Write the general equation of a vector-valued function in component form and unit-vector form.
Recognize parametric equations for a space curve.
Describe the shape of a helix and write its equation.
Define the limit of a vector-valued function.
Our study of vector-valued functions combines ideas from our earlier examination of single-variable calculus with our description of vectors in three dimensions from the preceding chapter. In this section we extend concepts from earlier chapters and also examine new ideas concerning curves in three-dimensional space. These definitions and theorems support the presentation of material in the rest of this chapter and also in the remaining chapters of the text.
Definition of a vector-valued function
Our first step in studying the calculus of vector-valued functions is to define what exactly a vector-valued function is. We can then look at graphs of vector-valued functions and see how they define curves in both two and three dimensions.
Definition
A
vector-valued function is a function of the form
where the
component functionsf, g, and
h , are real-valued functions of the parameter
t. Vector-valued functions are also written in the form
In both cases, the first form of the function defines a two-dimensional vector-valued function; the second form describes a three-dimensional vector-valued function.
The parameter
t can lie between two real numbers:
Another possibility is that the value of
t might take on all real numbers. Last, the component functions themselves may have domain restrictions that enforce restrictions on the value of
t. We often use
t as a parameter because
t can represent time.
Evaluating vector-valued functions and determining domains
For each of the following vector-valued functions, evaluate
Do any of these functions have domain restrictions?
To calculate each of the function values, substitute the appropriate value of
t into the function:
To determine whether this function has any domain restrictions, consider the component functions separately. The first component function is
and the second component function is
Neither of these functions has a domain restriction, so the domain of
is all real numbers.
To calculate each of the function values, substitute the appropriate value of
t into the function:
To determine whether this function has any domain restrictions, consider the component functions separately. The first component function is
the second component function is
and the third component function is
The first two functions are not defined for odd multiples of
so the function is not defined for odd multiples of
Therefore,
where
n is any integer.
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.