/*! 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 84 Given the following velocity fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 84

Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. $$v(t)=e^{-2 t}+4 ; s(0)=2$$

Short Answer

Expert verified
Answer: The position function s(t) of the object is given by $$s(t) = -\frac{1}{2}e^{-2t} + 4t + \frac{5}{2}$$.

Step by step solution

01

Identify the given functions and values

We are given the velocity function: $$v(t) = e^{-2t} + 4$$ and the initial position: $$s(0) = 2$$
02

Find the position function s(t) by integrating the velocity function

To find the position function, we need to integrate the velocity function: $$s(t) = \int v(t) dt$$ Substitute the given velocity function: $$s(t) = \int (e^{-2t} + 4) dt$$
03

Integrate the velocity function

Integrate each term separately: $$s(t) = \int e^{-2t} dt + \int 4 dt$$ The integration of the exponential term is: $$\int e^{-2t} dt = -\frac{1}{2}e^{-2t} + C_1$$ The integration of the constant term is: $$\int 4 dt = 4t + C_2$$ Combine the results to find the general position function: $$s(t) = -\frac{1}{2}e^{-2t} + 4t + (C_1 + C_2)$$
04

Use the initial position to find the constant C

Now, we will use the initial position s(0) to find the value of C: $$s(0) = -\frac{1}{2}e^{-2(0)} + 4(0) + (C_1 + C_2)$$ Substitute the given initial position: $$2 = -\frac{1}{2}(1) + (C_1 + C_2)$$ Solve for C: $$C_1 + C_2 = 2 + \frac{1}{2} = \frac{5}{2}$$ Now the position function is: $$s(t) = -\frac{1}{2}e^{-2t} + 4t + \frac{5}{2}$$
05

Graph the velocity and position functions

Graph the velocity function $$v(t) = e^{-2t} + 4$$ and the position function $$s(t) = -\frac{1}{2}e^{-2t} + 4t + \frac{5}{2}$$ to visualize the motion of the object. To graph the functions, we can use graphing software or draw them by hand, considering the shape and attributes of the functions. The velocity function is a sum of an exponential decay function and a horizontal line, resulting in a curve that starts from 5 and approaches 4 as t increases. The position function is a sum of a negative exponential decay function and a linear function with positive slope, giving us a curve that starts at (0,2) and increases steeply initially and then more slowly as time increases.

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!

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 the following indefinite integrals. Check your work by differentiation. $$\int\left(\csc ^{2} \theta+1\right) d \theta$$

The complexity of a computer algorithm is the number of operations or steps the algorithm needs to complete its task assuming there are \(n\) pieces of input (for example, the number of steps needed to put \(n\) numbers in ascending order). Four algorithms for doing the same task have complexities of A: \(n^{3 / 2},\) B: \(n \log _{2} n,\) C: \(n\left(\log _{2} n\right)^{2},\) and \(D: \sqrt{n} \log _{2} n .\) Rank the algorithms in order of increasing efficiency for large values of \(n\) Graph the complexities as they vary with \(n\) and comment on your observations.

The ranking of growth rates given in the text applies for \(x \rightarrow \infty .\) However, these rates may not be evident for small values of \(x .\) For example, an exponential grows faster than any power of \(x .\) However, for \(1

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=4 ; v(0)=-3, s(0)=2$$

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{array}{l}f^{\prime \prime}(x)>0 \text { on }(-\infty,-2) ; f^{\prime \prime}(x)<0 \text { on }(-2,1) ; f^{\prime \prime}(x)>0 \text { on } \\\\(1,3) ; f^{\prime \prime}(x)<0 \text { on }(3, \infty)\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.