

05:20

a bag contains 3 red 5 black and another bag 2 red ,5 black two balls are dr…

02:23

the sum of two digit number is 9 if 27 subtracted from the number its digits…

01:38

subtract : x² - 5/2x + 7 from 1/3 x² + 1/2x + 2

00:48

Under a given correspondence, two triangles are congruent if two sides and t…

03:48

Ted practices two types of swimming styles for a total of 50 minutes every d…

03:55

03:23

create a chart by calculating the heights of all the family members.converts…

the equation P = 16,250 + 75n can be used to determine the population, P, of…

00:52

The thickness of a notebook containing 300 sheet is 30 mm what is a thicknes…

01:00

If x = 24, then find the value of (2x)^x