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 $P$.$$f(x)=-x^{2}+2 x-1$$$\quad$$$P(1,0)$$

A. (1,0)C. 0

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Campbell University

University of Michigan - Ann Arbor

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.

02:00

a) sketch the graph of the…

02:22

02:33

02:03

02:25

Sketch a graph of the func…

02:21

01:40

Find the tangent line to t…

02:24

Equations of tangent lines…

Alright, here we are given a function and asked to sketch it and the news basic calculus to determine the slope of it at 0.1 comma zero. I've got ahead and sketch this function for us. It looks something like this. We are given the 0.10 which exists right here, and we want to know what the slope is at that point. So doing that, let's first draw a tangent line through it might look something like that we don't know. And let's start by finding a second point summer on there so that we can go ahead. News. Our known slope function where M is equal to y tu minus y one Oliver x two minus X one Let's plug a value really close to one into our function. Let's say f of 1.1 If we were to plug it into the function we would get that is equal to negative. 0.8 Very close to zero, it looks like so it's going to be our Y value when we have an X equal to 1.1 Now that we have both the coordinate we just found and 10 which was given to us. We can plug these values into our slope formula up here. In doing that, we end up getting a slope is equal to negative 0.1 So we're still really close to zero. But I think that it actually should be zero, not almost zero, because looking at our problem, we know that it has the 00.1 common zero. So we know that it hits that X axis, and I believe that it's going to be its maximum. No problems. At their minimum maximum, they have a slope equal to zero. So to confirm this suspicion, we're gonna start. Let's take the derivative of our function. Alpha vax is equal to negative X squared plus two X minus one. We're just gonna use the basic power role here to get f prime of X, which is equal to negative two X plus two. We know that the derivative is what the slope is at that point, but let's go ahead. We have an X equal to zero plugging that into the function we have f prime of our X is equal to one. I've probably one goal to negative two times one plus two. She gives us negative two plus two, which means that our slope at X equals one is actually equal to zero, which confirms the suspicion that we had.

View More Answers From This Book

Find Another Textbook

Numerade Educator

02:45

$$f(x)=\left\{\begin{array}{l}x^{2} \text { if } x \geq 0 \\x \text { if } x…

Given the curve whose equation is $f(x)=x^{0.3} .$ Let $P$ be the point (1,1…

03:25

Find the equations of the lines and plot the lines from Exercise $53$.

01:00

In each of the following, solve the given quadratic equation exactly using t…

06:47

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

03:58

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

06:43

Under what conditions on $a, b, c,$ and $d$ is $a x^{2}+a y^{2}+b x+c y+d=0$…

01:05

Find the indicated limit.$$\lim _{t \rightarrow 0}\left(\frac{1}{2} t^{2…

02:46

sketch the graph of the given ellipse, labeling all intercepts.$$\frac{x…

01:30

The sum of the squares of three consecutive odd integers is $515,$ find the …