/*! 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 2 Suppose that \(f(t)\) is a solut... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 2

Suppose that \(f(t)\) is a solution of \(y^{\prime}=t^{2}-y^{2}\) and the graph of \(f(t)\) passes through the point \((2,3) .\) Find the slope of the graph when \(t=2.\)

Short Answer

Expert verified
The slope at \( t = 2 \) is \( -5 \).

Step by step solution

01

Identify the Given Information

The differential equation provided is \( y' = t^2 - y^2 \). It is also given that the graph of \( f(t) \) passes through the point \( (2, 3) \). This means \( f(2) = 3 \).
02

Evaluate the Differential Equation at the Given Point

To find the slope at \( t = 2 \), substitute \( t = 2 \) and \( y = 3 \) into the differential equation \( y' = t^2 - y^2 \).
03

Substitute Values

Replace \( t \) with 2 and \( y \) with 3 in the equation \( y' = t^2 - y^2 \). This gives: \[ y' = 2^2 - 3^2 \]
04

Perform the Calculations

Simplify the equation: \[ y' = 4 - 9 \] \[ y' = -5 \]
05

Interpret the Result

The slope of the graph of \( f(t) \) when \( t = 2 \) is \( -5 \).

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.

differential equation
A differential equation involves an unknown function and its derivatives. In this exercise, the differential equation is given by \(\frac{dy}{dt} = t^2 - y^2\). This equation tells us how the rate of change of the function, noted as \(y'\) or \( \frac{dy}{dt}\), depends on the variable \(t\) and the function \(y(t)\). These types of equations are used to model real-world phenomena like population growth, heat transfer, and motion of objects. The key is to solve for \(y(t)\), which gives us the function that describes the behavior of the system over time.
slope
The slope of a graph at a particular point helps us understand how steep the graph is at that specific location. Mathematically, the slope of a curve defined by a function \(f(t)\) at a point \(t\) is given by the derivative \(f'(t)\). For example, in the given problem, we need to find the slope when \(t = 2\). We do this by evaluating the derivative using the provided differential equation. After substituting \(t = 2\) and \(y = 3\) into \(\frac{dy}{dt} = t^2 - y^2\), we get \(\frac{dy}{dt} = 2^2 - 3^2 = 4 - 9 = -5\). Hence, the slope of the graph at \(t = 2\) is -5.
initial condition
The initial condition provides us with specific information about the function at a particular point. This helps us find a unique solution to a differential equation. In this problem, the initial condition is the point \( (2, 3) \), meaning \( f(2) = 3 \). By knowing this, we can substitute these values into the differential equation to find specific information about the solution, such as the slope at \( t = 2 \). Without the initial condition, we would have an infinite number of possible solutions.
calculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. The two primary tools of calculus are derivatives and integrals. Derivatives measure how a quantity changes over time, which is crucial in understanding the slope of curves and solving differential equations. In our exercise, we use the derivative provided in the differential equation \( \frac{dy}{dt} = t^2 - y^2 \) to find the rate of change of \( y(t) \). Calculus helps us analyze dynamic systems by providing methods to describe and predict their behavior over time.

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

The Los Angeles Zoo plans to transport a California sea lion to the San Diego Zoo. The animal will be wrapped in a wet blanket during the trip. At any time t, the blanket will lose water (due to evaporation) at a rate proportional to the amount \(f(t)\) of water in the blanket, with constant of proportionality \(k=-.3 .\) Initially, the blanket will contain 2 gallons of seawater. (a) Set up the differential equation satisfied by \(f(t).\) (b) Use Euler's method with \(n=2\) to estimate the amount of moisture in the blanket after 1 hour. (c) Solve the differential equation in part (a) and compute \(f(1).\) (d) Compare the answers in parts (b) and (c) and approximate the error in using Euler's method.

Let \(f(t)\) be the solution of \(y^{\prime}=-(t+1) y^{2}, y(0)=1 .\) Use Euler's method with \(n=5\) to estimate \(f(1) .\) Then, solve the differential equation, find an explicit formula for \(f(t),\) and compute \(f(1) .\) How accurate is the estimated value of \(f(1) ?\).

Solve the given equation using an integrating factor. Take \(t>0\). $$y^{\prime}+y=2-e^{t}$$

Probability of Accidents Let \(t\) represent the total number of hours that a truck driver spends during a year driving on a certain highway connecting two cities, and let \(p(t)\) represent the probability that the driver will have at least one accident during these \(t\) hours. Then, \(0 \leq p(t) \leq 1,\) and \(1-p(t)\) represents the probability of not having an accident. Under ordinary conditions, the rate of increase in the probability of an accident (as a function of \(t\) ) is proportional to the probability of not having an accident. Construct and solve a differential equation for this situation.

Suppose that \(f(t)\) satisfies the initial-value problem \(y^{\prime}=y^{2}+t y-7, y(0)=3 .\) Is \(f(t)\) increasing or decreasing at \(t=0 ?\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

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.