Definition of vector-valued function and ordinary function

Reference no: EM13983822

1. Briefly compare and contrast the definition of "vector-valued function" and the definition of an ordinary function y = f (x).

2. How do you determine whether a curve C determined by r(t) is "smooth"? Give an example of a function r(t) that determines a non-smooth curve C, and explain why it fails to be smooth. Your example should be different from the examples we discussed in class.

3. Give the domain of r(t) = ln(t + 2) i + sin^-1(t/5) j + (1/t-1)k. Justify your answers.

4. If r'(t) = (tcos2t, e^-3t, √5t+4) and r(0) = (2,4,6) , find r(t).

5. Let r(t) = (sin 2t,3cos2t,e^t/4), with - ∞ <t < ∞.

(a) Explain why it is important to consider the domain of r(t) when sketching the graph of C, the space curve determined by r(t).

(b) Is C smooth? Explain how you know.

(c) Sketch the space curve C determined by 4(t), and show r(π), v(π), and a(π) on a BIG graph of C. You'll need lots of space for this graph - make sure your scale is appropriate. Be sure your graph of C displays enough of the important characteristics of the curve to be able to infer the shape of the entire graph of C for all values oft in the domain of r. If you wish, you may include a sentence or two to further describe C in addition to your graph.

6. Consider the vector-valued function r(t) = at + b sinωt j + b cosωt k, where a, b, and co are positive constants, that describes the position (in units of length from the origin) of a particle t units of time after the object starts moving.

(a) Describe the graph of C, the space curve determined by r(t).

(b) How does changing the value of a (while leaving b and ω fixed) change the graph of C?

(d) How does changing the value of ω (while leaving a and b fixed) change the graph of C?

(e) Find the speed of the particle moving along Cat any time t.

(f) Find the arclength of C for one coil. (Note that your answer will involve a, b, and ω.)

(g) Based on your answer to part (e), is the particle speeding up or slowing down as t increases? Explain.

Reference no: EM13983822

Questions Cloud

An argument to present to the judge : Cited for driving 70 miles per hour in a 55 mile-per-hour zone. You request to make an appearance in traffic court, and you construct an argument to present to the judge. Some facts are as follows: a. You drive a 1989 Volkswagen.

Compute the heat for the whole cycle : A heat engine carries 4.20 mol of an ideal monatomic gas around the cycle shown in the figure. Compute the heat Q for the whole cycle.

Cause and effect argument essay assignment : Choose a trend or phenomenon related to the sale, trade, or donation of human organs. The trend or phenomenon should have a definable set of causes and effects. Write a 1,500-1,750-word argument using five to seven academic resources that persuades a..

Why did you choose that profession : Who are you personally? Where are you from? How have your background and family values shaped who you are today? What are some of your greatest strengths and weaknesses?

Definition of vector-valued function and ordinary function : Briefly compare and contrast the definition of "vector-valued function" and the definition of an ordinary function y = f (x). How do you determine whether a curve C determined by r(t) is "smooth"? Give an example of a function r(t) that determines a..

Write program that implements lu factorisation algorithm : Write a program that implements the LU factorisation algorithm 2.3, with partial pivoting, given in the Text and in the supplementary section with MATLAB versions of the algorithms.

For annotated bibliographies : For annotated bibliographies when your teacher asks you to summarize what the book is about do you take the summarization from the internet or paraphrase it?

The defense of free speech : How might we argue that "the defense of free speech and the fight against racism to be complementary causes, and not at odds with each other"?

What is the cold reservior temperature in degrees celsius : If a Carnot engine operating with a hot reservior temperature of 596.8 oC has the same efficiency, what is the cold reservior temperature in degrees Celsius?

User Account

All Pages