Chapter 11: Q. 47 (page 872)

Use the given velocity vectors $v (t) = r' (t)$ and initial positions in exercise to find the position function $r (t)$ .
localid="1650736930792" $v (t) = e^{t} i + \ln t j, r (1) = i - 6 j$

Short Answer

Expert verified

$r (t) = (e^{t} + 1 - e) i + (t \ln t - t - 5) j$

Step by step solution

Given Information

Consider $v (t) = e^{t} i + \ln t j, r (1) = i - 6 j$

The objective is to find the position function $r (t)$ .

since $v (t) = r' (t)$

we have

$r (t) = \int v (t) d t = \int (e^{t} i + \ln t j) d t = i \int e^{t} d t + j \int \ln t d t = e^{t} i + (t \ln t - t) j + c$

Where c is a vector constant.

$= (e^{t} + c_{1}) i + (t \ln t - t + c_{2}) j$ where $c_{1}$ and $c_{2}$ are scalars.

Expression

$r (t) = (e^{t} + c_{1}) i + (t \ln t - t + c_{2}) j r (1) = (e^{1} + c_{1}) i + (1 \ln 1 - 1 + c_{2}) j r (1) = (e + c_{1}) i + (- 1 + c_{2}) j . . . (1)$

but the given initial position $r (t) = i - 6 j . . . . . . . . . (2)$

Comparing (1) and (2)

$e + c_{1} = 1, - 1 + c_{2} = - 6 c_{1} = 1 - e, c_{2} = - 6 + 1 c_{1} = 1 - e, c_{2} = - 5 r (t) = (e^{t} + 1 - e) i + (t \ln t - t - 5) j$

Calculation

Check: Taking the derivative of $r (t) = (e^{t} + 1 - e) i + (t \ln t - t - 5) j$ ,

We obtain $r' (t) = e^{t} i + \ln t j$ , which is the correct tangent vector function. Next, we evaluate

$r (1) = (e^{t} + 1 - e) t + (1 \ln 1 - 1 - 5) j = j - 6 j$

Which is true according to the problem.

Thus $r (t) = (e^{t} + 1 - e) i + (t \ln t - t - 5) j$

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Use the given velocity vectors $v (t) = r' (t)$ and initial positions in exercise to find the position function $r (t)$ .
localid="1650736930792" $v (t) = e^{t} i + \ln t j, r (1) = i - 6 j$

Short Answer

Step by step solution

Given Information

Expression

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Use the given velocity vectors v(t)=r'(t)and initial positions in exercise to find the position function r(t).localid="1650736930792" v(t)=eti+lntj,r(1)=i-6j

Short Answer

Step by step solution

Given Information

Expression

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Calculus

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Study anywhere. Anytime. Across all devices.

Use the given velocity vectors $v (t) = r' (t)$ and initial positions in exercise to find the position function $r (t)$ .
localid="1650736930792" $v (t) = e^{t} i + \ln t j, r (1) = i - 6 j$