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

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.$$f(x)=\sqrt{x+1} \quad P(3,2)$$

(c) $1 / 4$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Campbell University

Harvey Mudd College

Idaho State University

Boston College

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.

02:03

a) sketch the graph of the…

02:22

02:00

02:43

04:20

(a) find the slope of the …

Sketch a graph of the func…

01:22

Use a graphing calculator …

10:11

Write an equation of the t…

all right here. We're gonna be using some basic calculus to determine the slope of the function. Square root, X plus one at 10.32 I've sketched the function here. It looks something like this. 0.32 sits in here and we want to determine the slope of the function here so we could draw a tangent line. Something like that. We need another point somewhere on in order to use our soap function of em is equal to y tu minus y one Oliver x two minus x one. Through this, we can choose a value really, really close to three comma, two for X. So how would we use X is equal to 3.1 and plug that into our function. So we'll have f of 3.1 is equal to the square root of 4.1 I just went ahead and added our X value to the one in here right away. Taking the square root of that, we end up finding that we get a Y value equal to 2.25 So we now have another coordinate point of 3.1 comma 2.25 Plugging these values into our slope function we have em is equal to Are y two minus ry one all over x two minus X one. Let's go ahead and calculate this to get that is going to be equal to about 0.24 9998 just to truncate it there. Mhm. All right, And now we could assume that this is our slope, But let's let's simplify it a little bit. Or let's just round to make it a little easier to give us a slope at point three, Comma two is equal to about 0.25

View More Answers From This Book

Find Another Textbook

01:40

Find $f^{\prime}(x)$ if $f(x)=\frac{2 x+3}{3 x-2}$

02:14

Where is the function $f(x)=1 /(x-5)$ discontinuous. Why?

01:23

Use the vertical line test to determine if the given graph may represent a f…

03:32

Determine (a) $f(x)$ ) and the domain of the composite function, (b) $g(f(x)…

02:59

Given $f(x)=x^{3}-12 x .$ At which points will its tangent line (a) be horiz…

05:56

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

01:30

02:07

$f(x)=\left(x^{2}+2\right)\left(x^{4}-7\right) .$ Find $f^{\prime}(x)$ in tw…

01:17

When the price of a New York City bus ride was 35 cents, the number of payin…

02:51