/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Modify the argument of Example 1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 28

Modify the argument of Example 1 to find the equation of the tangent line to the specified graph at the point given. $$ \text { the graph of } y=x^{2} \text { at }(0,0) $$

Short Answer

Expert verified
The equation of the tangent line is \( y = 0 \).

Step by step solution

01

Recognize the given function

The given function is \( y = x^2 \). We need to find the equation of the tangent line at the point \((0,0)\).
02

Find the derivative of the function

The derivative of a function gives us the slope of the tangent line. To find the derivative, apply the power rule to \( y = x^2 \): \[ \frac{dy}{dx} = 2x \].
03

Evaluate the derivative at the given point

Substitute \( x = 0 \) into the derivative \( \frac{dy}{dx} = 2x \) to find the slope of the tangent line at \((0,0)\). This yields \( m = 2 \times 0 = 0 \).
04

Use the point-slope form to find the equation of the tangent line

The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point on the line, and \( m \) is the slope. Here, \( (x_1, y_1) = (0,0) \). Substitute these values and the slope from Step 3 into the equation: \[ y - 0 = 0(x - 0) \].
05

Simplify the equation

Since the slope is zero, the equation becomes \( y = 0 \), which represents a horizontal line through the point \((0,0)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
In calculus, a derivative is a fundamental concept that represents the rate at which a function is changing at any given point. Think of it as a tool that helps us understand how a function behaves as its input changes. When you calculate a derivative, you're essentially finding the "instantaneous" rate of change.
  • The derivative is denoted by \( \frac{dy}{dx} \), which means the change in \( y \) with respect to \( x \).
  • If a function is given as \( y = f(x) \), then its derivative \( f'(x) \) shows how \( y \) changes with \( x \).
Derivatives are widely used in various applications, including physics for motion, economics for elasticity, and biology for growth rates. By learning to find derivatives, you gain powerful insight into how different variables interact.
Power Rule
The power rule is a nifty shortcut for finding the derivative of functions of the form \( y = x^n \), where \( n \) is any real number. It's one of the first derivative rules you typically learn because it simplifies the process significantly.
  • The rule states: the derivative of \( x^n \) is \( nx^{n-1} \).
  • For example, if \( y = x^2 \), applying the power rule gives \( \frac{dy}{dx} = 2x^{2-1} = 2x \).
This rule makes it quick and easy to find the derivative of polynomials. Instead of using limits every time, the power rule allows for an efficient calculation by reducing the exponent by one and multiplying by the original exponent.
Point-Slope Form
Once you have the slope from the derivative, the point-slope form is a great way to find the equation of a line. Especially when you know one point on the line and its slope. The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \).
  • Here, \( (x_1, y_1) \) is the known point, and \( m \) is the slope.
  • In our exercise, at the point \( (0,0) \), the slope is 0.
Using the point-slope form, you can plug these values in to get the tangent line, which guides how the line behaves relative to the curve at that specific point. It鈥檚 an immensely useful form when dealing with tangents.
Horizontal Line
A horizontal line in graphing is a straight line that goes from left to right, parallel to the x-axis. It has a constant y-value everywhere. Key characteristics of a horizontal line include:
  • A slope of zero because there is no vertical change as you move along the line.
  • The equation of a horizontal line is of the form \( y = c \), where \( c \) is a constant.
In the context of our exercise, since the slope of the tangent line at the point \( (0,0) \) is zero, the equation is simply \( y = 0 \). Horizontal lines often represent steady or unchanged values in various contexts, and here it represents the horizontal nature of the tangent line at the point of tangency.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether the statement is true or false. Explain your answer. If \(f\) and \(g\) are discontinuous at \(x=c,\) then so is \(f g\)

The population \(p\) of the United States (in millions) in year \(t\) may be modeled by the function $$ p=\frac{50371.7}{151.3+181.626 e^{-0.031636(t-1950)}} $$ (a) Based on this model, what was the U.S. population in \(1950 ?\) (b) Plot \(p\) versus \(t\) for the 200 -year period from 1950 to 2150 . (c) By evaluating an appropriate limit, show that the graph of \(p\) versus \(t\) has a horizontal asymptote \(p=c\) for an appropriate constant \(c\). (d) What is the significance of the constant \(c\) in part (b) for population predicted by this model?

Find the limits. $$ \lim _{h \rightarrow 0} \frac{\sin h}{1-\cos h} $$

Sketch the graphs of the curves \(y=1 / x, y=-1 / x\) and \(y=f(x),\) where \(f\) is a function that satisfies the inequalities $$ -\frac{1}{x} \leq f(x) \leq \frac{1}{x} $$ for all \(x\) in the interval \([1,+\infty) .\) What can you say about the limit of \(f(x)\) as \(x \rightarrow+\infty ?\) Explain your reasoning.

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\sqrt[3]{x-8} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.