Download the App!

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

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Consider the two functions: $f(x)=x^{1 / 3}$ and $g(x)=x^{4 / 3}$ near $x=0 .$ Using $h=-0.1,-.0 .01,-0.001$ and $-0.0001,(\text { as } h \text { approaches } 0$ from the left), and $h=0.1, .0 .01,0.001$ and 0.0001 (as $h$ approaches 0 from the right). Find the slope of the secant lines passing through $P(0,0)$ and $Q(h, f h)$ ). Does $m_{\tan }(x)$ exist at (0,0)$?$ Why not? (b) Now repeat the process for $g(x) .$ What is the difference in the behavior at $P(0,0)$ for the two functions?

$$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \mathbf{h} & \mathbf{- . 1} & \mathbf{- . 0 1} & \mathbf{- . 0 0 1} & \mathbf{- . 0 0 0 1} & \mathbf{. 0 0 0 1} & \mathbf{. 0 0 1} & \mathbf{. 0 1} & \mathbf{. 1} \\\hline m_{P Q} & \mathbf{4 . 6 4 1 6} & \mathbf{2 1 . 5 4 4} & 100.00 & 464.16 & 464.16 & 100.00 & \mathbf{2 1 . 5 4 4} & \mathbf{4 . 6 4 1 6} \\m_{P Q} & -0.4642 & -0.2154 & -0.1 & -.0464 & .0464 & 0.1 & 0.2154 & 0.4642 \\\hline\end{array}$$in $f(x)=x^{1 / 3},$ the slope is becoming infinite as $h$ approaches zero for $f(x)=$ $x^{4 / 3}$ it approaches 0

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Missouri State University

Baylor University

University of Nottingham

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.

01:06

(a) If $f(x)=x^{2 / 3}+1$,…

03:23

For the function $f(x)=x^{…

01:09

(a) Graph $f(x)=\frac{1}{2…

01:03

01:19

Consider the function $f(…

01:04

a. Given the graph of $f$ …

02:18

Comparing Functions Consid…

01:08

01:01

(a) If $f(x)=x^{3}-2 x+2,$…

Steep secant linesa. G…

Yeah for the functioning fx echoes X to the power of a third. This is the value of X. And this city is the value of healthy edge. And this color is the slope of the second night. You can say that when h girls joe zero the slope of the second line goes to positive infinity no matter from the left hand side from the right side. But for the function FX equals x to the power of 4/3. The edge goes to zero. The slope of the second light goes to zero as well. This is also true. The edge goes to the arrow from the right enhance F. X equals X to the power of a third. Is not a differential about at the point of their but the function F X equals X to the power of 4 3rd is differential. But we at some point of view, if you draw a sketch of the quality of these two functions, they look like this. This is a graph of FX equals extra power of Western. And this is a graph or function FX Echoes X to the power of 4/3

View More Answers From This Book

Find Another Textbook

02:57

Determine the derivative at the given point on the curve using equation (2).…

01:53

Use the appropriate rules to determine the derivative.$$f(x)=\sqrt[3]{x}…

01:18

Compute the indicated limit.$$\text { (a) } \lim _{x \rightarrow 2^{+}} …

02:36

Find the points on the curve $y=x^{4}-8 x^{2}+3$ at which the tangent line i…

09:53

Given $f(x, y)=8 x^{3} y^{2}$ determine (a) $f(3,2),$ (b) $f(2,5)$. If neith…

02:08

Decide whether or not the function is continuous. If it is not continuous, i…

03:13

Show that a line parallel to the base of a triangle intersecting the other t…

04:04

(a) Graph the function defined by $f(x)=\left\{\begin{array}{ll}1 & \tex…

Sometimes when one plots a set of data, it appears to be piece wise linear, …

02:30

Show how to "build," by composition, the following function $f(x)=…